0.11/0.13 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.11/0.14 % Command : tptp2X_and_run_prover9 %d %s 0.13/0.35 % Computer : n023.cluster.edu 0.13/0.35 % Model : x86_64 x86_64 0.13/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.13/0.35 % Memory : 8042.1875MB 0.13/0.35 % OS : Linux 3.10.0-693.el7.x86_64 0.13/0.35 % CPULimit : 180 0.13/0.35 % DateTime : Thu Aug 29 15:13:07 EDT 2019 0.13/0.36 % CPUTime : 5.75/5.87 ============================== Prover9 =============================== 5.75/5.87 Prover9 (32) version 2009-11A, November 2009. 5.75/5.87 Process 18054 was started by sandbox on n023.cluster.edu, 5.75/5.87 Thu Aug 29 15:13:12 2019 5.75/5.87 The command was "/export/starexec/sandbox/solver/bin/prover9 -t 180 -f /tmp/Prover9_17899_n023.cluster.edu". 5.75/5.87 ============================== end of head =========================== 5.75/5.87 5.75/5.87 ============================== INPUT ================================= 5.75/5.87 5.75/5.87 % Reading from file /tmp/Prover9_17899_n023.cluster.edu 5.75/5.87 5.75/5.87 set(prolog_style_variables). 5.75/5.87 set(auto2). 5.75/5.87 % set(auto2) -> set(auto). 5.75/5.87 % set(auto) -> set(auto_inference). 5.75/5.87 % set(auto) -> set(auto_setup). 5.75/5.87 % set(auto_setup) -> set(predicate_elim). 5.75/5.87 % set(auto_setup) -> assign(eq_defs, unfold). 5.75/5.87 % set(auto) -> set(auto_limits). 5.75/5.87 % set(auto_limits) -> assign(max_weight, "100.000"). 5.75/5.87 % set(auto_limits) -> assign(sos_limit, 20000). 5.75/5.87 % set(auto) -> set(auto_denials). 5.75/5.87 % set(auto) -> set(auto_process). 5.75/5.87 % set(auto2) -> assign(new_constants, 1). 5.75/5.87 % set(auto2) -> assign(fold_denial_max, 3). 5.75/5.87 % set(auto2) -> assign(max_weight, "200.000"). 5.75/5.87 % set(auto2) -> assign(max_hours, 1). 5.75/5.87 % assign(max_hours, 1) -> assign(max_seconds, 3600). 5.75/5.87 % set(auto2) -> assign(max_seconds, 0). 5.75/5.87 % set(auto2) -> assign(max_minutes, 5). 5.75/5.87 % assign(max_minutes, 5) -> assign(max_seconds, 300). 5.75/5.87 % set(auto2) -> set(sort_initial_sos). 5.75/5.87 % set(auto2) -> assign(sos_limit, -1). 5.75/5.87 % set(auto2) -> assign(lrs_ticks, 3000). 5.75/5.87 % set(auto2) -> assign(max_megs, 400). 5.75/5.87 % set(auto2) -> assign(stats, some). 5.75/5.87 % set(auto2) -> clear(echo_input). 5.75/5.87 % set(auto2) -> set(quiet). 5.75/5.87 % set(auto2) -> clear(print_initial_clauses). 5.75/5.87 % set(auto2) -> clear(print_given). 5.75/5.87 assign(lrs_ticks,-1). 5.75/5.87 assign(sos_limit,10000). 5.75/5.87 assign(order,kbo). 5.75/5.87 set(lex_order_vars). 5.75/5.87 clear(print_given). 5.75/5.87 5.75/5.87 % formulas(sos). % not echoed (5233 formulas) 5.75/5.87 5.75/5.87 ============================== end of input ========================== 5.75/5.87 5.75/5.87 % From the command line: assign(max_seconds, 180). 5.75/5.87 5.75/5.87 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 5.84/5.87 5.84/5.87 % Formulas that are not ordinary clauses: 5.84/5.87 1 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_c (V_b_2 = V_c_2 & V_a_2 = V_d_2 | V_b_2 = V_d_2 & V_c_2 = V_a_2 <-> hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_b_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) = hAPP(hAPP(c_Set_Oinsert(T_c),V_c_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_d_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))))) # label(fact_doubleton__eq__iff) # label(axiom) # label(non_clause). [assumption]. 5.84/5.87 2 (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]. 5.84/5.87 3 (all V_l_2 all V_k_2 all T_c (class_Orderings_Oorder(T_c) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_k_2),V_l_2)) <-> c_SetInterval_Oord__class_OgreaterThanAtMost(T_c,V_k_2,V_l_2) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))) # label(fact_greaterThanAtMost__empty__iff) # label(axiom) # label(non_clause). [assumption]. 5.84/5.87 4 (all V_x_2 all T_c (V_x_2 = c_List_Olist_ONil(T_c) <-> 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]. 5.84/5.87 5 (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]. 5.84/5.87 6 (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]. 5.84/5.87 7 (all V_com2_2 all V_com1_2 all V_fun_2 all V_com_H_2 all V_fun_H_2 c_Com_Ocom_OWhile(V_fun_H_2,V_com_H_2) != c_Com_Ocom_OCond(V_fun_2,V_com1_2,V_com2_2)) # label(fact_com_Osimps_I53_J) # label(axiom) # label(non_clause). [assumption]. 5.84/5.87 8 (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]. 5.84/5.87 9 (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]. 5.84/5.87 10 (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]. 5.84/5.87 11 (all V_n_2 all V_l_2 all T_c hAPP(hAPP(c_List_Otake(T_c),V_n_2),V_l_2) = c_List_Osublist(T_c,V_l_2,hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),V_n_2))) # label(fact_sublist__upt__eq__take) # label(axiom) # label(non_clause). [assumption]. 5.84/5.87 12 (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]. 5.84/5.87 13 (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]. 5.84/5.87 14 (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]. 5.84/5.87 15 (all V_u_2 all V_m_2 all V_l_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_l_2),V_m_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_m_2),V_u_2)) -> c_SetInterval_Oord__class_OgreaterThanAtMost(T_c,V_l_2,V_u_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanAtMost(T_c,V_l_2,V_m_2)),c_SetInterval_Oord__class_OgreaterThanAtMost(T_c,V_m_2,V_u_2)))))) # label(fact_ivl__disj__un_I20_J) # label(axiom) # label(non_clause). [assumption]. 5.84/5.87 16 (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]. 5.84/5.87 17 (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]. 5.84/5.87 18 (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]. 5.84/5.87 19 (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]. 5.84/5.87 20 (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]. 5.84/5.87 21 (all V_j_2 all V_i_2 hAPP(c_List_Oset(tc_Int_Oint),c_List_Oupto(hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_i_2),c_Groups_Oone__class_Oone(tc_Int_Oint)),V_j_2)) = c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Int_Oint,V_i_2,V_j_2)) # label(fact_greaterThanAtMost__upto) # label(axiom) # label(non_clause). [assumption]. 5.84/5.87 22 (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]. 5.84/5.87 23 (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_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_n),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2))) -> hBOOL(hAPP(V_Pa_2,hAPP(hAPP(c_List_Onth(T_c),V_xs_2),B_n))))))) # label(fact_list__all__length) # label(axiom) # label(non_clause). [assumption]. 5.84/5.87 24 (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]. 5.84/5.87 25 (all V_xs_2 all V_f_2 all T_d all T_c hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),hAPP(c_List_Orev(T_d),V_xs_2)) = hAPP(c_List_Orev(T_c),hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_xs_2))) # label(fact_rev__map) # label(axiom) # label(non_clause). [assumption]. 5.84/5.87 26 (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]. 5.84/5.87 27 (all V_ys all V_xs all T_a hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),c_List_Osplice(T_a,V_xs,V_ys)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_ys))) # label(fact_length__splice) # label(axiom) # label(non_clause). [assumption]. 5.84/5.87 28 (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]. 5.84/5.87 29 (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]. 5.84/5.87 30 (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(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))) # label(fact_Int__left__absorb) # label(axiom) # label(non_clause). [assumption]. 5.84/5.87 31 (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]. 5.84/5.87 32 (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]. 5.84/5.87 33 (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]. 5.84/5.87 34 (all V_xs_2 all T_c ((all B_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_i),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2))) -> (all B_j (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_j),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2))) -> (B_j != B_i -> hAPP(hAPP(c_List_Onth(T_c),V_xs_2),B_i) != hAPP(hAPP(c_List_Onth(T_c),V_xs_2),B_j)))))) <-> hBOOL(hAPP(c_List_Odistinct(T_c),V_xs_2)))) # label(fact_distinct__conv__nth) # label(axiom) # label(non_clause). [assumption]. 5.84/5.87 35 (all V_b all V_a all T_a c_Nitpick_OAbs__Frac(T_a,c_Nitpick_Onorm__frac(V_a,V_b)) = hAPP(hAPP(c_Nitpick_Ofrac(T_a),V_a),V_b)) # label(fact_Nitpick_Ofrac__def) # label(axiom) # label(non_clause). [assumption]. 5.84/5.87 36 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),V_y)) = V_x)) # label(fact_min__max_Oinf__sup__absorb) # label(axiom) # label(non_clause). [assumption]. 5.84/5.87 37 (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]. 5.84/5.87 38 (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 -> hAPP(hAPP(c_List_Oappend(T_c),B_us),hAPP(hAPP(c_List_Olist_OCons(T_c),B_y),B_vs)) != V_ys_2)))))) # label(fact_listrel1E) # label(axiom) # label(non_clause). [assumption]. 5.84/5.87 39 (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]. 5.84/5.87 40 (all V_y all V_x all T_a (class_Lattices_Oboolean__algebra(T_a) -> hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_x)),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_y)))) # label(fact_compl__sup) # label(axiom) # label(non_clause). [assumption]. 5.84/5.87 41 (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]. 5.84/5.87 42 (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_zs_2),V_ts_2) = hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_ys_2) <-> (exists B_us (hAPP(hAPP(c_List_Oappend(T_c),B_us),V_ts_2) = V_ys_2 & hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),B_us) = V_zs_2 | V_ts_2 = hAPP(hAPP(c_List_Oappend(T_c),B_us),V_ys_2) & V_xs_2 = hAPP(hAPP(c_List_Oappend(T_c),V_zs_2),B_us))))) # label(fact_append__eq__append__conv2) # label(axiom) # label(non_clause). [assumption]. 5.84/5.87 43 (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]. 5.84/5.87 44 (all V_ys all V_y all V_xs all V_x all T_b all T_a hAPP(hAPP(c_List_Ozip(T_a,T_b),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)),hAPP(hAPP(c_List_Olist_OCons(T_b),V_y),V_ys)) = hAPP(hAPP(c_List_Olist_OCons(tc_prod(T_a,T_b)),hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_x),V_y)),hAPP(hAPP(c_List_Ozip(T_a,T_b),V_xs),V_ys))) # label(fact_zip__Cons__Cons) # label(axiom) # label(non_clause). [assumption]. 5.84/5.87 45 (all V_list all V_a 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_list)),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),hAPP(hAPP(c_List_Olist_OCons(T_a),V_a),V_list))) # label(fact_list_Osize_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.84/5.87 46 (all V_x_2 all V_g_2 all V_f_2 all T_d all T_c (class_Lattices_Olattice(T_c) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_c),hAPP(V_f_2,V_x_2)),hAPP(V_g_2,V_x_2)) = hAPP(hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_d,T_c)),V_f_2),V_g_2),V_x_2))) # label(fact_sup__apply) # label(axiom) # label(non_clause). [assumption]. 5.84/5.87 47 (all V_b all V_a all T_a (class_Groups_Ogroup__add(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)) = V_b)) # label(fact_minus__add__cancel) # label(axiom) # label(non_clause). [assumption]. 5.84/5.87 48 (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]. 5.84/5.87 49 (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]. 5.84/5.87 50 (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]. 5.84/5.87 51 (all V_x_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> hAPP(c_Hilbert__Choice_Oinv__into(T_c,T_d,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)),V_f_2),hAPP(V_f_2,V_x_2)) = V_x_2)) # label(fact_inv__f__f) # label(axiom) # label(non_clause). [assumption]. 5.84/5.87 52 (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]. 5.84/5.87 53 (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]. 5.84/5.87 54 (all V_n hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n) = V_n) # label(fact_max__0L) # label(axiom) # label(non_clause). [assumption]. 5.84/5.87 55 (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]. 5.84/5.87 56 (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]. 5.84/5.87 57 (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_x_2),V_z_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_y_2),V_z_2))))) # label(fact_min__max_Ole__sup__iff) # label(axiom) # label(non_clause). [assumption]. 5.84/5.87 58 (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]. 5.84/5.87 59 (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]. 5.84/5.87 60 (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]. 5.84/5.87 61 (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_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)) = 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))))) # label(fact_min__max_Ofold__inf__insert) # label(axiom) # label(non_clause). [assumption]. 5.84/5.87 62 (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]. 5.84/5.87 63 (all T_c all V_c_2 all V_Pa_2 ((exists B_x1 hBOOL(hAPP(V_Pa_2,B_x1))) -> (V_c_2 = c_Hilbert__Choice_OEps(T_c,V_Pa_2) -> hBOOL(hAPP(V_Pa_2,V_c_2))))) # label(fact_exE__some) # label(axiom) # label(non_clause). [assumption]. 5.84/5.87 64 (all T_c (class_Orderings_Olinorder(T_c) -> c_List_Olist_ONil(T_c) = c_List_Olinorder__class_Osorted__list__of__set(T_c,c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))) # label(fact_sorted__list__of__set__empty) # label(axiom) # label(non_clause). [assumption]. 5.84/5.87 65 (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]. 5.84/5.87 66 (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]. 5.84/5.87 67 (all V_c all V_a all V_b all T_a (class_Lattices_Osemilattice__sup(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_a),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_b),V_c)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_b),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_a),V_c)))) # label(fact_sup_Oleft__commute) # label(axiom) # label(non_clause). [assumption]. 5.84/5.87 68 (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]. 5.84/5.87 69 (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]. 5.84/5.87 70 (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(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))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_A_2)))) # label(fact_Compl__subset__Compl__iff) # label(axiom) # label(non_clause). [assumption]. 5.84/5.87 71 (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]. 5.84/5.87 72 (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]. 5.84/5.87 73 (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]. 5.84/5.87 74 (all V_j all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),V_j)) -> c_List_Oupt(V_i,V_j) = hAPP(hAPP(c_List_Olist_OCons(tc_Nat_Onat),V_i),c_List_Oupt(hAPP(c_Nat_OSuc,V_i),V_j)))) # label(fact_upt__conv__Cons) # label(axiom) # label(non_clause). [assumption]. 5.84/5.87 75 (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]. 5.84/5.87 76 (all V_C_2 all V_J_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_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_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_I_2),V_J_2)),V_C_2)) # label(fact_Sigma__Diff__distrib1) # label(axiom) # label(non_clause). [assumption]. 5.84/5.87 77 (all V_y_2 all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_x_2),V_y_2)) -> (V_y_2 = V_x_2 <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_x_2),V_y_2)))))) # label(fact_linorder__antisym__conv1) # label(axiom) # label(non_clause). [assumption]. 5.84/5.87 78 (all V_a all T_a (class_Groups_Omonoid__mult(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Groups_Oone__class_Oone(T_a)) = V_a)) # label(fact_mult__1__right) # label(axiom) # label(non_clause). [assumption]. 5.84/5.87 79 (all V_b all V_a all T_a (class_Divides_Osemiring__div(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_a),V_b)),V_b)),c_Divides_Odiv__class_Omod(T_a,V_a,V_b)) = V_a)) # label(fact_mod__div__decomp) # label(axiom) # label(non_clause). [assumption]. 5.84/5.87 80 (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]. 5.84/5.87 81 (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]. 5.84/5.87 82 (all V_f_2 all V_B_2 all V_A_2 all T_c all T_d (class_Fields_Ofield(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))) -> hAPP(V_f_2,B_x) != c_Groups_Ozero__class_Ozero(T_d))) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_f_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_d),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_d),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_f_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_f_2),V_B_2))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_f_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)))))))) # label(fact_setprod__Un) # label(axiom) # label(non_clause). [assumption]. 5.84/5.87 83 (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]. 5.84/5.88 84 (all V_x all T_a (class_Lattices_Oboolean__algebra(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_x)) = c_Orderings_Obot__class_Obot(T_a))) # label(fact_inf__compl__bot) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 85 (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]. 5.84/5.88 86 (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) & (all B_y (c_List_Olinorder__class_Osorted(T_c,B_y) & hBOOL(hAPP(c_List_Odistinct(T_c),B_y)) & V_A_2 = hAPP(c_List_Oset(T_c),B_y) -> B_x = B_y)) & hBOOL(hAPP(c_List_Odistinct(T_c),B_x))))))) # label(fact_finite__sorted__distinct__unique) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 87 (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]. 5.84/5.88 88 (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]. 5.84/5.88 89 (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]. 5.84/5.88 90 (all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> V_A_2 = hAPP(c_List_Oset(T_c),c_List_Olinorder__class_Osorted__list__of__set(T_c,V_A_2)) & 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))))) # label(fact_sorted__list__of__set) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 91 (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]. 5.84/5.88 92 (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]. 5.84/5.88 93 (all V_R_2 all V_y_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_y_2)),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_c,T_c),tc_HOL_Obool),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_R_2))) -> V_y_2 = V_x_2)) # label(fact_rel__pow__0__E) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 94 (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]. 5.84/5.88 95 (all V_f_2 all T_e all T_d all T_c hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,T_e),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_prod(T_c,T_d)),tc_fun(T_d,T_e),T_c),hAPP(c_COMBB(tc_prod(T_c,T_d),T_e,T_d),V_f_2)),c_Product__Type_OPair(T_c,T_d))) = V_f_2) # label(fact_split__eta) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 96 (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]. 5.84/5.88 97 (all V_xs_2 all V_f_2 all T_d all T_c hAPP(hAPP(c_List_Omap(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)),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))) # label(fact_concat__map__singleton) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 98 (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]. 5.84/5.88 99 (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]. 5.84/5.88 100 (all V_y all V_x all V_z all T_a (class_Fields_Ofield(T_a) -> (V_z != c_Groups_Ozero__class_Ozero(T_a) -> hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_z),V_y))),V_z) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),V_z)),V_y)))) # label(fact_divide__add__eq__iff) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 101 (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),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))) <-> hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_c),V_b_2),V_c_2))))))) # label(fact_div__dvd__div) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 102 (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]. 5.84/5.88 103 (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]. 5.84/5.88 104 (all V_x all T_a (class_Fields_Ofield__inverse__zero(T_a) & class_Int_Onumber__ring(T_a) -> hAPP(c_Groups_Ouminus__class_Ouminus(T_a),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]. 5.84/5.88 105 (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]. 5.84/5.88 106 (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]. 5.84/5.88 107 (all V_m all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n),V_m)) -> hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n)))) # label(fact_zdiff__int) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 108 (all V_r_2 all V_ys_2 all V_y_2 all V_xs_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2)),hAPP(hAPP(c_List_Olist_OCons(T_c),V_y_2),V_ys_2))),c_List_Olex(T_c,V_r_2))) <-> hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_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)) | V_y_2 = V_x_2 & hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_xs_2),V_ys_2)),c_List_Olex(T_c,V_r_2))))) # label(fact_Cons__in__lex) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 109 (all V_A_2 all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> c_Complete__Lattice_OInf__class_OInf(T_c,V_A_2) = 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))))))) # label(fact_Inf__Sup) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 110 (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(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_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_f_2),hAPP(c_List_Oset(T_c),V_xs_2))))) # label(fact_listsum__distinct__conv__setsum__set) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 111 (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]. 5.84/5.88 112 (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]. 5.84/5.88 113 (all V_a_2 all V_f_2 all T_c all T_d V_a_2 = c_List_Ofoldr(T_d,T_c,V_f_2,c_List_Olist_ONil(T_d),V_a_2)) # label(fact_foldr_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 114 (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]. 5.84/5.88 115 (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))) <-> V_xs_2 = V_ys_2 & hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_y_2)),V_r_2)) | 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)) # label(fact_Cons__listrel1__Cons) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 116 (all V_b all V_a all T_a (class_Divides_Osemiring__div(T_a) -> (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_a),V_a),V_b)) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_b),V_a)),V_a) = V_b))) # label(fact_dvd__div__mult__self) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 117 (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]. 5.84/5.88 118 (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_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)) = 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))))))))))))) # label(fact_min__max_Oinf__Sup2__distrib) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 119 (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]. 5.84/5.88 120 (all V_A_2 all V_b_2 all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_b_2),B_x)))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_b_2),c_Complete__Lattice_OInf__class_OInf(T_c,V_A_2)))))) # label(fact_le__Inf__iff) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 121 (all V_A_2 all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> c_Complete__Lattice_OSup__class_OSup(T_c,V_A_2) = 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)))))))) # label(fact_Sup__Inf) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 122 (all V_k_2 all V_l_2 all T_c (class_Orderings_Oorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_l_2),V_k_2)) -> c_SetInterval_Oord__class_OgreaterThanAtMost(T_c,V_k_2,V_l_2) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))) # label(fact_greaterThanAtMost__empty) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 123 (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(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_Opred(V_v))))) # label(fact_nat__number__of__diff__1) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 124 (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]. 5.84/5.88 125 (all T_a all V_xs (V_xs != c_List_Olist_ONil(T_a) -> hAPP(c_List_Orotate1(T_a),V_xs) = hAPP(hAPP(c_List_Oappend(T_a),hAPP(c_List_Otl(T_a),V_xs)),hAPP(hAPP(c_List_Olist_OCons(T_a),hAPP(c_List_Ohd(T_a),V_xs)),c_List_Olist_ONil(T_a))))) # label(fact_rotate1__hd__tl) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 126 (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]. 5.84/5.88 127 (all V_B_2 all V_t_2 all V_D_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_D_2)) -> (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),V_t_2),V_B_2)) -> (all B_x ((all B_xa (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xa),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),V_D_2))) -> (all B_xb (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xb),V_B_2)) -> B_x != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B_xb),B_xa))))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_t_2),B_x)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_t_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B_x),V_D_2))))))))) # label(fact_bset_I7_J) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 128 (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]. 5.84/5.88 129 (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]. 5.84/5.88 130 (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_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_c_2)))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_c_2)) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_c_2),c_Groups_Ozero__class_Ozero(T_c))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_a_2))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_c_2),c_Groups_Ozero__class_Ozero(T_c))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_c_2)),V_b_2))))))) # label(fact_divide__less__eq) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 131 (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]. 5.84/5.88 132 (all V_b all V_a hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b)) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_b))) # label(fact_zmod__zminus__zminus) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 133 (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]. 5.84/5.88 134 (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_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)) = 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)))))) # label(fact_setprod__insert) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 135 (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]. 5.84/5.88 136 (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]. 5.84/5.88 137 (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]. 5.84/5.88 138 (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]. 5.84/5.88 139 (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]. 5.84/5.88 140 (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]. 5.84/5.88 141 (all V_h_2 all V_g_2 all V_f_2 all T_h all T_g all T_d all T_f all T_e all T_c hAPP(hAPP(c_Product__Type_Oscomp(T_c,T_e,T_f,T_d),hAPP(hAPP(c_Product__Type_Oscomp(T_c,T_g,T_h,tc_prod(T_e,T_f)),V_f_2),V_g_2)),V_h_2) = hAPP(hAPP(c_Product__Type_Oscomp(T_c,T_g,T_h,T_d),V_f_2),hAPP(hAPP(c_COMBC(T_g,tc_fun(T_e,tc_fun(T_f,T_d)),tc_fun(T_h,T_d)),hAPP(hAPP(c_COMBB(tc_fun(T_h,tc_prod(T_e,T_f)),tc_fun(tc_fun(T_e,tc_fun(T_f,T_d)),tc_fun(T_h,T_d)),T_g),c_Product__Type_Oscomp(T_h,T_e,T_f,T_d)),V_g_2)),V_h_2))) # label(fact_scomp__scomp) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 142 (all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> c_Orderings_Obot__class_Obot(T_c) = c_Complete__Lattice_OSup__class_OSup(T_c,c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))) # label(fact_Sup__empty) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 143 (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]. 5.84/5.88 144 (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(V_F_2,V_A_2),V_s_2) = hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_d,V_f_2),V_s_2),V_A_2)))) # label(fact_folding_Oeq__fold) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 145 (all V_f_2 all T_c all T_d (c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)) = hAPP(c_Set_Oimage(T_d,T_c,V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool))) <-> (all B_x 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))))) # label(fact_surj__iff__all) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 146 (all V_j all V_i ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_i),V_j)) -> c_List_Oupto(V_i,V_j) = hAPP(hAPP(c_List_Olist_OCons(tc_Int_Oint),V_i),c_List_Oupto(hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_i),c_Groups_Oone__class_Oone(tc_Int_Oint)),V_j))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_i),V_j)) -> c_List_Olist_ONil(tc_Int_Oint) = c_List_Oupto(V_i,V_j)))) # label(fact_upto_Osimps) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 147 (all V_y_2 all V_x_2 all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,V_A_2) -> (hAPP(V_f_2,V_y_2) = hAPP(V_f_2,V_x_2) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_y_2),V_A_2)) -> V_y_2 = V_x_2))))) # label(fact_inj__onD) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 148 (all V_n_2 all V_m_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_m_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_n_2))) <-> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_k_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2))))) # label(fact_mult__le__cancel1) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 149 (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]. 5.84/5.88 150 (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]. 5.84/5.88 151 (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]. 5.84/5.88 152 (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]. 5.84/5.88 153 (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]. 5.84/5.88 154 (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]. 5.84/5.88 155 (all V_n_2 all V_m_2 all V_u_2 all V_i_2 all V_j_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_j_2),V_i_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_i_2),V_j_2)),V_u_2)),V_m_2)),V_n_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_i_2),V_u_2)),V_m_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_j_2),V_u_2)),V_n_2)))))) # label(fact_nat__less__add__iff1) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 156 (all V_x_2 all T_c (class_Int_Onumber__ring(T_c) & class_Rings_Olinordered__idom(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_x_2),c_Int_OBit1(c_Int_OPls))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_x_2)),c_Groups_Oone__class_Oone(T_c)))))) # label(fact_le__special_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 157 (all V_B_2 all V_A_2 all V_f_2 all T_c all T_d hAPP(c_Set_Oimage(T_d,T_c,V_f_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_d,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_d,T_c,V_f_2),V_A_2)),hAPP(c_Set_Oimage(T_d,T_c,V_f_2),V_B_2))) # label(fact_image__Un) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 158 (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]. 5.84/5.88 159 (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]. 5.84/5.88 160 (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]. 5.84/5.88 161 (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 | hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = V_n)) # label(fact_less__2__cases) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 162 (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]. 5.84/5.88 163 (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]. 5.84/5.88 164 (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]. 5.84/5.88 165 (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]. 5.84/5.88 166 (all V_B_2 all V_A_2 all V_I_2 all T_c all T_d hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_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)) = 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))) # label(fact_UN__Un__distrib) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 167 (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_x = B_y))))) # label(fact_antisym__def) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 168 (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]. 5.84/5.88 169 (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]. 5.84/5.88 170 (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_x1 hBOOL(hAPP(V_Pa_2,B_x1))) <-> (exists B_x (hBOOL(hAPP(V_Pa_2,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))))))))) # label(fact_periodic__finite__ex) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 171 (all V_n_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m_2)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n_2))))) # label(fact_zless__int) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 172 (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_List_Omap_Ocomp) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 173 (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]. 5.84/5.88 174 (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]. 5.84/5.88 175 (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]. 5.84/5.88 176 (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]. 5.84/5.88 177 (all V_a_2 all V_c_2 all V_b_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_a_2 = c_Groups_Ozero__class_Ozero(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))))) # label(fact_divide__eq__eq) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 178 (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]. 5.84/5.88 179 (all V_w all V_x all V_z all V_y all T_a (class_Fields_Ofield(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) != V_y -> (V_z != c_Groups_Ozero__class_Ozero(T_a) -> hAPP(hAPP(c_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)) = 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)))))) # label(fact_add__frac__eq) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 180 (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]. 5.84/5.88 181 (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]. 5.84/5.88 182 (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]. 5.84/5.88 183 (all V_x_2 all V_z_2 all V_f_2 all T_d all T_c (hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_c,T_d,V_f_2,V_z_2,c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))),V_x_2)) -> V_z_2 = V_x_2)) # label(fact_empty__fold__graphE) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 184 (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]. 5.84/5.88 185 (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]. 5.84/5.88 186 (all V_m all V_j all V_k (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_k),V_j)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_j)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_k),V_m)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j),V_k))),V_m))) # label(fact_diff__Suc__diff__eq2) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 187 (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)) <-> 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__iff) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 188 (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]. 5.84/5.88 189 (all V_b all V_a all T_a (class_Fields_Olinordered__field__inverse__zero(T_a) -> hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_b)) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_b)))) # label(fact_abs__divide) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 190 (all V_B_2 all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,V_A_2) & hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2) = V_B_2 <-> c_Fun_Obij__betw(T_c,T_d,V_f_2,V_A_2,V_B_2))) # label(fact_bij__betw__def) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 191 (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]. 5.84/5.88 192 (all V_a all V_r all V_q all V_c all V_b (hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_b,V_c),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_q),V_r))) -> (V_c != c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> hBOOL(hAPP(c_Divides_Odivmod__int__rel(hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_a),V_b),V_c),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_a),V_q)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_a),V_r)),V_c))),c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_a),V_r),V_c))))))) # label(fact_zmult1__lemma) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 193 (all V_a_2 all T_c (class_Groups_Olinordered__ab__group__add(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_double__zero__sym) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 194 (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_d,T_c,V_Abs_2),V_A_2) = c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)))) # label(fact_type__definition_OAbs__image) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 195 (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]. 5.84/5.88 196 (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_H = V_q)))) # label(fact_unique__quotient) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 197 (all V_Y_2 all T_c V_Y_2 = hAPP(c_Set_Oimage(T_c,T_c,c_COMBI(T_c)),V_Y_2)) # label(fact_image__ident) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 198 (all V_xs_2 all V_n_2 all T_c (c_List_Olist_ONil(T_c) = V_xs_2 <-> hAPP(c_List_Orotate(T_c,V_n_2),V_xs_2) = c_List_Olist_ONil(T_c))) # label(fact_rotate__is__Nil__conv) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 199 (all V_a_2 all V_b_2 all T_c (class_Rings_Odivision__ring(T_c) -> (c_Groups_Ozero__class_Ozero(T_c) != V_b_2 -> (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]. 5.84/5.88 200 (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]. 5.84/5.88 201 (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]. 5.84/5.88 202 (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]. 5.84/5.88 203 (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_j_2 = V_i_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]. 5.84/5.88 204 (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]. 5.84/5.88 205 (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]. 5.84/5.88 206 (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]. 5.84/5.88 207 (all T_c all T_d all V_y_2 all V_x_2 all V_f_2 (V_y_2 = hAPP(V_f_2,V_x_2) -> c_Fun_Ofun__upd(T_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]. 5.84/5.88 208 (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]. 5.84/5.88 209 (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(tc_Int_Oint),V_x_2),V_y_2)) <-> 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)))))) # label(fact_less__number__of) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 210 (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]. 5.84/5.88 211 (all T_a (class_Int_Onumber__ring(T_a) & class_Rings_Olinordered__idom(T_a) -> c_Groups_Oone__class_Oone(T_a) = hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OMin)))) # label(fact_abs__minus__one) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 212 (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]. 5.84/5.88 213 (all V_x all T_a hAPP(c_List_OremoveAll(T_a,V_x),c_List_Olist_ONil(T_a)) = c_List_Olist_ONil(T_a)) # label(fact_removeAll_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 214 (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]. 5.84/5.88 215 (all V_z all T_a (class_Int_Onumber__ring(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OMin)),V_z) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_z))) # label(fact_mult__minus1) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 216 (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]. 5.84/5.88 217 (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]. 5.84/5.88 218 (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_OINFI(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__inf__class_Oinf(T_c)),V_f_2)),c_Orderings_Otop__class_Otop(T_c)),V_xs_2))) # label(fact_INFI__set__fold) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 219 (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]. 5.84/5.88 220 (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_ORange(T_d,T_c),V_r_2))) <-> (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))))) # label(fact_Range__iff) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 221 (all V_c all V_b all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Divides_Odiv__class_Omod(T_a,V_b,V_c)),V_c))) # label(fact_mod__mult__right__eq) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 222 (all V_l (c_List_Onat__list(V_l) -> V_l = c_List_Oembed__list(c_List_Oreturn__list(V_l)))) # label(fact_transfer__nat__int__list__return__embed) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 223 (all V_x_2 all T_d all T_c (class_Orderings_Obot(T_c) -> c_Orderings_Obot__class_Obot(T_c) = hAPP(c_Orderings_Obot__class_Obot(tc_fun(T_d,T_c)),V_x_2))) # label(fact_bot__apply) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 224 (all V_j_2 all V_i_2 all V_Pa_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)))) <-> c_List_Oall__interval__int(V_Pa_2,V_i_2,V_j_2))) # label(fact_all__interval__int__def) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 225 (all V_C_2 all V_B_2 all V_A_2 all T_c (c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)) = hAPP(hAPP(c_Relation_Orel__comp(T_c,T_c,T_c),V_A_2),V_B_2) -> (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]. 5.84/5.88 226 (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)) <-> hBOOL(hAPP(V_A_2,V_x_2)) | V_y_2 = V_x_2)) # label(fact_insert__code) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 227 (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]. 5.84/5.88 228 (all V_x all V_k all V_i all T_a hAPP(hAPP(c_List_Odrop(T_a),V_i),c_List_Oreplicate(T_a,V_k,V_x)) = c_List_Oreplicate(T_a,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_k),V_i),V_x)) # label(fact_drop__replicate) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 229 (all V_l c_Int_OMin != c_Int_OBit0(V_l)) # label(fact_rel__simps_I42_J) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 230 (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]. 5.84/5.88 231 (all V_A_2 all V_B_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_A_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2) = V_B_2)) # label(fact_Int__absorb1) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 232 (all V_B_2 all V_A_2 all T_d all T_c all V_C_2 ((V_C_2 = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_d,tc_HOL_Obool)),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) & (V_C_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool),T_c),hAPP(c_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))))) # label(fact_UN__extend__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 233 (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]. 5.84/5.88 234 (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_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__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_C_2))) # label(fact_Un__insert__left) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 235 (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]. 5.84/5.88 236 (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]. 5.84/5.88 237 (all V_B_2 all V_A_2 all V_F_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofolding__idem(T_c,T_d,V_f_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> hAPP(V_F_2,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(c_Fun_Ocomp(T_d,T_d,T_d,hAPP(V_F_2,V_A_2)),hAPP(V_F_2,V_B_2)))))) # label(fact_folding__idem_Ounion__idem) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 238 (all V_xs_2 all V_f_2 all T_d all V_Pa_2 all T_c hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),hAPP(c_List_Ofilter(T_d,hAPP(c_Fun_Ocomp(T_c,tc_HOL_Obool,T_d,V_Pa_2),V_f_2)),V_xs_2)) = hAPP(c_List_Ofilter(T_c,V_Pa_2),hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_xs_2))) # label(fact_filter__map) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 239 (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]. 5.84/5.88 240 (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]. 5.84/5.88 241 (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)))) -> c_Groups_Ozero__class_Ozero(T_d) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_f_2),V_A_2))))) # label(fact_setprod__zero) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 242 (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]. 5.84/5.88 243 (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]. 5.84/5.88 244 (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) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n) != c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_field__power__not__zero) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 245 (all V_y_2 all V_x_2 all T_c (class_Orderings_Oorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_x_2),V_y_2)) & V_y_2 != V_x_2 <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_x_2),V_y_2))))) # label(fact_order__less__le) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 246 (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]. 5.84/5.88 247 (all V_R_2 all V_b_2 all V_a_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_b_2)),c_Transitive__Closure_Ortrancl(T_c,V_R_2))) -> V_b_2 != V_a_2 & hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_b_2)),c_Transitive__Closure_Otrancl(T_c,V_R_2))) | V_b_2 = V_a_2)) # label(fact_rtranclD) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 248 (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(V_Pa_2,V_x_2)) & hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2))))) # label(fact_set__takeWhileD) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 249 (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]. 5.84/5.88 250 (all V_xs_2 all V_f_2 all T_d all T_c (class_Groups_Oab__group__add(T_c) -> hAPP(c_Groups_Ouminus__class_Ouminus(T_c),hAPP(c_List_Omonoid__add__class_Olistsum(T_c),hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_xs_2))) = hAPP(c_List_Omonoid__add__class_Olistsum(T_c),hAPP(hAPP(c_List_Omap(T_d,T_c),hAPP(c_Fun_Ocomp(T_c,T_c,T_d,c_Groups_Ouminus__class_Ouminus(T_c)),V_f_2)),V_xs_2)))) # label(fact_uminus__listsum__map) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 251 (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]. 5.84/5.88 252 (all V_b all V_a all V_c all T_a (class_Rings_Odivision__ring(T_a) -> (V_c != c_Groups_Ozero__class_Ozero(T_a) -> (hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c) = V_b -> 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]. 5.84/5.88 253 (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]. 5.84/5.88 254 (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]. 5.84/5.88 255 (all V_w all V_v all T_a (class_Int_Onumber__ring(T_a) -> hAPP(c_Int_Onumber__class_Onumber__of(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_v),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_w))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w)))) # label(fact_diff__number__of__eq) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 256 (all V_x_2 all V_S_2 all V_R_2 all T_c (hBOOL(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_R_2)),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_S_2)),V_x_2)) <-> hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_R_2),V_S_2))))) # label(fact_inf__Int__eq) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 257 (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]. 5.84/5.88 258 (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]. 5.84/5.88 259 (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]. 5.84/5.88 260 (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]. 5.84/5.88 261 (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]. 5.84/5.88 262 (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]. 5.84/5.88 263 (all V_x_2 all V_f_2 all T_e all T_d all T_c hAPP(c_Product__Type_Ofst(T_c,T_d),hAPP(c_Product__Type_Oapfst(T_e,T_c,T_d,V_f_2),V_x_2)) = hAPP(V_f_2,hAPP(c_Product__Type_Ofst(T_e,T_d),V_x_2))) # label(fact_fst__apfst) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 264 (all V_n_2 all V_a_2 all T_c (class_Power_Opower(T_c) & class_Rings_Ono__zero__divisors(T_c) & class_Rings_Ozero__neq__one(T_c) & class_Rings_Omult__zero(T_c) -> (hAPP(hAPP(c_Power_Opower__class_Opower(T_c),V_a_2),V_n_2) = c_Groups_Ozero__class_Ozero(T_c) <-> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != V_n_2 & V_a_2 = c_Groups_Ozero__class_Ozero(T_c)))) # label(fact_power__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 265 (all V_B_2 all V_A_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2) = hAPP(c_Relation_OImage(T_c,T_c,c_Relation_OId__on(T_c,V_A_2)),V_B_2)) # label(fact_Image__Id__on) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 266 (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]. 5.84/5.88 267 (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]. 5.84/5.88 268 (all V_a all V_b 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_b),V_a),V_b))) # label(fact_mod__mult__self1__is__0) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 269 (all V_l_2 all V_k_2 (-(hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_k_2),V_l_2)) & -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_l_2),V_k_2))) <-> c_SetInterval_Oord_OgreaterThanAtMost(tc_Nat_Onat,c_Rings_Odvd__class_Odvd(tc_Nat_Onat),hAPP(hAPP(c_COMBS(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),tc_Nat_Onat),c_COMBS(tc_Nat_Onat,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_Nat_Onat),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Nat_Onat),c_fconj)),c_Rings_Odvd__class_Odvd(tc_Nat_Onat)))),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_Nat_Onat),hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Nat_Onat),c_fNot)),hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_HOL_Obool),c_Rings_Odvd__class_Odvd(tc_Nat_Onat)))),V_k_2,V_l_2) = c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) # label(fact_dvd_OgreaterThanAtMost__empty__iff) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 270 (all V_x_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_x_2),hAPP(c_Finite__Set_Ofold1(T_c,c_Orderings_Oord__class_Omin(T_c)),V_A_2))) <-> (all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_x_2),B_x))))))))) # label(fact_min__max_Obelow__fold1__iff) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 271 (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_b_2 = V_c_2))) # label(fact_add__right__cancel) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 272 (all V_y all V_x all T_a (class_Lattices_Olattice(T_a) -> V_x = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y)))) # label(fact_sup__inf__absorb) # label(axiom) # label(non_clause). [assumption]. 5.84/5.88 273 (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]. 5.84/5.88 274 (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]. 5.84/5.88 275 (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]. 5.84/5.89 276 (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]. 5.84/5.89 277 (all V_Pa_2 ((all B_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B_x)) -> hBOOL(hAPP(V_Pa_2,hAPP(c_Int_Onat,B_x))))) <-> (all B_x1 hBOOL(hAPP(V_Pa_2,B_x1))))) # label(fact_all__nat) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 278 (all V_b all V_a ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_a),V_b))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_b),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> 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),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]. 5.84/5.89 279 (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__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))) <-> V_x_2 = c_Groups_Ozero__class_Ozero(T_c) & V_y_2 = c_Groups_Ozero__class_Ozero(T_c)))) # label(fact_sum__squares__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 280 (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]. 5.84/5.89 281 (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]. 5.84/5.89 282 (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]. 5.84/5.89 283 (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_Ozero__class_Ozero(tc_Nat_Onat) <-> V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_m_2)) # label(fact_mult__is__0) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 284 (all V_n all V_m all T_a (class_Rings_Osemiring__1(T_a) -> hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),V_m),V_n)) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),V_m)),V_n))) # label(fact_of__nat__power) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 285 (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]. 5.84/5.89 286 (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]. 5.84/5.89 287 (all V_zs all V_xs all V_ys all V_xs1 all V_x all T_a (hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs1) = V_ys -> (hAPP(hAPP(c_List_Oappend(T_a),V_xs1),V_zs) = V_xs -> hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs) = hAPP(hAPP(c_List_Oappend(T_a),V_ys),V_zs)))) # label(fact_Cons__eq__appendI) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 288 (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]. 5.84/5.89 289 (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]. 5.84/5.89 290 (all V_a_2 all V_b_2 all T_c (class_Orderings_Oorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_b_2),V_a_2)) -> c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = c_SetInterval_Oord__class_OatLeastAtMost(T_c,V_a_2,V_b_2)))) # label(fact_atLeastatMost__empty) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 291 (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]. 5.84/5.89 292 (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]. 5.84/5.89 293 (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]. 5.84/5.89 294 (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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_v),V_w))),V_z) = 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)))) # label(fact_mult__number__of__left) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 295 (all V_r_H all V_q_H all V_r all V_q all V_b all V_a (hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a,V_b),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_q),V_r))) -> (hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a,V_b),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_q_H),V_r_H))) -> (V_b != c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> V_r_H = V_r)))) # label(fact_unique__remainder) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 296 (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)) -> c_SetInterval_Oord__class_OgreaterThanLessThan(T_c,V_l_2,V_u_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanAtMost(T_c,V_l_2,V_m_2)),c_SetInterval_Oord__class_OgreaterThanLessThan(T_c,V_m_2,V_u_2)))))) # label(fact_ivl__disj__un_I16_J) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 297 (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]. 5.84/5.89 298 (all V_g_2 all V_x_2 all V_A_2 all T_c all T_d (class_Groups_Ocomm__monoid__add(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_g_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_d),hAPP(V_g_2,V_x_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_g_2),V_A_2)))))) # label(fact_setsum_Oinsert) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 299 (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]. 5.84/5.89 300 (all V_y all V_x all T_a (class_Lattices_Osemilattice__inf(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y)) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y) = V_x))) # label(fact_inf__absorb1) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 301 (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]. 5.84/5.89 302 (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]. 5.84/5.89 303 (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]. 5.84/5.89 304 (all V_u_2 hAPP(c_Nat_OSuc,V_u_2) = hAPP(c_Finite__Set_Ocard(tc_Nat_Onat),hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),V_u_2))) # label(fact_card__atMost) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 305 (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)) -> (V_N_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> c_Big__Operators_Olinorder__class_OMin(T_c,hAPP(c_Set_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]. 5.84/5.89 306 (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]. 5.84/5.89 307 (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_d,T_c),V_f_2),hAPP(hAPP(c_List_Omap(T_e,T_d),V_g_2),V_xs_2)) = hAPP(hAPP(c_List_Omap(T_e,T_c),hAPP(c_Fun_Ocomp(T_d,T_c,T_e,V_f_2),V_g_2)),V_xs_2)) # label(fact_map__map) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 308 (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]. 5.84/5.89 309 (all V_ys_2 all V_xs_2 all T_c (hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_ys_2) = V_xs_2 <-> c_List_Olist_ONil(T_c) = V_ys_2)) # label(fact_append__self__conv) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 310 (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]. 5.84/5.89 311 (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]. 5.84/5.89 312 (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),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))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_c),V_A_2)),hAPP(c_Finite__Set_Ocard(T_c),V_B_2))))) # label(fact_card__Un__Int) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 313 (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)) -> 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)) & (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)))) # label(fact_z3div__def) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 314 (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]. 5.84/5.89 315 (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]. 5.84/5.89 316 (all V_Pa_2 all T_d all V_R_2 all T_c (c_Equiv__Relations_Oequivp(T_c,V_R_2) -> V_Pa_2 = c_Quotient_OBabs(T_c,T_d,c_Quotient_ORespects(T_c,V_R_2),V_Pa_2))) # label(fact_babs__reg__eqv) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 317 (all V_B_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_A_2 -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_B_2 -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2) -> 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__disjoint) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 318 (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]. 5.84/5.89 319 (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]. 5.84/5.89 320 (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)))) & (-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]. 5.84/5.89 321 (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))) -> hAPP(V_f_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_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)) = 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))))))) # label(fact_setsum__Un__zero) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 322 (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]. 5.84/5.89 323 (all V_B_2 all V_A_2 (c_Nat__Transfer_Onat__set(V_A_2) -> (c_Nat__Transfer_Onat__set(V_B_2) -> (V_A_2 = V_B_2 <-> hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),V_B_2) = hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),V_A_2))))) # label(fact_transfer__int__nat__set__relations_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 324 (all V_x_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_A_2 -> hAPP(hAPP(c_Orderings_Oord__class_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,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2))))))) # label(fact_Max_Oinsert) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 325 (all V_x all V_y all T_a (class_Divides_Oring__div(T_a) -> (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_a),V_y),V_x)) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_x)),V_y) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_x),V_y))))) # label(fact_dvd__neg__div) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 326 (all V_n all V_a all T_a (class_Groups_Omonoid__mult(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)),V_a))) # label(fact_power__commutes) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 327 (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]. 5.84/5.89 328 (all V_r_2 all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_Nat_Onat,tc_fun(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c)),tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(tc_Nat_Onat,tc_HOL_Obool))),c_List_Olexn(T_c,V_r_2)) = c_List_Olex(T_c,V_r_2)) # label(fact_lex__def) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 329 (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]. 5.84/5.89 330 (all V_ys_2 all V_xs_2 all V_f_2 all T_c all T_d (hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_xs_2) = hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_ys_2) -> (c_Fun_Oinj__on(T_d,T_c,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool))) -> V_xs_2 = V_ys_2))) # label(fact_map__injective) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 331 (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]. 5.84/5.89 332 (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]. 5.84/5.89 333 (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]. 5.84/5.89 334 (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]. 5.84/5.89 335 (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]. 5.84/5.89 336 (all V_x_2 all V_xs_2 all T_c (hBOOL(hAPP(c_List_Odistinct(T_c),V_xs_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2))) -> hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),hAPP(c_List_Orev(T_c),c_List_OtakeWhile(T_c,hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_c),c_fNot),hAPP(hAPP(c_COMBC(T_c,T_c,tc_HOL_Obool),c_fequal),V_x_2)),V_xs_2))) = c_List_OdropWhile(T_c,hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_c),c_fNot),hAPP(hAPP(c_COMBC(T_c,T_c,tc_HOL_Obool),c_fequal),V_x_2)),hAPP(c_List_Orev(T_c),V_xs_2))))) # label(fact_dropWhile__neq__rev) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 337 (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]. 5.84/5.89 338 (all V_x_2 all V_A_2 all V_F_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofolding__idem(T_c,T_d,V_f_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> hAPP(c_Fun_Ocomp(T_d,T_d,T_d,hAPP(V_F_2,V_A_2)),hAPP(V_f_2,V_x_2)) = hAPP(V_F_2,V_A_2))))) # label(fact_folding__idem_Oin__comp__idem) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 339 (all V_xs_2 all V_n_2 all T_c c_Enum_On__lists(T_c,hAPP(c_Nat_OSuc,V_n_2),V_xs_2) = c_List_Oconcat(tc_List_Olist(T_c),hAPP(hAPP(c_List_Omap(tc_List_Olist(T_c),tc_List_Olist(tc_List_Olist(T_c))),hAPP(hAPP(c_COMBC(tc_List_Olist(T_c),tc_List_Olist(T_c),tc_List_Olist(tc_List_Olist(T_c))),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_List_Olist(T_c)),tc_fun(tc_List_Olist(T_c),tc_List_Olist(tc_List_Olist(T_c))),tc_List_Olist(T_c)),c_List_Omap(T_c,tc_List_Olist(T_c))),hAPP(c_COMBC(T_c,tc_List_Olist(T_c),tc_List_Olist(T_c)),c_List_Olist_OCons(T_c)))),V_xs_2)),c_Enum_On__lists(T_c,V_n_2,V_xs_2)))) # label(fact_n__lists_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 340 (all V_l_2 all V_k_2 (-(-hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_l_2),V_k_2)) & hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_k_2),V_l_2))) <-> c_SetInterval_Oord_OgreaterThanAtMost(tc_Nat_Onat,c_Rings_Odvd__class_Odvd(tc_Nat_Onat),hAPP(hAPP(c_COMBS(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),tc_Nat_Onat),c_COMBS(tc_Nat_Onat,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_Nat_Onat),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Nat_Onat),c_fconj)),c_Rings_Odvd__class_Odvd(tc_Nat_Onat)))),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_Nat_Onat),hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Nat_Onat),c_fNot)),hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_HOL_Obool),c_Rings_Odvd__class_Odvd(tc_Nat_Onat)))),V_k_2,V_l_2) = c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) # label(fact_dvd_OgreaterThanAtMost__empty__iff2) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 341 (all V_xs all T_a all V_n (c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_n,hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> V_xs = hAPP(c_List_Orotate(T_a,V_n),V_xs))) # label(fact_rotate__id) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 342 (all V_m c_Divides_Odivmod__nat(V_m,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_m)) # label(fact_divmod__nat__zero) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 343 (all V_Qa_2 all V_Pa_2 (-hBOOL(V_Pa_2) | hBOOL(hAPP(hAPP(c_fconj,V_Pa_2),V_Qa_2)) | -hBOOL(V_Qa_2))) # label(help_c__fconj__1) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 344 (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)),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)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a_2),V_b_2)))))) # label(fact_nonneg1__imp__zdiv__pos__iff) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 345 (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]. 5.84/5.89 346 (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),V_f_2),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))) # label(fact_map__fun__upd) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 347 (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]. 5.84/5.89 348 (all V_b c_Divides_OposDivAlg(c_Groups_Ozero__class_Ozero(tc_Int_Oint),V_b) = hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) # label(fact_posDivAlg__0) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 349 (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]. 5.84/5.89 350 (all V_B_2 all V_A_2 all V_R_2 all T_d all T_c (c_Relation_Osingle__valued(T_c,T_d,hAPP(c_Relation_Oconverse(T_d,T_c),V_R_2)) -> hAPP(c_Relation_OImage(T_d,T_c,V_R_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_d,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_d,T_c,V_R_2),V_A_2)),hAPP(c_Relation_OImage(T_d,T_c,V_R_2),V_B_2)))) # label(fact_Image__Int__eq) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 351 (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]. 5.84/5.89 352 (all V_z hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_z),c_Groups_Ozero__class_Ozero(tc_Int_Oint)) = V_z) # label(fact_zadd__0__right) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 353 (all V_z V_z = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_z))) # label(fact_zminus__zminus) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 354 (all V_n all V_m (hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n) = V_m -> c_Groups_Oone__class_Oone(tc_Nat_Onat) = V_n | V_m = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_mult__eq__self__implies__10) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 355 (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]. 5.84/5.89 356 (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]. 5.84/5.89 357 (all V_g_2 all T_c all T_d hAPP(c_Fun_Ocomp(T_d,T_d,T_c,c_Fun_Oid(T_d)),V_g_2) = V_g_2) # label(fact_id__o) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 358 (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]. 5.84/5.89 359 (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]. 5.84/5.89 360 (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 & (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)))) & V_xs_2 = hAPP(c_List_Ofilter(T_c,V_Pa_2),B_vs) & 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_Cons__eq__filter__iff) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 361 (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]. 5.84/5.89 362 (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]. 5.84/5.89 363 (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)) = V_c_2) & (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = V_A_2 -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_HOL_Obool)),V_A_2),hAPP(c_COMBK(tc_fun(T_d,tc_HOL_Obool),T_c),V_c_2)) = c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool))))) # label(fact_UN__constant) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 364 (all V_A_H_H_2 all V_f_H_2 all T_e all V_A_H_2 all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Obij__betw(T_c,T_d,V_f_2,V_A_2,V_A_H_2) -> (c_Fun_Obij__betw(T_d,T_e,V_f_H_2,V_A_H_2,V_A_H_H_2) <-> c_Fun_Obij__betw(T_c,T_e,hAPP(c_Fun_Ocomp(T_d,T_e,T_c,V_f_H_2),V_f_2),V_A_2,V_A_H_H_2)))) # label(fact_bij__betw__comp__iff) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 365 (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]. 5.84/5.89 366 (all V_m all V_n hAPP(hAPP(c_Code__Numeral_Osubtract__code__numeral,hAPP(c_Code__Numeral_Oof__nat,V_n)),hAPP(c_Code__Numeral_Oof__nat,V_m)) = hAPP(c_Code__Numeral_Oof__nat,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),V_m))) # label(fact_subtract__code__numeral__code) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 367 (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]. 5.84/5.89 368 (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]. 5.84/5.89 369 (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]. 5.84/5.89 370 (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]. 5.84/5.89 371 (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]. 5.84/5.89 372 (all V_Pa_2 all V_n_2 ((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)))) <-> (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)))))) # label(fact_all__nat__less__eq) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 373 (all V_N_2 (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),V_N_2)) <-> (exists B_m all B_x (hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),V_N_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_x),B_m)))))) # label(fact_finite__nat__set__iff__bounded) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 374 (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]. 5.84/5.89 375 (all V_m_2 all V_x_2 (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_m_2 | V_x_2 = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) <-> hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),V_x_2),V_m_2) = 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]. 5.84/5.89 376 (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]. 5.84/5.89 377 (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]. 5.84/5.89 378 (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]. 5.84/5.89 379 (all T_1 all T_2 (class_Finite__Set_Ofinite(T_1) & class_Finite__Set_Ofinite(T_2) -> class_Finite__Set_Ofinite(tc_sum(T_2,T_1)))) # label(arity_sum__Finite__Set_Ofinite) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 380 (all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2) = V_A_2)) # label(fact_Int__absorb2) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 381 (all V_y all V_xs all V_x all T_a hAPP(hAPP(c_List_Olist__update(T_a,hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_y) = hAPP(hAPP(c_List_Olist_OCons(T_a),V_y),V_xs)) # label(fact_list__update__code_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 382 (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_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)) = 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))) # label(fact_rel__comp__UNION__distrib) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 383 (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]. 5.84/5.89 384 (all V_x all T_a (class_Int_Onumber__ring(T_a) & class_Rings_Olinordered__idom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_x)),c_Groups_Ozero__class_Ozero(T_a))) -> hAPP(c_Groups_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))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_x)),c_Groups_Ozero__class_Ozero(T_a))) -> hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_x)) = hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_x)))) # label(fact_abs__number__of) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 385 (all V_z all V_y all V_x all T_a (class_Groups_Oordered__ab__semigroup__add__imp__le(T_a) -> hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_x),V_z)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_y),V_z)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),V_y)),V_z))) # label(fact_max__add__distrib__left) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 386 (all V_m_2 all V_n_2 (V_m_2 = V_n_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]. 5.84/5.89 387 (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]. 5.84/5.89 388 (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_g_2,B_a) = hAPP(V_h_2,hAPP(V_f_2,B_a)))) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_e),V_h_2),V_B_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_e),V_g_2),V_A_2)))))) # label(fact_setsum__reindex__cong) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 389 (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]. 5.84/5.89 390 (all V_w (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w))) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)) = hAPP(c_Product__Type_Ofst(tc_Int_Oint,tc_Int_Oint),c_Divides_OposDivAlg(c_Groups_Oone__class_Oone(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w))))) # label(fact_div__pos__pos__1__number__of) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 391 (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]. 5.84/5.89 392 (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__sup__class_Osup(T_c),V_x_2),c_Big__Operators_Olattice__class_OInf__fin(T_c,V_A_2)) = c_Big__Operators_Olattice__class_OInf__fin(T_c,hAPP(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))))))))) # label(fact_sup__Inf1__distrib) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 393 (all T_c all V_ys_2 all V_xs_2 (V_xs_2 = 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))) -> hAPP(hAPP(c_List_Onth(T_c),V_ys_2),B_i) = hAPP(hAPP(c_List_Onth(T_c),V_xs_2),B_i))) & hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_ys_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2))) # label(fact_list__eq__iff__nth__eq) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 394 (all V_A_2 all V_ys_2 all V_xs_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_c)),V_xs_2),c_List_Olists(T_c,V_A_2))) & hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_c)),V_ys_2),c_List_Olists(T_c,V_A_2))) <-> hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_c)),hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_ys_2)),c_List_Olists(T_c,V_A_2))))) # label(fact_append__in__lists__conv) # label(axiom) # label(non_clause). [assumption]. 5.84/5.89 395 (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]. 5.86/5.89 396 (all V_n all V_m all V_k hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_m)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n))) # label(fact_diff__mult__distrib2) # label(axiom) # label(non_clause). [assumption]. 5.86/5.89 397 (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]. 5.86/5.89 398 (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 V_n_2 = hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m_2),B_k))))) # label(fact_less__iff__Suc__add) # label(axiom) # label(non_clause). [assumption]. 5.86/5.89 399 (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))) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)))),c_Groups_Oone__class_Oone(tc_Int_Oint)) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit1(V_v)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(V_w)))))) # label(fact_zmod__number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 5.86/5.89 400 (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]. 5.86/5.89 401 (all V_xs_2 all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> 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),c_List_Olinorder__class_Oinsort__insert__key(T_c,T_c,c_COMBI(T_c),V_x_2,V_xs_2)))) # label(fact_set__insort__insert) # label(axiom) # label(non_clause). [assumption]. 5.86/5.89 402 (all V_a_2 all V_b_2 all V_c_2 all T_c (class_Rings_Odivision__ring(T_c) -> (V_c_2 != c_Groups_Ozero__class_Ozero(T_c) -> (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]. 5.86/5.89 403 (all V_m all V_n all V_a all T_a (class_Fields_Ofield(T_a) -> (V_a != c_Groups_Ozero__class_Ozero(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n),V_m)) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_m)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)))))) # label(fact_power__diff) # label(axiom) # label(non_clause). [assumption]. 5.86/5.89 404 (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]. 5.86/5.89 405 (all V_k2_2 all V_k1_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OBit0(V_k1_2)),c_Int_OBit1(V_k2_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k1_2),V_k2_2)))) # label(fact_less__eq__int__code_I14_J) # label(axiom) # label(non_clause). [assumption]. 5.86/5.89 406 (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]. 5.86/5.89 407 (all V_n hAPP(c_Int_Onat,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n))) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_nat__zminus__int) # label(axiom) # label(non_clause). [assumption]. 5.86/5.89 408 (all V_x all T_a (class_Rings_Osemiring__1(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x)) -> hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),hAPP(c_Int_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_x))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),hAPP(c_Int_Onat,V_x))),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),hAPP(c_Int_Onat,V_x)))))) # label(fact_of__nat__double) # label(axiom) # label(non_clause). [assumption]. 5.86/5.89 409 (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]. 5.86/5.89 410 (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]. 5.86/5.89 411 (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]. 5.86/5.89 412 (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,V_A_2) = hAPP(c_Finite__Set_Ofold1(T_c,c_Lattices_Osemilattice__inf__class_Oinf(T_c)),V_A_2)))) # label(fact_Inf__fin_OF__eq) # label(axiom) # label(non_clause). [assumption]. 5.86/5.89 413 (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]. 5.86/5.89 414 (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]. 5.86/5.89 415 (all V_v all V_m c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(c_Int_OBit1(c_Int_OPls)))),V_m),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,V_m))),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v))) # label(fact_Suc__mod__eq__add3__mod__number__of) # label(axiom) # label(non_clause). [assumption]. 5.86/5.89 416 (all V_A_2 all T_c V_A_2 = hAPP(c_Relation_ODomain(T_c,T_c),c_Relation_OId__on(T_c,V_A_2))) # label(fact_Domain__Id__on) # label(axiom) # label(non_clause). [assumption]. 5.86/5.89 417 (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]. 5.86/5.89 418 (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]. 5.86/5.89 419 (all V_b_H all V_b all V_a_H all V_c all V_a all T_a (class_Divides_Osemiring__div(T_a) -> (c_Divides_Odiv__class_Omod(T_a,V_a,V_c) = c_Divides_Odiv__class_Omod(T_a,V_a_H,V_c) -> (c_Divides_Odiv__class_Omod(T_a,V_b_H,V_c) = c_Divides_Odiv__class_Omod(T_a,V_b,V_c) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_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]. 5.86/5.89 420 (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]. 5.86/5.89 421 (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]. 5.86/5.89 422 (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]. 5.86/5.89 423 (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]. 5.86/5.89 424 (all V_B_2 all V_A_2 all T_c (class_Lattices_Olattice(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_c),c_Big__Operators_Olattice__class_OInf__fin(T_c,V_A_2)),c_Big__Operators_Olattice__class_OInf__fin(T_c,V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_c),c_Big__Operators_Olattice__class_OInf__fin(T_c,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))),c_Big__Operators_Olattice__class_OInf__fin(T_c,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)))))))) # label(fact_Inf__fin_Ounion__inter) # label(axiom) # label(non_clause). [assumption]. 5.86/5.89 425 (all V_b all V_a all T_a all T_b V_b = hAPP(c_Product__Type_Osnd(T_b,T_a),hAPP(hAPP(c_Product__Type_OPair(T_b,T_a),V_a),V_b))) # label(fact_snd__conv) # label(axiom) # label(non_clause). [assumption]. 5.86/5.89 426 (all V_B_2 all V_A_2 all V_I_2 all T_d all T_c hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),V_I_2),hAPP(hAPP(c_COMBS(T_c,tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool)),T_c),c_Groups_Ominus__class_Ominus(tc_fun(T_d,tc_HOL_Obool))),V_A_2)),V_B_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),V_I_2),V_A_2)),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),V_I_2),V_B_2))) # label(fact_Sigma__Diff__distrib2) # label(axiom) # label(non_clause). [assumption]. 5.86/5.89 427 (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]. 5.86/5.89 428 (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]. 5.86/5.89 429 (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]. 5.86/5.89 430 (all V_b_H_2 all V_a_H_2 all V_b_2 all V_a_2 all T_d all T_c (c_Nitpick_Opair__box_OPairBox(T_c,T_d,V_a_2,V_b_2) = c_Nitpick_Opair__box_OPairBox(T_c,T_d,V_a_H_2,V_b_H_2) <-> V_b_H_2 = V_b_2 & V_a_2 = V_a_H_2)) # label(fact_pair__box_Oinject) # label(axiom) # label(non_clause). [assumption]. 5.86/5.89 431 (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]. 5.86/5.89 432 (all V_r_2 all V_q_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Nitpick_Onum(T_c),c_Nitpick_Oplus__frac(T_c,V_q_2,c_Nitpick_Ouminus__frac(T_c,V_r_2)))),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) <-> c_Nitpick_Oless__eq__frac(T_c,V_q_2,V_r_2))) # label(fact_Nitpick_Oless__eq__frac__def) # label(axiom) # label(non_clause). [assumption]. 5.86/5.89 433 (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]. 5.86/5.89 434 (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)),V_A_2) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) # label(fact_Compl__disjoint2) # label(axiom) # label(non_clause). [assumption]. 5.86/5.89 435 (all V_Pa_2 all V_a1_2 all V_a0_2 (hBOOL(hAPP(c_Wellfounded_Oaccp(tc_prod(tc_Int_Oint,tc_Int_Oint),c_Divides_OnegDivAlg__rel),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_a0_2),V_a1_2))) -> ((all B_a all B_b (hBOOL(hAPP(c_Wellfounded_Oaccp(tc_prod(tc_Int_Oint,tc_Int_Oint),c_Divides_OnegDivAlg__rel),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),B_a),B_b))) -> ((-(hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B_b),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B_a),B_b)))) -> hBOOL(hAPP(hAPP(V_Pa_2,B_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),B_b)))) -> hBOOL(hAPP(hAPP(V_Pa_2,B_a),B_b))))) -> hBOOL(hAPP(hAPP(V_Pa_2,V_a0_2),V_a1_2))))) # label(fact_negDivAlg_Opinduct) # label(axiom) # label(non_clause). [assumption]. 5.86/5.89 436 (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]. 5.86/5.89 437 (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]. 5.86/5.89 438 (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_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]. 5.86/5.89 439 (all T_c (class_Orderings_Oord(T_c) -> c_Orderings_Oord__class_Omin(T_c) = c_Orderings_Oord_Omin(T_c,c_Orderings_Oord__class_Oless__eq(T_c)))) # label(fact_min__ord__min) # label(axiom) # label(non_clause). [assumption]. 5.86/5.89 440 (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]. 5.86/5.89 441 (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]. 5.86/5.89 442 (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]. 5.86/5.89 443 (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]. 5.86/5.89 444 (all V_c all V_b all V_a hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_a),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_b),V_c))),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_a),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_b,V_c))),V_c)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_a),V_b)),V_c)) # label(fact_zdiv__zmult1__eq) # label(axiom) # label(non_clause). [assumption]. 5.86/5.89 445 (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]. 5.86/5.89 446 (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]. 5.86/5.89 447 (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]. 5.86/5.89 448 (all V_Pa_2 all V_f_2 all T_d all T_c (c_Fun_Obij__betw(T_c,T_d,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool))) -> hAPP(c_Set_OCollect(T_d),hAPP(hAPP(c_COMBB(T_c,tc_HOL_Obool,T_d),V_Pa_2),c_Hilbert__Choice_Oinv__into(T_c,T_d,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)),V_f_2))) = hAPP(c_Set_Oimage(T_c,T_d,V_f_2),hAPP(c_Set_OCollect(T_c),V_Pa_2)))) # label(fact_bij__image__Collect__eq) # label(axiom) # label(non_clause). [assumption]. 5.86/5.89 449 (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_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_inf__sup__aci_I7_J) # label(axiom) # label(non_clause). [assumption]. 5.86/5.89 450 (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]. 5.86/5.89 451 (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]. 5.86/5.89 452 (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),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)))) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),V_A_2)))))) # label(fact_finite__compl) # label(axiom) # label(non_clause). [assumption]. 5.86/5.89 453 (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)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_M_2 -> (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]. 5.86/5.89 454 (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]. 5.86/5.89 455 (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]. 5.86/5.89 456 (all V_r_2 all V_A_2 all T_c 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)))) = hAPP(hAPP(c_Equiv__Relations_Oquotient(T_c),V_A_2),V_r_2)) # label(fact_quotient__def) # label(axiom) # label(non_clause). [assumption]. 5.86/5.89 457 (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]. 5.86/5.89 458 (all V_ys_2 all V_xs_2 all T_c (hAPP(c_List_Orev(T_c),V_xs_2) = hAPP(c_List_Orev(T_c),V_ys_2) <-> V_xs_2 = V_ys_2)) # label(fact_rev__is__rev__conv) # label(axiom) # label(non_clause). [assumption]. 5.86/5.89 459 (all V_n_2 all V_k_2 all V_m_2 all V_f_2 all T_c (class_Groups_Ocomm__monoid__add(T_c) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_c),V_f_2),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m_2),V_k_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n_2),V_k_2))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_c),hAPP(hAPP(c_COMBB(tc_Nat_Onat,T_c,tc_Nat_Onat),V_f_2),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),V_k_2))),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2),V_n_2)))) # label(fact_setsum__shift__bounds__nat__ivl) # label(axiom) # label(non_clause). [assumption]. 5.86/5.89 460 (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]. 5.86/5.89 461 (all V_f_2 all V_A_2 all T_d all V_a_2 all T_c (class_Orderings_Olinorder(T_c) & class_Complete__Lattice_Ocomplete__lattice(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),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]. 5.86/5.89 462 (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]. 5.86/5.89 463 (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]. 5.86/5.89 464 (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]. 5.86/5.89 465 (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_Orderings_Oord__class_Omax(T_c),c_Big__Operators_Olinorder__class_OMax(T_c,V_A_2)),c_Big__Operators_Olinorder__class_OMax(T_c,V_B_2)) = c_Big__Operators_Olinorder__class_OMax(T_c,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)))))))) # label(fact_Max__Un) # label(axiom) # label(non_clause). [assumption]. 5.86/5.89 466 (all V_b_2 all V_a_2 hAPP(c_Product__Type_Oprod_Oprod__case(tc_Int_Oint,tc_Int_Oint,tc_HOL_Obool),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_HOL_Obool),tc_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Int_Oint,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_Int_Oint,tc_HOL_Obool),tc_fun(tc_Int_Oint,tc_HOL_Obool)),tc_Int_Oint),c_COMBS(tc_Int_Oint,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Int_Oint,tc_HOL_Obool),tc_fun(tc_Int_Oint,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_Int_Oint),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Int_Oint),c_fconj)),hAPP(hAPP(c_COMBB(tc_fun(tc_Int_Oint,tc_Int_Oint),tc_fun(tc_Int_Oint,tc_HOL_Obool),tc_Int_Oint),hAPP(c_COMBB(tc_Int_Oint,tc_HOL_Obool,tc_Int_Oint),hAPP(c_fequal,V_a_2))),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint),tc_Int_Oint),c_Groups_Oplus__class_Oplus(tc_Int_Oint)),hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b_2)))))),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Int_Oint),c_fconj),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Int_Oint),hAPP(c_fimplies,hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b_2))),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Int_Oint),c_fconj),hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Int_Oint,tc_HOL_Obool),c_Orderings_Oord__class_Oless(tc_Int_Oint)),V_b_2))))),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Int_Oint),hAPP(c_fimplies,hAPP(c_fNot,hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b_2)))),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Int_Oint),c_fconj),hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b_2))),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Int_Oint,tc_HOL_Obool),c_Orderings_Oord__class_Oless__eq(tc_Int_Oint)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))))))) = c_Divides_Odivmod__int__rel(V_a_2,V_b_2)) # label(fact_divmod__int__rel__def) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 467 (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]. 5.86/5.90 468 (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]. 5.86/5.90 469 (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]. 5.86/5.90 470 (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]. 5.86/5.90 471 (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]. 5.86/5.90 472 (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]. 5.86/5.90 473 (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),hAPP(hAPP(c_Set_Oinsert(T_d),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool)))))) <-> 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)))) # label(fact_Image__singleton__iff) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 474 (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]. 5.86/5.90 475 (all V_ys_2 all V_y_2 all V_xs_2 all T_d all T_c hAPP(hAPP(c_List_Ozip(T_c,T_d),V_xs_2),hAPP(hAPP(c_List_Olist_OCons(T_d),V_y_2),V_ys_2)) = hAPP(c_List_Olist_Olist__case(tc_List_Olist(tc_prod(T_c,T_d)),T_c,c_List_Olist_ONil(tc_prod(T_c,T_d)),hAPP(hAPP(c_COMBC(T_c,tc_fun(tc_List_Olist(T_c),tc_List_Olist(tc_prod(T_c,T_d))),tc_fun(tc_List_Olist(T_c),tc_List_Olist(tc_prod(T_c,T_d)))),hAPP(hAPP(c_COMBB(tc_fun(tc_List_Olist(tc_prod(T_c,T_d)),tc_List_Olist(tc_prod(T_c,T_d))),tc_fun(tc_fun(tc_List_Olist(T_c),tc_List_Olist(tc_prod(T_c,T_d))),tc_fun(tc_List_Olist(T_c),tc_List_Olist(tc_prod(T_c,T_d)))),T_c),c_COMBB(tc_List_Olist(tc_prod(T_c,T_d)),tc_List_Olist(tc_prod(T_c,T_d)),tc_List_Olist(T_c))),hAPP(hAPP(c_COMBB(tc_prod(T_c,T_d),tc_fun(tc_List_Olist(tc_prod(T_c,T_d)),tc_List_Olist(tc_prod(T_c,T_d))),T_c),c_List_Olist_OCons(tc_prod(T_c,T_d))),hAPP(hAPP(c_COMBC(T_c,T_d,tc_prod(T_c,T_d)),c_Product__Type_OPair(T_c,T_d)),V_y_2)))),hAPP(hAPP(c_COMBC(tc_List_Olist(T_c),tc_List_Olist(T_d),tc_List_Olist(tc_prod(T_c,T_d))),c_List_Ozip(T_c,T_d)),V_ys_2))),V_xs_2)) # label(fact_zip__Cons) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 476 (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]. 5.86/5.90 477 (all V_g_2 all V_f_2 all V_K_2 all T_c all T_d (class_Groups_Oordered__ab__semigroup__add(T_d) & class_Groups_Ocomm__monoid__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]. 5.86/5.90 478 (all T_a (class_Int_Onumber__ring(T_a) -> c_Groups_Oone__class_Oone(T_a) = hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OBit1(c_Int_OPls)))) # label(fact_numeral__1__eq__1) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 479 (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]. 5.86/5.90 480 (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]. 5.86/5.90 481 (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]. 5.86/5.90 482 (all V_b_2 all V_a_2 all T_c (class_Groups_Oab__group__add(T_c) -> (c_Groups_Ozero__class_Ozero(T_c) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_c),V_a_2),V_b_2) <-> V_a_2 = V_b_2))) # label(fact_eq__iff__diff__eq__0) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 483 (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]. 5.86/5.90 484 (all V_i all V_d ((-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_d),V_i)) -> c_Lazy__Sequence_Oappend(tc_Int_Oint,hAPP(c_Lazy__Sequence_Osingle(tc_Int_Oint),V_i),c_Lazy__Sequence_Osmall__lazy_H(V_d,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_i),c_Groups_Oone__class_Oone(tc_Int_Oint)))) = c_Lazy__Sequence_Osmall__lazy_H(V_d,V_i)) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_d),V_i)) -> c_Lazy__Sequence_Oempty(tc_Int_Oint) = c_Lazy__Sequence_Osmall__lazy_H(V_d,V_i)))) # label(fact_small__lazy_H_Osimps) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 485 (all V_S_2 ((all B_x (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_x),V_S_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B_x)))) <-> c_Nat__Transfer_Onat__set(V_S_2))) # label(fact_nat__set__def) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 486 (all V_n all V_m (V_m = V_n -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m),V_n)))) # label(fact_eq__imp__le) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 487 (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]. 5.86/5.90 488 (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]. 5.86/5.90 489 (all V_B_2 all V_A_2 all T_d all T_c all V_C_2 ((V_C_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> hAPP(hAPP(c_Lattices_Osemilattice__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))) & (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)) = c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool))))) # label(fact_INT__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 490 (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]. 5.86/5.90 491 (all V_f_2 all T_d all T_c (class_Groups_Ocomm__monoid__mult(T_c) -> c_Groups_Oone__class_Oone(T_c) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_d,T_c),V_f_2),c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool))))) # label(fact_setprod__empty) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 492 (all V_xs all T_a all V_y all V_x ((V_x != V_y -> hAPP(c_List_OremoveAll(T_a,V_x),hAPP(hAPP(c_List_Olist_OCons(T_a),V_y),V_xs)) = hAPP(hAPP(c_List_Olist_OCons(T_a),V_y),hAPP(c_List_OremoveAll(T_a,V_x),V_xs))) & (V_y = V_x -> hAPP(c_List_OremoveAll(T_a,V_x),V_xs) = hAPP(c_List_OremoveAll(T_a,V_x),hAPP(hAPP(c_List_Olist_OCons(T_a),V_y),V_xs))))) # label(fact_removeAll_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 493 (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]. 5.86/5.90 494 (all V_A_2 all T_c ((hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,T_c,c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),hAPP(c_COMBK(tc_Nat_Onat,T_c),c_Groups_Oone__class_Oone(tc_Nat_Onat))),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_A_2) = hAPP(c_Finite__Set_Ocard(T_c),V_A_2)) & (-hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(c_Finite__Set_Ocard(T_c),V_A_2)))) # label(fact_card__def) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 495 (all V_b all V_a all T_a (class_Groups_Ogroup__add(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_b)),V_b) = V_a)) # label(fact_diff__add__cancel) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 496 (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]. 5.86/5.90 497 (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]. 5.86/5.90 498 (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_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))) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_b)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_a)))) # label(fact_pos__zmod__mult__2) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 499 (all V_x all V_n all T_a hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),c_List_Oreplicate(T_a,V_n,V_x)) = V_n) # label(fact_length__replicate) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 500 (all T_c hAPP(c_Set_OPow(T_c),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) = hAPP(hAPP(c_Set_Oinsert(tc_fun(T_c,tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))),c_Orderings_Obot__class_Obot(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool)))) # label(fact_Pow__empty) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 501 (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]. 5.86/5.90 502 (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]. 5.86/5.90 503 (all V_B_2 all V_A_2 all T_e all V_f_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_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_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]. 5.86/5.90 504 (all V_B_2 all V_C_2 all V_a_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_C_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_B_2)),V_C_2) = hAPP(hAPP(c_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__if1) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 505 (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__equivp__refl__symp__transp) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 506 (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]. 5.86/5.90 507 (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]. 5.86/5.90 508 (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]. 5.86/5.90 509 (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]. 5.86/5.90 510 (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]. 5.86/5.90 511 (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]. 5.86/5.90 512 (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(V_R_2,B_x),B_xa)) & hBOOL(hAPP(hAPP(c_member(T_c),B_xa),V_Y_2)))))) -> 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]. 5.86/5.90 513 (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]. 5.86/5.90 514 (all V_ys_2 all V_Pa_2 all V_xs_2 all T_d all T_c hAPP(hAPP(c_List_Ozip(T_c,T_d),V_xs_2),c_List_OtakeWhile(T_d,V_Pa_2,V_ys_2)) = c_List_OtakeWhile(tc_prod(T_c,T_d),hAPP(c_Fun_Ocomp(T_d,tc_HOL_Obool,tc_prod(T_c,T_d),V_Pa_2),c_Product__Type_Osnd(T_c,T_d)),hAPP(hAPP(c_List_Ozip(T_c,T_d),V_xs_2),V_ys_2))) # label(fact_zip__takeWhile__snd) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 515 (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]. 5.86/5.90 516 (all V_x all V_k all T_a all V_xs (V_xs != c_List_Olist_ONil(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_k -> c_List_Olast(T_a,hAPP(hAPP(c_List_Olist__update(T_a,V_xs),V_k),V_x)) = c_List_Olast(T_a,V_xs)) & (hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = V_k -> c_List_Olast(T_a,hAPP(hAPP(c_List_Olist__update(T_a,V_xs),V_k),V_x)) = V_x))) # label(fact_last__list__update) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 517 (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]. 5.86/5.90 518 (all V_m hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = V_m) # label(fact_minus__nat_Odiff__0) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 519 (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]. 5.86/5.90 520 (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]. 5.86/5.90 521 (all V_xs_2 all V_f_2 all T_d all T_c hAPP(c_Set_Oimage(T_d,T_c,V_f_2),hAPP(c_List_Oset(T_d),V_xs_2)) = hAPP(c_List_Oset(T_c),hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_xs_2))) # label(fact_set__map) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 522 (all V_v all V_u all V_y all V_a all V_x all T_a (class_Rings_Olinordered__semiring__1__strict(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_u)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_v)) -> (c_Groups_Oone__class_Oone(T_a) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_u),V_v) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_u),V_x)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_v),V_y))),V_a))))))))) # label(fact_convex__bound__lt) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 523 (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]. 5.86/5.90 524 (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]. 5.86/5.90 525 (all V_b_2 all V_a_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_b_2)),c_Relation_OId(T_c))) <-> V_b_2 = V_a_2)) # label(fact_pair__in__Id__conv) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 526 (all V_a_2 all T_c (class_Groups_Oordered__ab__group__add(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_a_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_a_2),c_Groups_Ozero__class_Ozero(T_c)))))) # label(fact_neg__0__less__iff__less) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 527 (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_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n)) = 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))) # label(fact_add__mult__distrib2) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 528 (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]. 5.86/5.90 529 (all V_n all V_m hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),hAPP(c_Nat_OSuc,V_n)) = hAPP(c_Nat_OSuc,hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),V_m),V_n))) # label(fact_max__Suc__Suc) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 530 (all V_x_2 all T_c V_x_2 = 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))))) # label(fact_the__elem__eq) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 531 (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]. 5.86/5.90 532 (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]. 5.86/5.90 533 (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]. 5.86/5.90 534 (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]. 5.86/5.90 535 (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]. 5.86/5.90 536 (all V_xs_2 all T_c 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)) = c_Random_Oselect__weight(T_c,V_xs_2)) # label(fact_select__weigth__drop__zero) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 537 (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]. 5.86/5.90 538 (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]. 5.86/5.90 539 (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]. 5.86/5.90 540 (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]. 5.86/5.90 541 (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]. 5.86/5.90 542 (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]. 5.86/5.90 543 (all V_v hAPP(c_Int_Onat,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)) # label(fact_nat__number__of__def) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 544 (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]. 5.86/5.90 545 (all V_vs_2 all V_us_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_us_2),V_vs_2)),c_List_Olistrel1(T_c,V_r_2))) & V_xs_2 = V_ys_2 | hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_xs_2),V_ys_2)),c_List_Olistrel1(T_c,V_r_2))) & V_us_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)),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]. 5.86/5.90 546 (all V_B_2 all V_A_2 hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),V_A_2),V_B_2) = hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),V_A_2)),hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),V_B_2)))) # label(fact_Nat__Transfer_Otransfer__nat__int__set__functions_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 547 (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]. 5.86/5.90 548 (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]. 5.86/5.90 549 (all V_z all V_y all V_x hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_y),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_x),V_z)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_x),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_y),V_z))) # label(fact_zadd__left__commute) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 550 (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_OBit1(c_Int_OBit1(c_Int_OPls)))),V_n) = hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,V_n)))) # label(fact_Suc3__eq__add__3) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 551 (all V_x all T_a (class_Lattices_Oboolean__algebra(T_a) -> c_Orderings_Otop__class_Otop(T_a) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_x)))) # label(fact_sup__compl__top) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 552 (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]. 5.86/5.90 553 (all V_r_2 all V_A_2 all T_c (c_Relation_Ototal__on(T_c,V_A_2,V_r_2) <-> (all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)) -> (all B_xa (hBOOL(hAPP(hAPP(c_member(T_c),B_xa),V_A_2)) -> (B_xa != B_x -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),B_x),B_xa)),V_r_2)) | hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),B_xa),B_x)),V_r_2))))))))) # label(fact_total__on__def) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 554 (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]. 5.86/5.90 555 (all V_r_2 all V_b_2 all V_a_2 all T_d all T_c hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),hAPP(c_Relation_ODomain(T_c,T_d),V_r_2)) = hAPP(c_Relation_ODomain(T_c,T_d),hAPP(hAPP(c_Set_Oinsert(tc_prod(T_c,T_d)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_a_2),V_b_2)),V_r_2))) # label(fact_Domain__insert) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 556 (all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,V_A_2) -> hAPP(c_Finite__Set_Ocard(T_d),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2)) = hAPP(c_Finite__Set_Ocard(T_c),V_A_2))) # label(fact_card__image) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 557 (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]. 5.86/5.90 558 (all V_n all V_a all T_a (class_Rings_Olinordered__idom(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)),V_n) = hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)))) # label(fact_power__abs) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 559 (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]. 5.86/5.90 560 (all V_k all T_a (class_Int_Onumber__ring(T_a) -> hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_Osucc(V_k)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Groups_Oone__class_Oone(T_a)),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_k)))) # label(fact_number__of__succ) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 561 (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]. 5.86/5.90 562 (all V_a_2 all V_f_2 all T_c all V_g_2 (c_Finite__Set_Ofold1(T_c,V_f_2) = V_g_2 -> V_a_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)))))) # label(fact_fold1__singleton__def) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 563 (all V_A_2 all V_f_2 all T_d all T_c (class_Groups_Oordered__ab__group__add__abs(T_c) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_c),hAPP(hAPP(c_COMBB(T_c,T_c,T_d),c_Groups_Oabs__class_Oabs(T_c)),V_f_2)),V_A_2) = hAPP(c_Groups_Oabs__class_Oabs(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_c),hAPP(hAPP(c_COMBB(T_c,T_c,T_d),c_Groups_Oabs__class_Oabs(T_c)),V_f_2)),V_A_2)))) # label(fact_abs__setsum__abs) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 564 (all V_w_2 (hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(V_w_2)))) <-> hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w_2))))) # label(fact_neg__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 565 (all V_y all V_i all T_a c_List_Olist_ONil(T_a) = hAPP(hAPP(c_List_Olist__update(T_a,c_List_Olist_ONil(T_a)),V_i),V_y)) # label(fact_list__update__code_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 566 (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]. 5.86/5.90 567 (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]. 5.86/5.90 568 (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)),V_xs_2) -> hBOOL(hAPP(c_List_Odistinct(T_c),V_xs_2)))) # label(fact_card__distinct) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 569 (all V_i_2 all V_xs_2 all V_A_2 all T_c hAPP(hAPP(c_List_Omap(tc_prod(T_c,tc_Nat_Onat),T_c),c_Product__Type_Ofst(T_c,tc_Nat_Onat)),hAPP(c_List_Ofilter(tc_prod(T_c,tc_Nat_Onat),hAPP(hAPP(c_COMBC(tc_prod(T_c,tc_Nat_Onat),tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool),tc_prod(T_c,tc_Nat_Onat)),c_member(tc_Nat_Onat)),hAPP(hAPP(c_COMBC(tc_prod(T_c,tc_Nat_Onat),tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat),tc_prod(T_c,tc_Nat_Onat)),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),c_Product__Type_Osnd(T_c,tc_Nat_Onat))),V_i_2))),V_A_2)),hAPP(hAPP(c_List_Ozip(T_c,tc_Nat_Onat),V_xs_2),c_List_Oupt(c_Groups_Ozero__class_Ozero(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2))))) = hAPP(hAPP(c_List_Omap(tc_prod(T_c,tc_Nat_Onat),T_c),c_Product__Type_Ofst(T_c,tc_Nat_Onat)),hAPP(c_List_Ofilter(tc_prod(T_c,tc_Nat_Onat),hAPP(hAPP(c_COMBC(tc_prod(T_c,tc_Nat_Onat),tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool),tc_prod(T_c,tc_Nat_Onat)),c_member(tc_Nat_Onat)),c_Product__Type_Osnd(T_c,tc_Nat_Onat))),V_A_2)),hAPP(hAPP(c_List_Ozip(T_c,tc_Nat_Onat),V_xs_2),c_List_Oupt(V_i_2,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i_2),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2))))))) # label(fact_sublist__shift__lemma) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 570 (all V_q all V_p all V_x all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_p)),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(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_p),V_q)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I26_J) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 571 (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_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)) = 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)))) # label(fact_UN__insert__distrib) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 572 (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]. 5.86/5.90 573 (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]. 5.86/5.90 574 (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]. 5.86/5.90 575 (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]. 5.86/5.90 576 (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]. 5.86/5.90 577 (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]. 5.86/5.90 578 (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]. 5.86/5.90 579 (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]. 5.86/5.90 580 (all V_ys_2 all V_y_2 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)),hAPP(hAPP(c_List_Olist_OCons(T_c),V_y_2),V_ys_2)) & -hBOOL(hAPP(V_Pa_2,V_y_2)) <-> c_List_OdropWhile(T_c,V_Pa_2,V_xs_2) = hAPP(hAPP(c_List_Olist_OCons(T_c),V_y_2),V_ys_2))) # label(fact_dropWhile__eq__Cons__conv) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 581 (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]. 5.86/5.90 582 (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]. 5.86/5.90 583 (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]. 5.86/5.90 584 (all V_f_2 all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> ((exists B_x ((all B_xa (hBOOL(hAPP(hAPP(c_member(T_c),B_xa),V_A_2)) -> (B_x != B_xa -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(V_f_2,B_xa)))) & c_Groups_Oone__class_Oone(tc_Nat_Onat) = hAPP(V_f_2,B_x) & hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)))) <-> 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]. 5.86/5.90 585 (all V_fs_2 all V_f_2 all V_y_2 all V_x_2 all T_c (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) <-> 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_in__measures_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 586 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_y),V_x)))) # label(fact_linorder__le__less__linear) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 587 (all V_b_2 all V_a_2 (c_SetInterval_Oord_OatLeastLessThan(tc_Nat_Onat,c_Rings_Odvd__class_Odvd(tc_Nat_Onat),hAPP(hAPP(c_COMBS(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),tc_Nat_Onat),c_COMBS(tc_Nat_Onat,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_Nat_Onat),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Nat_Onat),c_fconj)),c_Rings_Odvd__class_Odvd(tc_Nat_Onat)))),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_Nat_Onat),hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Nat_Onat),c_fNot)),hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_HOL_Obool),c_Rings_Odvd__class_Odvd(tc_Nat_Onat)))),V_a_2,V_b_2) = c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) <-> -(-hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_b_2),V_a_2)) & hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_a_2),V_b_2))))) # label(fact_dvd_OatLeastLessThan__empty__iff2) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 588 (all V_a2_2 all V_r_2 all T_d all T_c (hBOOL(hAPP(c_Predicate_ORangeP(T_c,T_d,V_r_2),V_a2_2)) <-> (exists B_a hBOOL(hAPP(hAPP(V_r_2,B_a),V_a2_2))))) # label(fact_RangeP_Oequations) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 589 (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)))) <-> hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_neg__number__of__pred__iff__0) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 590 (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]. 5.86/5.90 591 (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]. 5.86/5.90 592 (all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> c_Groups_Ozero__class_Ozero(T_a) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_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]. 5.86/5.90 593 (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]. 5.86/5.90 594 (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]. 5.86/5.90 595 (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]. 5.86/5.90 596 (all V_B_2 all V_A_2 all T_c (hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2) = V_A_2)) # label(fact_Diff__triv) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 597 (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]. 5.86/5.90 598 (all V_c all V_b all V_a all T_a (class_Orderings_Oord(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_b)) -> (V_b = V_c -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_c)))))) # label(fact_ord__le__eq__trans) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 599 (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]. 5.86/5.90 600 (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]. 5.86/5.90 601 (all V_l_2 all V_k_2 (V_k_2 = V_l_2 <-> c_Int_OBit0(V_k_2) = c_Int_OBit0(V_l_2))) # label(fact_rel__simps_I48_J) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 602 (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]. 5.86/5.90 603 (all V_z_2 all V_y_2 all V_x_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofun__left__comm(T_c,T_d,V_f_2) -> hAPP(hAPP(V_f_2,V_y_2),hAPP(hAPP(V_f_2,V_x_2),V_z_2)) = hAPP(hAPP(V_f_2,V_x_2),hAPP(hAPP(V_f_2,V_y_2),V_z_2)))) # label(fact_fun__left__comm_Ofun__left__comm) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 604 (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]. 5.86/5.90 605 (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]. 5.86/5.90 606 (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]. 5.86/5.90 607 (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]. 5.86/5.90 608 (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]. 5.86/5.90 609 (all V_Pa_2 all T_c c_Enum_Oenum__the(T_c,V_Pa_2) = c_HOL_OThe(T_c,V_Pa_2)) # label(fact_enum__the__def) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 610 (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]. 5.86/5.90 611 (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]. 5.86/5.90 612 (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]. 5.86/5.90 613 (all V_k_2 hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),V_k_2),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),V_k_2)) = hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_k_2))) # label(fact_lessThan__Suc) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 614 (all T_a c_List_Oremdups(T_a,c_List_Olist_ONil(T_a)) = c_List_Olist_ONil(T_a)) # label(fact_remdups_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 615 (all V_p all V_m ((c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_m -> 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]. 5.86/5.90 616 (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]. 5.86/5.90 617 (all V_A_2 all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> c_Complete__Lattice_OInf__class_OInf(T_c,V_A_2) = hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_c,c_Lattices_Osemilattice__inf__class_Oinf(T_c)),c_Orderings_Otop__class_Otop(T_c)),V_A_2)))) # label(fact_Inf__fold__inf) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 618 (all V_a all V_b all V_c all T_a (class_Rings_Odivision__ring(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) != V_c -> (hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c) = V_b -> V_a = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_b),V_c))))) # label(fact_divide__eq__imp) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 619 (all V_a_2 all T_c (class_Groups_Olinordered__ab__group__add(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_double__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 620 (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]. 5.86/5.90 621 (all V_y_2 all V_x_2 all V_c_2 all T_c all T_e all T_d hAPP(hAPP(c_Product__Type_Ocurry(T_d,T_e,T_c,V_c_2),V_x_2),V_y_2) = hAPP(V_c_2,hAPP(hAPP(c_Product__Type_OPair(T_d,T_e),V_x_2),V_y_2))) # label(fact_curry__def) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 622 (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]. 5.86/5.90 623 (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_Groups_Ozero__class_Ozero(T_d) = c_Nitpick_Osetsum_H(T_c,T_d,V_f_2,V_A_2)))) # label(fact_Nitpick_Osetsum_H__def) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 624 (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]. 5.86/5.90 625 (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]. 5.86/5.90 626 (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]. 5.86/5.90 627 (all V_i all V_b ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),V_b),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),V_i),V_b)) -> c_Random_Olog(V_b,V_i) = c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)) & (-(hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),V_i),V_b)) | 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)))))) # label(fact_log_Osimps) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 628 (all V_f_2 all V_a_2 all V_A_2 all T_c all T_d (class_Rings_Oring(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_d),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_f_2),V_A_2)),hAPP(V_f_2,V_a_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_f_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))))))) # label(fact_setsum__diff1__ring) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 629 (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]. 5.86/5.90 630 (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]. 5.86/5.90 631 (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(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__idem_Oinsert__idem) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 632 (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)) & hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),V_A_2)))) # label(fact_Diff__iff) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 633 (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]. 5.86/5.90 634 (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(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_A_2)) # label(fact_Int__commute) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 635 (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]. 5.86/5.90 636 (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]. 5.86/5.90 637 (all V_b_2 all V_a_2 all V_f_2 all T_c (V_a_2 = V_b_2 <-> hBOOL(hAPP(c_Finite__Set_Ofold1Set(T_c,V_f_2,hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))),V_b_2)))) # label(fact_fold1Set__sing) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 638 (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]. 5.86/5.90 639 (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]. 5.86/5.90 640 (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]. 5.86/5.90 641 (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]. 5.86/5.90 642 (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(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) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_g_2),V_A_2)) & (-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_OF__eq) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 643 (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]. 5.86/5.90 644 (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_Finite__Set_Ocard(T_c),hAPP(c_List_Oset(T_c),V_xs_2))) # label(fact_length__remdups__card__conv) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 645 (all V_x_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> V_x_2 = 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)))) -> hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),V_x_2),c_Big__Operators_Olinorder__class_OMin(T_c,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))))) = c_Big__Operators_Olinorder__class_OMin(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)))))) # label(fact_Min_Oinsert__remove) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 646 (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]. 5.86/5.90 647 (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]. 5.86/5.90 648 (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]. 5.86/5.90 649 (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(T_c),V_a_2),V_b_2)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_b_2),V_d_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_c_2),V_a_2)) <-> 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)))))) # label(fact_atLeastatMost__subset__iff) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 650 (all V_y_2 all V_k_2 all V_xs_2 all V_f_2 all T_c all T_d hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),hAPP(hAPP(c_List_Olist__update(T_d,V_xs_2),V_k_2),V_y_2)) = hAPP(hAPP(c_List_Olist__update(T_c,hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_xs_2)),V_k_2),hAPP(V_f_2,V_y_2))) # label(fact_map__update) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 651 (all V_r all V_q all T_a c_Nitpick_Otimes__frac(T_a,V_q,V_r) = hAPP(hAPP(c_Nitpick_Ofrac(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Nitpick_Onum(T_a),V_q)),hAPP(c_Nitpick_Onum(T_a),V_r))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Nitpick_Odenom(T_a),V_q)),hAPP(c_Nitpick_Odenom(T_a),V_r)))) # label(fact_Nitpick_Otimes__frac__def) # label(axiom) # label(non_clause). [assumption]. 5.86/5.90 652 (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]. 5.86/5.90 653 (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]. 5.86/5.90 654 (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]. 5.86/5.90 655 (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]. 5.86/5.90 656 (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]. 5.86/5.90 657 (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]. 5.86/5.90 658 (all V_z all T_a (class_Int_Onumber__ring(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_z) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_z),V_z))) # label(fact_mult__2) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 659 (all V_ys all T_a all V_xs (c_List_Olist_ONil(T_a) != V_xs -> hAPP(c_List_Ohd(T_a),V_xs) = hAPP(c_List_Ohd(T_a),hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys)))) # label(fact_hd__append2) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 660 (all V_j_2 all V_i_2 hAPP(c_List_Oset(tc_Int_Oint),c_List_Oupto(V_i_2,V_j_2)) = c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,V_i_2,V_j_2)) # label(fact_set__upto) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 661 (all V_y all V_x all T_a (class_Lattices_Oboolean__algebra(T_a) -> hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y)) = 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)))) # label(fact_compl__inf) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 662 (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]. 5.86/5.91 663 (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]. 5.86/5.91 664 (all V_c all V_b all V_a all T_a (class_Divides_Oring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_b),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),c_Divides_Odiv__class_Omod(T_a,V_a,V_c)),V_b),V_c))) # label(fact_mod__diff__left__eq) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 665 (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]. 5.86/5.91 666 (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]. 5.86/5.91 667 (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]. 5.86/5.91 668 (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]. 5.86/5.91 669 (all V_a_2 all T_c (class_Rings_Oring__1__no__zero__divisors(T_c) -> (c_Groups_Ozero__class_Ozero(T_c) = V_a_2 <-> 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]. 5.86/5.91 670 (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]. 5.86/5.91 671 (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]. 5.86/5.91 672 (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),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(T_c),V_a_2),V_b_2))))) # label(fact_add__less__cancel__right) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 673 (all V_u_2 all V_l_2 hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_u_2),hAPP(c_Nat_OSuc,V_l_2)) = hAPP(c_Finite__Set_Ocard(tc_Nat_Onat),c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Nat_Onat,V_l_2,V_u_2))) # label(fact_card__greaterThanLessThan) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 674 (all T_d all T_c (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)))) <-> 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)))))) # label(fact_finite__Plus__UNIV__iff) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 675 (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]. 5.86/5.91 676 (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]. 5.86/5.91 677 (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]. 5.86/5.91 678 (all V_xs_2 all V_x_2 all T_c ((exists B_i (hAPP(hAPP(c_List_Onth(T_c),V_xs_2),B_i) = V_x_2 & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_i),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2))))) <-> hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2))))) # label(fact_in__set__conv__nth) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 679 (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]. 5.86/5.91 680 (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]. 5.86/5.91 681 (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]. 5.86/5.91 682 (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),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__eq(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_c_2)),V_b_2)))))) # label(fact_pos__le__divide__eq) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 683 (all V_com_H_2 all V_fun_H_2 all V_com_2 all V_fun_2 (c_Com_Ocom_OWhile(V_fun_2,V_com_2) = c_Com_Ocom_OWhile(V_fun_H_2,V_com_H_2) <-> V_com_H_2 = V_com_2 & V_fun_2 = V_fun_H_2)) # label(fact_com_Osimps_I5_J) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 684 (all T_c c_Equiv__Relations_Oequivp(T_c,c_fequal)) # label(fact_identity__equivp) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 685 (all V_u_2 all V_l_2 hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,V_l_2),V_u_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)))) # label(fact_image__add__int__atLeastLessThan) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 686 (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]. 5.86/5.91 687 (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]. 5.86/5.91 688 (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]. 5.86/5.91 689 (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]. 5.86/5.91 690 (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]. 5.86/5.91 691 (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]. 5.86/5.91 692 (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]. 5.86/5.91 693 (all V_A_2 all V_f_2 all T_d all V_z_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_z_2),hAPP(c_Set_Oimage(T_d,T_c,V_f_2),V_A_2))) <-> (exists B_x (V_z_2 = hAPP(V_f_2,B_x) & hBOOL(hAPP(hAPP(c_member(T_d),B_x),V_A_2)))))) # label(fact_image__iff) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 694 (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]. 5.86/5.91 695 (all V_y_2 all V_x_2 all V_r_2 all V_A_2 all T_c (c_Equiv__Relations_Oequiv(T_c,V_A_2,V_r_2) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_y_2),V_A_2)) & hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) & hAPP(c_Relation_OImage(T_c,T_c,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_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(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]. 5.86/5.91 696 (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]. 5.86/5.91 697 (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]. 5.86/5.91 698 (all V_x all V_xs all T_a all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))) -> V_x = hAPP(hAPP(c_List_Onth(T_a),hAPP(hAPP(c_List_Olist__update(T_a,V_xs),V_i),V_x)),V_i))) # label(fact_nth__list__update__eq) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 699 (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_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,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_Olinorder__class_OMin(T_c,V_A_2)))))) # label(fact_Min_Oremove) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 700 (all V_f_2 all V_p_2 all V_n_2 all V_m_2 all T_c (class_Groups_Ocomm__monoid__add(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n_2),V_p_2)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_c),V_f_2),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2),V_n_2))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_c),V_f_2),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_n_2),V_p_2))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_c),V_f_2),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2),V_p_2)))))) # label(fact_setsum__add__nat__ivl) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 701 (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]. 5.86/5.91 702 (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]. 5.86/5.91 703 (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]. 5.86/5.91 704 (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]. 5.86/5.91 705 (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)) -> c_Groups_Oone__class_Oone(T_d) = 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)),V_S_2))))) # label(fact_fold__image__1) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 706 (all V_A_2 all V_l_2 all V_x_2 all T_c c_List_Osublist(T_c,hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_l_2),V_A_2) = hAPP(hAPP(c_List_Oappend(T_c),hAPP(hAPP(hAPP(c_If(tc_List_Olist(T_c)),hAPP(hAPP(c_member(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_A_2)),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),c_List_Olist_ONil(T_c))),c_List_Olist_ONil(T_c))),c_List_Osublist(T_c,V_l_2,hAPP(c_Set_OCollect(tc_Nat_Onat),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool),tc_Nat_Onat),c_member(tc_Nat_Onat)),c_Nat_OSuc)),V_A_2))))) # label(fact_sublist__Cons) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 707 (all V_f_2 all V_A_2 all V_a_2 all T_c ((-hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,tc_Nat_Onat),V_f_2),V_A_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,tc_Nat_Onat),V_f_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))))) & (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,tc_Nat_Onat),V_f_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,tc_Nat_Onat),V_f_2),V_A_2)),hAPP(V_f_2,V_a_2))))) # label(fact_setsum__diff1__nat) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 708 (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]. 5.86/5.91 709 (all V_A_2 all V_l_H_2 all V_l_2 all T_c c_List_Osublist(T_c,hAPP(hAPP(c_List_Oappend(T_c),V_l_2),V_l_H_2),V_A_2) = 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))))) # label(fact_sublist__append) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 710 (all V_x_2 all V_n_2 all T_c (c_List_Oreplicate(T_c,V_n_2,V_x_2) = c_List_Olist_ONil(T_c) <-> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n_2)) # label(fact_empty__replicate) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 711 (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(V_Pa_2,B_x)) & hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2))))))) # label(fact_ex__nat__less) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 712 (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]. 5.86/5.91 713 (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]. 5.86/5.91 714 (all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a))) -> hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)))) # label(fact_abs__of__neg) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 715 (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]. 5.86/5.91 716 (all V_r_2 all T_c ((all B_x1 hBOOL(hAPP(c_Wellfounded_Oaccp(T_c,V_r_2),B_x1))) <-> c_Wellfounded_OwfP(T_c,V_r_2))) # label(fact_wfP__accp__iff) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 717 (all V_B_2 all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> (hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_B_2) = hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2) <-> V_B_2 = V_A_2))) # label(fact_inj__image__eq__iff) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 718 (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),V_xs_2) = hAPP(c_List_Ofoldl(tc_List_Olist(T_c),T_c,hAPP(c_COMBC(T_c,tc_List_Olist(T_c),tc_List_Olist(T_c)),c_List_Olinorder__class_Oinsort__key(T_c,T_c,c_COMBI(T_c))),c_List_Olist_ONil(T_c)),V_xs_2))) # label(fact_sort__foldl__insort) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 719 (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]. 5.86/5.91 720 (all V_r_2 all T_c (c_Wellfounded_Owf(T_c,hAPP(c_Set_OCollect(tc_prod(T_c,T_c)),hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_c,tc_HOL_Obool),V_r_2))) <-> c_Wellfounded_OwfP(T_c,V_r_2))) # label(fact_wfP__def) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 721 (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]. 5.86/5.91 722 (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]. 5.86/5.91 723 (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__number__of__0) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 724 (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]. 5.86/5.91 725 (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]. 5.86/5.91 726 (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]. 5.86/5.91 727 (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]. 5.86/5.91 728 (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]. 5.86/5.91 729 (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]. 5.86/5.91 730 (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]. 5.86/5.91 731 (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]. 5.86/5.91 732 (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]. 5.86/5.91 733 (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]. 5.86/5.91 734 (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]. 5.86/5.91 735 (all V_f_2 all V_A_2 all T_c (class_Groups_Ocomm__monoid__mult(T_c) & class_Groups_Ocomm__monoid__add(T_c) -> (c_Nat__Transfer_Onat__set(V_A_2) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_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)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,T_c),V_f_2),V_A_2)))) # label(fact_transfer__int__nat__sum__prod_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 736 (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(c_Finite__Set_Ocard(T_c),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_c),V_A_2)),hAPP(c_Finite__Set_Ocard(T_c),V_B_2)))))) # label(fact_card__Un__disjoint) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 737 (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(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_k),V_n_2)) & (all B_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B_i),B_k)) -> -hBOOL(hAPP(V_Pa_2,B_i)))) & hBOOL(hAPP(V_Pa_2,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B_k),c_Groups_Oone__class_Oone(tc_Nat_Onat))))))))) # label(fact_ex__least__nat__less) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 738 (all V_x_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) -> V_x_2 = c_Big__Operators_Olattice_OSup__fin(T_c,c_Orderings_Oord__class_Omax(T_c),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2))) & (hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> 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]. 5.86/5.91 739 (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]. 5.86/5.91 740 (all V_B_2 all T_c 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) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) # label(fact_Int__empty__left) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 741 (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]. 5.86/5.91 742 (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]. 5.86/5.91 743 (all T_a c_List_Olist_ONil(T_a) = c_List_Oconcat(T_a,c_List_Olist_ONil(tc_List_Olist(T_a)))) # label(fact_concat_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 744 (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]. 5.86/5.91 745 (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]. 5.86/5.91 746 (all V_s_2 all V_Z_2 all V_p_2 all V_Pa_2 all T_c (hBOOL(hAPP(V_p_2,V_s_2)) & hBOOL(hAPP(hAPP(V_Pa_2,V_Z_2),V_s_2)) <-> hBOOL(hAPP(hAPP(c_Hoare__Mirabelle_Opeek__and(T_c,V_Pa_2,V_p_2),V_Z_2),V_s_2)))) # label(fact_peek__and__def) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 747 (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]. 5.86/5.91 748 (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_Olist__all(T_c,V_Pa_2,V_xs_2))) # label(fact_Ball__set__list__all) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 749 (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]. 5.86/5.91 750 (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]. 5.86/5.91 751 (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]. 5.86/5.91 752 (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]. 5.86/5.91 753 (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]. 5.86/5.91 754 (all V_xs_2 all T_c (c_List_Olist_ONil(tc_List_Olist(T_c)) = c_List_Otranspose(T_c,V_xs_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_xs_2))) -> B_x = c_List_Olist_ONil(T_c))))) # label(fact_transpose__empty) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 755 (all V_As_2 all V_A_2 all T_c c_List_Olistset(T_c,hAPP(hAPP(c_List_Olist_OCons(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_As_2)) = c_List_Oset__Cons(T_c,V_A_2,c_List_Olistset(T_c,V_As_2))) # label(fact_listset_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 756 (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_OgreaterThanAtMost(T_c,V_x_2,V_y_2)) = 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)))) # label(fact_image__uminus__greaterThanAtMost) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 757 (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]. 5.86/5.91 758 (all V_list_2 all V_a_2 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),hAPP(hAPP(c_List_Olist_OCons(T_d),V_a_2),V_list_2)) = hAPP(hAPP(V_f2_2,V_a_2),V_list_2)) # label(fact_list_Osimps_I5_J) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 759 (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]. 5.86/5.91 760 (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]. 5.86/5.91 761 (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) & hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_d,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_A_2))) = c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool)) & c_Fun_Oinj__on(T_c,T_d,V_f_2,V_B_2))) # label(fact_inj__on__Un) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 762 (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]. 5.86/5.91 763 (all V_A_2 all V_f_2 all T_d all T_c (class_Rings_Osemiring__1(T_c) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_c),hAPP(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__add__class_Osetsum(T_d,tc_Nat_Onat),V_f_2),V_A_2)))) # label(fact_of__nat__setsum) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 764 (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]. 5.86/5.91 765 (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_OSup__fin(T_c,V_A_2)) & (hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_c),V_x_2),c_Big__Operators_Olattice__class_OSup__fin(T_c,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))))) = c_Big__Operators_Olattice__class_OSup__fin(T_c,V_A_2)))))) # label(fact_Sup__fin_Oremove) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 766 (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]. 5.86/5.91 767 (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]. 5.86/5.91 768 (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]. 5.86/5.91 769 (all V_j_2 all V_i_2 hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_i_2),V_j_2) = hAPP(c_List_Oset(tc_Nat_Onat),c_List_Oupt(V_i_2,V_j_2))) # label(fact_set__upt) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 770 (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(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) = hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2))) # label(fact_image__eq__fold__image) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 771 (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]. 5.86/5.91 772 (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]. 5.86/5.91 773 (all V_C_2 all V_B_2 all V_A_2 all T_d all T_c hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),V_A_2),hAPP(c_COMBK(tc_fun(T_d,tc_HOL_Obool),T_c),V_C_2))),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),V_B_2),hAPP(c_COMBK(tc_fun(T_d,tc_HOL_Obool),T_c),V_C_2))) = hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)),hAPP(c_COMBK(tc_fun(T_d,tc_HOL_Obool),T_c),V_C_2))) # label(fact_Times__Un__distrib1) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 774 (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]. 5.86/5.91 775 (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]. 5.86/5.91 776 (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]. 5.86/5.91 777 (all V_ys all V_x all V_xs all T_a hAPP(hAPP(c_List_Onth(T_a),hAPP(hAPP(c_List_Oappend(T_a),V_xs),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_ys))),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)) = V_x) # label(fact_nth__append__length) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 778 (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]. 5.86/5.91 779 (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]. 5.86/5.91 780 (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]. 5.86/5.91 781 (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__iff2) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 782 (all V_xs_2 all V_f_2 all T_d all V_Pa_2 all T_c c_List_OtakeWhile(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),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))) # label(fact_takeWhile__map) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 783 (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]. 5.86/5.91 784 (all V_B_2 all V_A_2 all T_c (class_Lattices_Olattice(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (V_B_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_A_2)) -> c_Big__Operators_Olattice__class_OInf__fin(T_c,V_A_2) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_c),c_Big__Operators_Olattice__class_OInf__fin(T_c,V_B_2)),c_Big__Operators_Olattice__class_OInf__fin(T_c,V_A_2))))))) # label(fact_Inf__fin_Osubset__idem) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 785 (all V_y_2 all V_x_2 all T_c (class_Rings_Ocomm__ring__1(T_c) -> (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_c),V_x_2),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_y_2))) <-> hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_c),V_x_2),V_y_2))))) # label(fact_dvd__minus__iff) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 786 (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]. 5.86/5.91 787 (all V_y_2 (hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),V_y_2),c_Orderings_Otop__class_Otop(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) -> -(all B_x V_y_2 != hAPP(c_Code__Numeral_Onat__of,B_x)))) # label(fact_nat__of__cases) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 788 (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]. 5.86/5.91 789 (all V_m all V_y all V_x c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_x),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_y,V_m)),V_m) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_x),V_y),V_m)) # label(fact_zdiff__zmod__right) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 790 (all V_b_2 all V_a_2 all V_r_2 all T_d all T_c (hBOOL(hAPP(hAPP(V_r_2,V_b_2),V_a_2)) <-> hBOOL(hAPP(hAPP(c_Predicate_Oconversep(T_c,T_d,V_r_2),V_a_2),V_b_2)))) # label(fact_conversep__iff) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 791 (all V_g_2 all T_e all V_f_2 all T_c all T_d (hAPP(c_Set_Oimage(T_d,T_c,V_f_2),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_c,T_e,V_g_2),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) = c_Orderings_Otop__class_Otop(tc_fun(T_e,tc_HOL_Obool)) -> c_Orderings_Otop__class_Otop(tc_fun(T_e,tc_HOL_Obool)) = hAPP(c_Set_Oimage(T_d,T_e,hAPP(c_Fun_Ocomp(T_c,T_e,T_d,V_g_2),V_f_2)),c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool)))))) # label(fact_comp__surj) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 792 (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))) -> (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))))) <-> (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))))))))) # label(fact_finite__Collect__bounded__ex) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 793 (all V_x_2 all V_f_2 all V_A_2 all T_d all T_c hAPP(c_Fun_Othe__inv__into(T_c,T_d,V_A_2,V_f_2),V_x_2) = 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)))) # label(fact_the__inv__into__def) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 794 (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]. 5.86/5.91 795 (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]. 5.86/5.91 796 (all V_u_2 all V_l_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_l_2),V_u_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_c),V_l_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))),c_SetInterval_Oord__class_OgreaterThanLessThan(T_c,V_l_2,V_u_2)) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_c,V_l_2),V_u_2)))) # label(fact_ivl__disj__un_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 797 (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]. 5.86/5.91 798 (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]. 5.86/5.91 799 (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]. 5.86/5.91 800 (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]. 5.86/5.91 801 (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]. 5.86/5.91 802 (all V_ys_2 all V_xs_2 all T_c hAPP(c_List_Oset(T_c),hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_ys_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_List_Oset(T_c),V_xs_2)),hAPP(c_List_Oset(T_c),V_ys_2))) # label(fact_set__append) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 803 (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_a_2 != V_b_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_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]. 5.86/5.91 804 (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]. 5.86/5.91 805 (all V_k_2 all V_n_2 all V_Pa_2 ((V_k_2 = c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> hBOOL(hAPP(V_Pa_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_k_2)) -> (all B_i ((exists B_j (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B_j)) & hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_k_2),B_i)),B_j) = V_n_2 & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B_j),V_k_2)))) -> hBOOL(hAPP(V_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 (hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_k_2),B_i)),B_j) = V_n_2 & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B_j),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),B_j)))) -> hBOOL(hAPP(V_Pa_2,B_i))))) <-> hBOOL(hAPP(V_Pa_2,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_n_2),V_k_2))))) # label(fact_split__zdiv) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 806 (all V_g_2 all V_x_2 all V_A_2 all T_c all T_d (class_Groups_Ocomm__monoid__mult(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_d),hAPP(V_g_2,V_x_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_g_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_g_2),V_A_2))))) # label(fact_setprod_Oremove) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 807 (all V_m all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_m)),V_m) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),c_Groups_Oone__class_Oone(T_a))),V_m))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 808 (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]. 5.86/5.91 809 (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]. 5.86/5.91 810 (all V_A_2 all V_y_2 all T_d all T_c (class_Rings_Osemiring__1(T_c) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_c),hAPP(c_COMBK(T_c,T_d),V_y_2)),V_A_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_c),hAPP(c_Finite__Set_Ocard(T_d),V_A_2))),V_y_2))) # label(fact_setsum__constant) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 811 (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]. 5.86/5.91 812 (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_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)))) = 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))) & (-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]. 5.86/5.91 813 (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]. 5.86/5.91 814 (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]. 5.86/5.91 815 (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]. 5.86/5.91 816 (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]. 5.86/5.91 817 (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]. 5.86/5.91 818 (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]. 5.86/5.91 819 (all V_x all T_a all V_xs ((V_xs = c_List_Olist_ONil(T_a) -> V_x = c_List_Olast(T_a,hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs))) & (V_xs != c_List_Olist_ONil(T_a) -> c_List_Olast(T_a,V_xs) = c_List_Olast(T_a,hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs))))) # label(fact_last_Osimps) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 820 (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]. 5.86/5.91 821 (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_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin)),V_b) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin))) # label(fact_div__eq__minus1) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 822 (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]. 5.86/5.91 823 (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]. 5.86/5.91 824 (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]. 5.86/5.91 825 (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]. 5.86/5.91 826 (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)) -> V_z_2 = hAPP(hAPP(V_F_2,V_g_2),V_A_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]. 5.86/5.91 827 (all V_l_2 all V_k_2 all T_c (class_Orderings_Oorder(T_c) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = c_SetInterval_Oord__class_OgreaterThanAtMost(T_c,V_k_2,V_l_2) <-> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_k_2),V_l_2))))) # label(fact_greaterThanAtMost__empty__iff2) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 828 (all V_n_2 all V_m_2 (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_m_2 & c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n_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))) # label(fact_add__is__0) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 829 (all V_v ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_v),c_Int_OPls)) -> c_Groups_Oone__class_Oone(tc_Nat_Onat) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),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)) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_Osucc(V_v))))) # label(fact_nat__1__add__number__of) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 830 (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),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),V_A_2))) # label(fact_Compl__disjoint) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 831 (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)) -> 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_A_2),hAPP(c_COMBK(tc_fun(T_d,tc_HOL_Obool),T_c),V_c_2))) & (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_A_2 -> 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))))) # label(fact_INT__constant) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 832 (all V_A_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_A_2) = V_A_2) # label(fact_Un__absorb) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 833 (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]. 5.86/5.91 834 (all V_xs all T_a hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_List_Olist(T_a))),c_Enum_Osublists(T_a,V_xs))) # label(fact_length__sublists) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 835 (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]. 5.86/5.91 836 (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]. 5.86/5.91 837 (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]. 5.86/5.91 838 (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]. 5.86/5.91 839 (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]. 5.86/5.91 840 (all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)) = hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a))) # label(fact_abs__idempotent) # label(axiom) # label(non_clause). [assumption]. 5.86/5.91 841 (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]. 5.86/5.91 842 (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]. 5.86/5.91 843 (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]. 5.86/5.91 844 (all V_f_2 all V_a_2 all V_F_2 all T_c all T_d (class_Groups_Ocomm__monoid__add(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_F_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_F_2)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_d),hAPP(V_f_2,V_a_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_f_2),V_F_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_f_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_F_2)))))) # label(fact_setsum__insert) # label(axiom) # label(non_clause). [assumption]. 5.86/5.92 845 (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]. 5.86/5.92 846 (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]. 5.86/5.92 847 (all V_b_2 all V_a_2 all V_c_2 all T_c (class_Fields_Olinordered__field(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_c_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_c_2)),V_b_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_a_2),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_b_2),V_c_2))))))) # label(fact_pos__less__divide__eq) # label(axiom) # label(non_clause). [assumption]. 5.86/5.92 848 (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]. 5.86/5.92 849 (all V_xs_2 all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> 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))) = hAPP(c_List_Ofoldl(T_c,T_c,c_Orderings_Oord__class_Omin(T_c),V_x_2),V_xs_2))) # label(fact_Min__fin__set__fold) # label(axiom) # label(non_clause). [assumption]. 5.86/5.92 850 (all V_z2 all V_z1 all V_w hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_w),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_z1),V_z2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_w),V_z1)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_w),V_z2))) # label(fact_zadd__zmult__distrib2) # label(axiom) # label(non_clause). [assumption]. 5.86/5.92 851 (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]. 5.86/5.92 852 (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]. 5.86/5.92 853 (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]. 5.86/5.92 854 (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]. 5.86/5.92 855 (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]. 5.86/5.92 856 (all T_a (class_Rings_Ozero__neq__one(T_a) -> c_Groups_Ozero__class_Ozero(T_a) != c_Groups_Oone__class_Oone(T_a))) # label(fact_zero__neq__one) # label(axiom) # label(non_clause). [assumption]. 5.86/5.92 857 (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__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))) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),V_A_2),V_B_2)) # label(fact_Nat__Transfer_Otransfer__nat__int__set__functions_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.86/5.92 858 (all V_B_2 all V_A_2 all T_d all T_c (V_B_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_Product__Type_OSigma(T_c,T_d),V_A_2),hAPP(c_COMBK(tc_fun(T_d,tc_HOL_Obool),T_c),V_B_2)) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)))) # label(fact_Times__empty) # label(axiom) # label(non_clause). [assumption]. 5.86/5.92 859 (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]. 5.86/5.92 860 (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]. 5.86/5.92 861 (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]. 5.86/5.92 862 (all V_y all V_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_y)) -> hAPP(c_Int_Onat,c_Divides_Odiv__class_Omod(tc_Int_Oint,V_x,V_y)) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Int_Onat,V_x),hAPP(c_Int_Onat,V_y))))) # label(fact_nat__mod__distrib) # label(axiom) # label(non_clause). [assumption]. 5.86/5.92 863 (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]. 5.86/5.92 864 (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]. 5.86/5.92 865 (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]. 5.86/5.92 866 (all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_A_2 -> ((all B_x all B_y hBOOL(hAPP(hAPP(c_member(T_c),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),B_x),B_y)),hAPP(hAPP(c_Set_Oinsert(T_c),B_x),hAPP(hAPP(c_Set_Oinsert(T_c),B_y),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))))) -> hBOOL(hAPP(hAPP(c_member(T_c),c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),V_A_2)),V_A_2))))))) # label(fact_min__max_OInf__fin_Oclosed) # label(axiom) # label(non_clause). [assumption]. 5.86/5.92 867 (all V_xs_2 all V_Pa_2 all T_c (c_List_OtakeWhile(T_c,V_Pa_2,V_xs_2) = V_xs_2 <-> (all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(c_List_Oset(T_c),V_xs_2))) -> hBOOL(hAPP(V_Pa_2,B_x)))))) # label(fact_takeWhile__eq__all__conv) # label(axiom) # label(non_clause). [assumption]. 5.86/5.92 868 (all V_b all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,V_a,V_b) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b),V_b))) # label(fact_mod__add__self2) # label(axiom) # label(non_clause). [assumption]. 5.86/5.92 869 (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]. 5.86/5.92 870 (all V_d c_Lazy__Sequence_Osmall__lazy_H(V_d,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_d)) = c_Lazy__Sequence_Osmall__lazy__class_Osmall__lazy(tc_Int_Oint,V_d)) # label(fact_small__lazy__int__def) # label(axiom) # label(non_clause). [assumption]. 5.86/5.92 871 (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 = hAPP(c_List_OremoveAll(T_c,V_x_2),V_xs_2))) # label(fact_removeAll__id) # label(axiom) # label(non_clause). [assumption]. 5.86/5.92 872 (all V_w all V_z all V_y all V_x all T_a (class_Fields_Ofield__inverse__zero(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),V_y)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_z),V_w)) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_z)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_y),V_w)))) # label(fact_times__divide__times__eq) # label(axiom) # label(non_clause). [assumption]. 5.86/5.92 873 (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]. 5.86/5.92 874 (all V_x all V_y ((V_y = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> c_Nitpick_Onat__gcd(V_x,V_y) = V_x) & (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != V_y -> c_Nitpick_Onat__gcd(V_y,c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_x,V_y)) = c_Nitpick_Onat__gcd(V_x,V_y)))) # label(fact_nat__gcd_Osimps) # label(axiom) # label(non_clause). [assumption]. 5.86/5.92 875 (all V_l_2 (c_Nat__Transfer_Onat__set(hAPP(c_List_Oset(tc_Int_Oint),V_l_2)) <-> c_List_Onat__list(V_l_2))) # label(fact_nat__list__def) # label(axiom) # label(non_clause). [assumption]. 5.86/5.92 876 (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]. 5.86/5.92 877 (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]. 5.86/5.92 878 (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]. 5.86/5.92 879 (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]. 5.86/5.92 880 (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]. 5.86/5.92 881 (all V_n_2 all V_m_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_k_2)),V_m_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_k_2)),V_n_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)))) # label(fact_Suc__mult__less__cancel1) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 882 (all T_a all V_xs (V_xs != c_List_Olist_ONil(T_a) -> hAPP(c_List_Ohd(T_a),V_xs) = c_List_Olast(T_a,hAPP(c_List_Orev(T_a),V_xs)))) # label(fact_last__rev) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 883 (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]. 5.89/5.92 884 (all V_n_2 all V_v_2 hAPP(hAPP(c_HOL_OLet(tc_Int_Oint,tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_Opred(V_v_2))),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat)),tc_Int_Oint),c_If(tc_Nat_Onat)),c_Nat__Numeral_Oneg)),hAPP(c_Nat_OSuc,V_n_2))),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Nat_Onat,tc_Int_Oint),c_Nat_OSuc),hAPP(hAPP(c_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)))) = 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))) # label(fact_max__Suc__number__of) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 885 (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]. 5.89/5.92 886 (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_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__Un__disjoint) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 887 (all V_r_2 all V_A_2 all V_X_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),V_X_2),hAPP(hAPP(c_Equiv__Relations_Oquotient(T_c),V_A_2),V_r_2))) -> -(all B_x (V_X_2 = hAPP(c_Relation_OImage(T_c,T_c,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_c),B_x),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) -> -hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)))))) # label(fact_quotientE) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 888 (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_Omin(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_min__le__iff__disj) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 889 (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]. 5.89/5.92 890 (all V_xqq all V_xq all T_a c_Lazy__Sequence_Oflat(T_a,c_Lazy__Sequence_Olazy__sequence_OInsert(tc_Lazy__Sequence_Olazy__sequence(T_a),V_xq,V_xqq)) = c_Lazy__Sequence_Oappend(T_a,V_xq,c_Lazy__Sequence_Oflat(T_a,V_xqq))) # label(fact__03) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 891 (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]. 5.89/5.92 892 (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_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),hAPP(hAPP(c_COMBC(T_c,T_d,T_d),hAPP(hAPP(c_COMBS(T_c,T_d,tc_fun(T_d,T_d)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(T_d,tc_fun(T_d,T_d)),T_c),c_If(T_d)),hAPP(hAPP(c_COMBC(T_c,T_c,tc_HOL_Obool),c_fequal),V_a_2))),V_b_2)),V_c_2)),V_S_2) = hAPP(hAPP(c_Power_Opower__class_Opower(T_d),V_c_2),hAPP(c_Finite__Set_Ocard(T_c),V_S_2))) & (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))))) # label(fact_setprod__gen__delta) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 893 (all V_b all V_c all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Divides_Odiv__class_Omod(T_a,V_a,V_b)),V_c))) # label(fact_mod__mult__mult2) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 894 (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]. 5.89/5.92 895 (all V_y all V_i all V_xs all V_x all T_a hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),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_OCons(T_a),V_x),V_xs)),hAPP(c_Nat_OSuc,V_i)),V_y)) # label(fact_list__update__code_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 896 (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]. 5.89/5.92 897 (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(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(V_h_2,B_x)),hAPP(V_h_2,B_y)) = hAPP(V_h_2,hAPP(hAPP(c_Groups_Otimes__class_Otimes(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(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)) = hAPP(V_h_2,hAPP(c_Finite__Set_Ofold1(T_c,c_Groups_Otimes__class_Otimes(T_c)),V_N_2))))))) # label(fact_hom__fold1__commute) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 898 (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]. 5.89/5.92 899 (all V_k_2 all V_A_2 all T_c (hAPP(c_Nat_OSuc,V_k_2) = hAPP(c_Finite__Set_Ocard(T_c),V_A_2) <-> (exists B_b exists B_B (V_A_2 = hAPP(hAPP(c_Set_Oinsert(T_c),B_b),B_B) & (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_k_2 -> B_B = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) & V_k_2 = hAPP(c_Finite__Set_Ocard(T_c),B_B) & -hBOOL(hAPP(hAPP(c_member(T_c),B_b),B_B)))))) # label(fact_card__Suc__eq) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 900 (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]. 5.89/5.92 901 (all V_l c_Int_OBit0(hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),c_Int_OMin),V_l)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),c_Int_OMin),c_Int_OBit1(V_l))) # label(fact_diff__bin__simps_I6_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 902 (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__right) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 903 (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_neg__equal__zero) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 904 (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]. 5.89/5.92 905 (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]. 5.89/5.92 906 (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_n_2 = V_m_2)) # label(fact_int__int__eq) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 907 (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),V_xs_2)) <-> hBOOL(hAPP(c_List_Odistinct(T_c),c_List_Olinorder__class_Osort__key(T_c,T_d,V_f_2,V_xs_2)))))) # label(fact_distinct__sort) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 908 (all V_z all T_a (class_Int_Onumber__ring(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_z),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_z),V_z))) # label(fact_mult__2__right) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 909 (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]. 5.89/5.92 910 (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]. 5.89/5.92 911 (all V_x_2 all V_A_2 all T_c (class_Lattices_Olattice(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_A_2 -> c_Big__Operators_Olattice__class_OInf__fin(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_c),V_x_2),c_Big__Operators_Olattice__class_OInf__fin(T_c,V_A_2))))))) # label(fact_Inf__fin_Oinsert) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 912 (all V_b_2 all V_a_2 all V_c_2 all T_c (class_Fields_Olinordered__field(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_c_2),c_Groups_Ozero__class_Ozero(T_c))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_a_2),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_b_2),V_c_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_c_2))))))) # label(fact_neg__less__divide__eq) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 913 (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]. 5.89/5.92 914 (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]. 5.89/5.92 915 (all 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_UNIV__not__empty) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 916 (all V_y_2 all V_x_2 all T_c (V_x_2 != V_y_2 <-> hBOOL(hAPP(hAPP(c_Predicate_Oconversep(T_c,T_c,hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_c),hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_c),c_fNot)),c_fequal)),V_x_2),V_y_2)))) # label(fact_conversep__noteq) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 917 (all T_c all V_B_2 all V_A_2 (V_B_2 = V_A_2 -> -(hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) -> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_A_2))))) # label(fact_equalityE) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 918 (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]. 5.89/5.92 919 (all V_b_2 all V_a_2 all T_c (class_Groups_Ogroup__add(T_c) -> (V_b_2 = V_a_2 <-> hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_a_2) = hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_b_2)))) # label(fact_neg__equal__iff__equal) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 920 (all V_ys_2 all V_xs_2 all V_x_2 all T_d all T_c hAPP(hAPP(c_List_Ozip(T_c,T_d),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2)),V_ys_2) = hAPP(c_List_Olist_Olist__case(tc_List_Olist(tc_prod(T_c,T_d)),T_d,c_List_Olist_ONil(tc_prod(T_c,T_d)),hAPP(hAPP(c_COMBC(T_d,tc_fun(tc_List_Olist(T_d),tc_List_Olist(tc_prod(T_c,T_d))),tc_fun(tc_List_Olist(T_d),tc_List_Olist(tc_prod(T_c,T_d)))),hAPP(hAPP(c_COMBB(tc_fun(tc_List_Olist(tc_prod(T_c,T_d)),tc_List_Olist(tc_prod(T_c,T_d))),tc_fun(tc_fun(tc_List_Olist(T_d),tc_List_Olist(tc_prod(T_c,T_d))),tc_fun(tc_List_Olist(T_d),tc_List_Olist(tc_prod(T_c,T_d)))),T_d),c_COMBB(tc_List_Olist(tc_prod(T_c,T_d)),tc_List_Olist(tc_prod(T_c,T_d)),tc_List_Olist(T_d))),hAPP(hAPP(c_COMBB(tc_prod(T_c,T_d),tc_fun(tc_List_Olist(tc_prod(T_c,T_d)),tc_List_Olist(tc_prod(T_c,T_d))),T_d),c_List_Olist_OCons(tc_prod(T_c,T_d))),hAPP(c_Product__Type_OPair(T_c,T_d),V_x_2)))),hAPP(c_List_Ozip(T_c,T_d),V_xs_2))),V_ys_2)) # label(fact_zip__Cons1) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 921 (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]. 5.89/5.92 922 (all V_R_2 all V_Qa_2 all V_Pa_2 all T_c all T_e all T_d hAPP(hAPP(V_Pa_2,V_R_2),V_Qa_2) = hAPP(hAPP(hAPP(c_COMBC(T_d,T_e,T_c),V_Pa_2),V_Qa_2),V_R_2)) # label(help_c__COMBC__1) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 923 (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]. 5.89/5.92 924 (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))) <-> 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_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__lex) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 925 (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]. 5.89/5.92 926 (all V_ys all V_xs all V_n all T_b all T_a 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)) = 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))) # label(fact_drop__zip) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 927 (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]. 5.89/5.92 928 (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]. 5.89/5.92 929 (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]. 5.89/5.92 930 (all V_B_2 all V_A_2 all V_I_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_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool)),V_I_2),V_A_2)),V_B_2)) <-> (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)))))) # label(fact_UN__subset__iff) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 931 (all V_y_2 all V_x_2 all V_r_2 all T_e all T_d all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_d,T_e)),hAPP(hAPP(c_Product__Type_OPair(T_d,T_e),V_x_2),V_y_2)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(tc_prod(T_d,T_e),tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))),V_r_2))) <-> hBOOL(hAPP(hAPP(hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_fun(T_e,tc_HOL_Obool))),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_prod(T_d,T_e),tc_HOL_Obool),tc_fun(T_d,tc_fun(T_e,tc_HOL_Obool)),T_c),hAPP(c_COMBC(T_d,tc_fun(tc_prod(T_d,T_e),tc_HOL_Obool),tc_fun(T_e,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_e,tc_fun(tc_fun(tc_prod(T_d,T_e),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_d,T_e),tc_HOL_Obool),tc_fun(T_e,tc_HOL_Obool)),T_d),c_COMBC(T_e,tc_fun(tc_prod(T_d,T_e),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_e,tc_prod(T_d,T_e)),tc_fun(T_e,tc_fun(tc_fun(tc_prod(T_d,T_e),tc_HOL_Obool),tc_HOL_Obool)),T_d),hAPP(c_COMBB(tc_prod(T_d,T_e),tc_fun(tc_fun(tc_prod(T_d,T_e),tc_HOL_Obool),tc_HOL_Obool),T_e),c_member(tc_prod(T_d,T_e)))),c_Product__Type_OPair(T_d,T_e))))),V_r_2)),V_x_2),V_y_2)))) # label(fact_INF__INT__eq2) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 932 (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_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(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))))))))))))) # label(fact_min__max_Osup__Inf2__distrib) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 933 (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]. 5.89/5.92 934 (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_Ortrancl(T_c,V_R_2)),c_Relation_OId(T_c)) = c_Transitive__Closure_Ortrancl(T_c,V_R_2)) # label(fact_rtrancl__reflcl__absorb) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 935 (all V_d all T_b all T_a (class_Lazy__Sequence_Osmall__lazy(T_b) & class_Lazy__Sequence_Osmall__lazy(T_a) -> c_Lazy__Sequence_Oproduct(T_a,T_b,c_Lazy__Sequence_Osmall__lazy__class_Osmall__lazy(T_a,V_d),c_Lazy__Sequence_Osmall__lazy__class_Osmall__lazy(T_b,V_d)) = c_Lazy__Sequence_Osmall__lazy__class_Osmall__lazy(tc_prod(T_a,T_b),V_d))) # label(fact_small__lazy__prod__def) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 936 (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) <-> 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)))) # label(fact_nat__eq__add__iff1) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 937 (all V_ys_2 all V_g_2 all T_f all V_xs_2 all V_f_2 all T_c all T_e all T_d hAPP(hAPP(c_List_Omap(tc_prod(T_d,T_e),T_c),V_f_2),hAPP(hAPP(c_List_Ozip(T_d,T_e),V_xs_2),hAPP(hAPP(c_List_Omap(T_f,T_e),V_g_2),V_ys_2))) = hAPP(hAPP(c_List_Omap(tc_prod(T_d,T_f),T_c),hAPP(c_Product__Type_Oprod_Oprod__case(T_d,T_f,T_c),hAPP(hAPP(c_COMBB(tc_fun(T_f,tc_prod(T_d,T_e)),tc_fun(T_f,T_c),T_d),hAPP(c_COMBB(tc_prod(T_d,T_e),T_c,T_f),V_f_2)),hAPP(hAPP(c_COMBC(T_d,tc_fun(T_f,T_e),tc_fun(T_f,tc_prod(T_d,T_e))),hAPP(hAPP(c_COMBB(tc_fun(T_e,tc_prod(T_d,T_e)),tc_fun(tc_fun(T_f,T_e),tc_fun(T_f,tc_prod(T_d,T_e))),T_d),c_COMBB(T_e,tc_prod(T_d,T_e),T_f)),c_Product__Type_OPair(T_d,T_e))),V_g_2)))),hAPP(hAPP(c_List_Ozip(T_d,T_f),V_xs_2),V_ys_2))) # label(fact_map__zip__map2) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 938 (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]. 5.89/5.92 939 (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]. 5.89/5.92 940 (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]. 5.89/5.92 941 (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]. 5.89/5.92 942 (all V_a (V_a != c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_a) = c_Groups_Oone__class_Oone(tc_Int_Oint))) # label(fact_zdiv__self) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 943 (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]. 5.89/5.92 944 (all V_r_2 all T_c (c_Relation_Otrans(T_c,hAPP(c_Relation_Oconverse(T_c,T_c),V_r_2)) <-> c_Relation_Otrans(T_c,V_r_2))) # label(fact_trans__converse) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 945 (all V_xs all T_b all T_a c_List_Olist_ONil(tc_prod(T_a,T_b)) = hAPP(hAPP(c_List_Ozip(T_a,T_b),V_xs),c_List_Olist_ONil(T_b))) # label(fact_zip_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 946 (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]. 5.89/5.92 947 (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]. 5.89/5.92 948 (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]. 5.89/5.92 949 (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]. 5.89/5.92 950 (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]. 5.89/5.92 951 (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]. 5.89/5.92 952 (all V_xs_2 all V_x_2 all V_f_2 all T_c all T_d (class_Orderings_Olinorder(T_d) -> hAPP(c_Nat_OSuc,hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2)) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_c,T_d,V_f_2),V_x_2),V_xs_2)))) # label(fact_length__insort) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 953 (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)) -> (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) <-> (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)))))))) # label(fact_setprod__zero__iff) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 954 (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_number__of__minus) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 955 (all V_z all V_y all V_x all T_a (class_Groups_Oordered__ab__semigroup__add__imp__le(T_a) -> hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_x),V_z)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_y),V_z)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),V_y)),V_z))) # label(fact_min__add__distrib__left) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 956 (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]. 5.89/5.92 957 (all V_A_2 all V_B_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_A_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_A_2)),V_A_2)) # label(fact_Un__Diff__cancel2) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 958 (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]. 5.89/5.92 959 (all V_f_2 all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,tc_Nat_Onat),V_f_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,tc_Nat_Onat),V_f_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,tc_Nat_Onat),V_f_2),V_B_2))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,tc_Nat_Onat),V_f_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)))))) # label(fact_setsum__Un__nat) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 960 (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]. 5.89/5.92 961 (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]. 5.89/5.92 962 (all T_c all V_xs_2 (c_List_Olist_ONil(T_c) != V_xs_2 -> c_Random_Oselect(T_c,V_xs_2) = 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)))) # label(fact_select__weigth__select) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 963 (all V_ys_2 all V_xs_2 all T_c (V_xs_2 = c_List_Olist_ONil(T_c) & V_ys_2 = c_List_Olist_ONil(T_c) <-> c_List_Olist_ONil(T_c) = hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_ys_2))) # label(fact_append__is__Nil__conv) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 964 (all V_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_x),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> (exists B_n hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Nat_OSuc,B_n))) = V_x))) # label(fact_negD) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 965 (all V_x_2 all V_A_2 all T_d all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> hAPP(c_Complete__Lattice_OInf__class_OInf(tc_fun(T_d,T_c),V_A_2),V_x_2) = c_Complete__Lattice_OInf__class_OInf(T_c,hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_d,T_c),tc_HOL_Obool),tc_HOL_Obool,T_c),c_HOL_OEx(tc_fun(T_d,T_c))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_d,T_c),tc_HOL_Obool),tc_fun(tc_fun(T_d,T_c),tc_HOL_Obool),T_c),hAPP(c_COMBS(tc_fun(T_d,T_c),tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_fun(T_d,T_c)),c_fconj),hAPP(hAPP(c_COMBC(tc_fun(T_d,T_c),tc_fun(tc_fun(T_d,T_c),tc_HOL_Obool),tc_HOL_Obool),c_member(tc_fun(T_d,T_c))),V_A_2)))),hAPP(hAPP(c_COMBC(T_c,tc_fun(tc_fun(T_d,T_c),T_c),tc_fun(tc_fun(T_d,T_c),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(tc_fun(T_d,T_c),T_c),tc_fun(tc_fun(T_d,T_c),tc_HOL_Obool)),T_c),c_COMBB(T_c,tc_HOL_Obool,tc_fun(T_d,T_c))),c_fequal)),hAPP(hAPP(c_COMBC(tc_fun(T_d,T_c),T_d,T_c),c_COMBI(tc_fun(T_d,T_c))),V_x_2)))))))) # label(fact_Inf__fun__def) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 966 (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]. 5.89/5.92 967 (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]. 5.89/5.92 968 (all V_xs_2 all V_a_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_a_2),V_xs_2) != c_List_Olist_ONil(T_c))) # label(fact_insort__not__Nil) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 969 (all T_c c_Transitive__Closure_Otrancl(T_c,c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool))) # label(fact_trancl__empty) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 970 (all V_ys all V_xs all V_n all T_a hAPP(hAPP(c_List_Otake(T_a),V_n),hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys)) = hAPP(hAPP(c_List_Oappend(T_a),hAPP(hAPP(c_List_Otake(T_a),V_n),V_xs)),hAPP(hAPP(c_List_Otake(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))),V_ys))) # label(fact_take__append) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 971 (all V_n all T_a hAPP(hAPP(c_List_Odrop(T_a),V_n),c_List_Olist_ONil(T_a)) = c_List_Olist_ONil(T_a)) # label(fact_drop__Nil) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 972 (all V_b all V_a all T_a (class_Orderings_Olinorder(T_a) -> hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_b),V_a) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_a),V_b))) # label(fact_min__max_Osup_Ocommute) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 973 (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]. 5.89/5.92 974 (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_x != B_y -> (hAPP(V_f_2,B_y) = hAPP(V_f_2,B_x) -> 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]. 5.89/5.92 975 (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]. 5.89/5.92 976 (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) = V_m)) # label(fact_mod__less) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 977 (all T_c hAPP(c_Relation_ODomain(T_c,T_c),c_Relation_OId(T_c)) = c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) # label(fact_Domain__Id) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 978 (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]. 5.89/5.92 979 (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_Omax(T_c),V_x_2),V_y_2)))))) # label(fact_less__max__iff__disj) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 980 (all V_i_2 all V_h_2 all V_g_2 all V_f_2 all T_d all T_c all T_f all T_e all T_h all T_g hAPP(c_Fun_Ocomp(tc_prod(T_g,T_h),tc_prod(T_e,T_f),tc_prod(T_c,T_d),c_Product__Type_Omap__pair(T_g,T_e,T_h,T_f,V_f_2,V_g_2)),c_Product__Type_Omap__pair(T_c,T_g,T_d,T_h,V_h_2,V_i_2)) = c_Product__Type_Omap__pair(T_c,T_e,T_d,T_f,hAPP(c_Fun_Ocomp(T_g,T_e,T_c,V_f_2),V_h_2),hAPP(c_Fun_Ocomp(T_h,T_f,T_d,V_g_2),V_i_2))) # label(fact_map__pair_Ocomp) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 981 (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))) = 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_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)))))) # label(fact_zpower__number__of__odd) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 982 (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]. 5.89/5.92 983 (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]. 5.89/5.92 984 (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]. 5.89/5.92 985 (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]. 5.89/5.92 986 (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]. 5.89/5.92 987 (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]. 5.89/5.92 988 (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)) = 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)) = hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),c_List_Oremdups(T_c,V_xs_2))))) # label(fact_remdups_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 989 (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]. 5.89/5.92 990 (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]. 5.89/5.92 991 (all V_b c_Divides_OnegDivAlg(hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin),V_b) = hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_b),c_Groups_Oone__class_Oone(tc_Int_Oint)))) # label(fact_negDivAlg__minus1) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 992 (all V_xs_2 all T_c (hAPP(c_List_Orev(T_c),V_xs_2) = c_List_Olist_ONil(T_c) <-> c_List_Olist_ONil(T_c) = V_xs_2)) # label(fact_rev__is__Nil__conv) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 993 (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)),V_ly) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),V_ly)),V_rx))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 994 (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)) -> 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)))))) # label(fact_sup__Inf__absorb) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 995 (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]. 5.89/5.92 996 (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]. 5.89/5.92 997 (all V_n_2 all V_m_2 all V_a_2 all T_c (class_Rings_Olinordered__semidom(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Oone__class_Oone(T_c)),V_a_2)) -> (hAPP(hAPP(c_Power_Opower__class_Opower(T_c),V_a_2),V_n_2) = hAPP(hAPP(c_Power_Opower__class_Opower(T_c),V_a_2),V_m_2) <-> V_n_2 = V_m_2)))) # label(fact_power__inject__exp) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 998 (all V_h_2 all V_g_2 all V_A_2 all T_c all T_d (class_Groups_Ocomm__monoid__mult(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),hAPP(hAPP(c_COMBS(T_c,T_d,T_d),hAPP(hAPP(c_COMBB(T_d,tc_fun(T_d,T_d),T_c),c_Groups_Otimes__class_Otimes(T_d)),V_g_2)),V_h_2)),V_A_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_d),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_g_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_h_2),V_A_2))))) # label(fact_setprod_Odistrib) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 999 (all V_y_2 all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_x_2),V_y_2)) <-> V_x_2 = V_y_2 | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_y_2),V_x_2))))) # label(fact_not__less__iff__gr__or__eq) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 1000 (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__ex1(T_c,V_Pa_2,V_xs_2)) & (hBOOL(hAPP(V_Pa_2,V_x_2)) -> c_List_Olist__all(T_c,hAPP(hAPP(c_COMBS(T_c,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_c),c_fdisj),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_c),c_fNot),V_Pa_2))),hAPP(c_fequal,V_x_2)),V_xs_2)) <-> c_List_Olist__ex1(T_c,V_Pa_2,hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2)))) # label(fact_list__ex1__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 1001 (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]. 5.89/5.92 1002 (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]. 5.89/5.92 1003 (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_y_2) = V_f_2 <-> hAPP(V_f_2,V_x_2) = V_y_2)) # label(fact_fun__upd__idem__iff) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 1004 (all V_vs all V_ys all V_us all T_b all V_xs all T_a (hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_us) -> (hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_ys) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),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)) = 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))))) # label(fact_zip__append) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 1005 (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]. 5.89/5.92 1006 (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]. 5.89/5.92 1007 (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]. 5.89/5.92 1008 (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]. 5.89/5.92 1009 (all V_R_2 all V_y_2 all V_x_2 all T_c (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))) & 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_rtrancl__eq__or__trancl) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 1010 (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]. 5.89/5.92 1011 (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]. 5.89/5.92 1012 (all V_xs_2 all V_Pa_2 all T_c ((exists B_x ((all B_y (hBOOL(hAPP(hAPP(c_member(T_c),B_y),hAPP(c_List_Oset(T_c),V_xs_2))) & hBOOL(hAPP(V_Pa_2,B_y)) -> B_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))))) <-> c_List_Olist__ex1(T_c,V_Pa_2,V_xs_2))) # label(fact_list__ex1__iff__raw) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 1013 (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]. 5.89/5.92 1014 (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_xs_2 = V_ys_2)))) # label(fact_concat__eq__concat__iff) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 1015 (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]. 5.89/5.92 1016 (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]. 5.89/5.92 1017 (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_Omin(T_c),V_x_2),V_y_2) = V_x_2))) # label(fact_min__max_Ole__iff__inf) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 1018 (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]. 5.89/5.92 1019 (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]. 5.89/5.92 1020 (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]. 5.89/5.92 1021 (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]. 5.89/5.92 1022 (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]. 5.89/5.92 1023 (all V_a all T_a (class_Orderings_Olinorder(T_a) -> hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_a),V_a) = V_a)) # label(fact_min__max_Osup_Oidem) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 1024 (all V_n all T_a (class_Power_Opower(T_a) & class_Rings_Osemiring__0(T_a) -> (V_n = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> 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]. 5.89/5.92 1025 (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]. 5.89/5.92 1026 (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]. 5.89/5.92 1027 (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]. 5.89/5.92 1028 (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))) -> c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b) = hAPP(c_Product__Type_Osnd(tc_Int_Oint,tc_Int_Oint),hAPP(c_Divides_OnegateSnd,c_Divides_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_mod__neg__neg) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 1029 (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)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> c_Complete__Lattice_OInf__class_OInf(T_c,V_A_2) = c_Big__Operators_Olattice__class_OInf__fin(T_c,V_A_2))))) # label(fact_Inf__fin__Inf) # label(axiom) # label(non_clause). [assumption]. 5.89/5.92 1030 (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),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]. 5.89/5.92 1031 (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]. 5.89/5.92 1032 (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]. 5.89/5.93 1033 (all V_r_2 all T_c all T_d 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_Range__def) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1034 (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]. 5.89/5.93 1035 (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_u)) & (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_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]. 5.89/5.93 1036 (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]. 5.89/5.93 1037 (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]. 5.89/5.93 1038 (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]. 5.89/5.93 1039 (all T_c all V_Pa_2 ((exists B_x1 hBOOL(hAPP(V_Pa_2,B_x1))) <-> hBOOL(hAPP(V_Pa_2,c_Hilbert__Choice_OEps(T_c,V_Pa_2))))) # label(fact_some__eq__ex) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1040 (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]. 5.89/5.93 1041 (all V_B_2 all V_A_2 all T_d all T_c all V_C_2 ((c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = V_C_2 -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_d,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),V_B_2)) = V_A_2) & (V_C_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool),T_c),hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_d,tc_HOL_Obool)),V_A_2)),V_B_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_d,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),V_B_2))))) # label(fact_INT__extend__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1042 (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]. 5.89/5.93 1043 (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]. 5.89/5.93 1044 (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]. 5.89/5.93 1045 (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]. 5.89/5.93 1046 (all V_A_2 all V_f_2 all T_c all T_d (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) <-> c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool)) = V_A_2)) # label(fact_image__is__empty) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1047 (all V_A_2 all T_d all V_I_2 all T_c ((exists B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_I_2)) & hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),hAPP(V_A_2,B_x))))) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_HOL_Obool)),V_I_2),V_A_2))))) # label(fact_finite__INT) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1048 (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]. 5.89/5.93 1049 (all V_B_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(c_Finite__Set_Ofold1(T_c,c_Orderings_Oord__class_Omin(T_c)),V_B_2)),hAPP(c_Finite__Set_Ofold1(T_c,c_Orderings_Oord__class_Omin(T_c)),V_A_2)))))))) # label(fact_fold1__antimono) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1050 (all V_y all V_x (hAPP(c_Nat_OSuc,V_y) = hAPP(c_Nat_OSuc,V_x) -> V_x = V_y)) # label(fact_Suc__inject) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1051 (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_c_2),V_d_2) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_c,V_a_2),V_b_2) <-> V_d_2 = V_b_2 & V_c_2 = V_a_2))))) # label(fact_atLeastLessThan__eq__iff) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1052 (all V_y_2 all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_x_2),V_y_2)) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_x_2),V_y_2)) <-> V_x_2 = V_y_2)))) # label(fact_linorder__antisym__conv2) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1053 (all V_X_2 all V_r_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_c,T_c,V_r_2),V_X_2)),V_X_2)) -> hAPP(c_Relation_OImage(T_c,T_c,c_Transitive__Closure_Ortrancl(T_c,V_r_2)),V_X_2) = V_X_2)) # label(fact_Image__closed__trancl) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1054 (all V_A_2 all V_f_2 all T_c all T_d (c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)) = hAPP(c_Set_Oimage(T_d,T_c,V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool))) -> V_A_2 = 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_surj__image__vimage__eq) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1055 (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]. 5.89/5.93 1056 (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_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]. 5.89/5.93 1057 (all V_y all V_x all T_a (class_Groups_Olinordered__ab__group__add(T_a) -> 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)) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),V_y)))) # label(fact_minus__max__eq__min) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1058 (all V_a_2 all V_A_2 all T_c (class_Lattices_Olattice(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_c,c_Lattices_Osemilattice__inf__class_Oinf(T_c)),V_a_2),V_A_2) = c_Big__Operators_Olattice__class_OInf__fin(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_A_2))))) # label(fact_Inf__fin_Oeq__fold__idem_H) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1059 (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_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]. 5.89/5.93 1060 (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]. 5.89/5.93 1061 (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]. 5.89/5.93 1062 (all V_A_2 hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),V_A_2)) = V_A_2) # label(fact_transfer__int__nat__set__return__embed) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1063 (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]. 5.89/5.93 1064 (all V_xss_2 all T_d all V_xs_2 all T_c 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))) = 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))))) # label(fact_transpose__aux__max) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1065 (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]. 5.89/5.93 1066 (all V_y_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),c_List_Oreplicate(T_c,V_n_2,V_y_2)))) <-> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != V_n_2 & V_y_2 = V_x_2)) # label(fact_in__set__replicate) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1067 (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))) -> V_y_2 = hAPP(V_f_2,hAPP(c_Hilbert__Choice_Oinv__into(T_d,T_c,V_A_2,V_f_2),V_y_2)))) # label(fact_f__inv__into__f) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1068 (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]. 5.89/5.93 1069 (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]. 5.89/5.93 1070 (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]. 5.89/5.93 1071 (all V_Pa_2 all T_d all T_c c_HOL_OThe(tc_prod(T_c,T_d),hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,tc_HOL_Obool),V_Pa_2)) = c_HOL_OThe(tc_prod(T_c,T_d),hAPP(hAPP(c_COMBS(tc_prod(T_c,T_d),T_d,tc_HOL_Obool),hAPP(hAPP(c_COMBB(T_c,tc_fun(T_d,tc_HOL_Obool),tc_prod(T_c,T_d)),V_Pa_2),c_Product__Type_Ofst(T_c,T_d))),c_Product__Type_Osnd(T_c,T_d)))) # label(fact_The__split) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1072 (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]. 5.89/5.93 1073 (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]. 5.89/5.93 1074 (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]. 5.89/5.93 1075 (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]. 5.89/5.93 1076 (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]. 5.89/5.93 1077 (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]. 5.89/5.93 1078 (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]. 5.89/5.93 1079 (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]. 5.89/5.93 1080 (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]. 5.89/5.93 1081 (all V_x_2 all V_g_2 all T_f all V_f_2 all T_d all T_c all T_e hAPP(c_Product__Type_Oapfst(T_f,T_c,T_d,hAPP(c_Fun_Ocomp(T_e,T_c,T_f,V_f_2),V_g_2)),V_x_2) = hAPP(c_Product__Type_Oapfst(T_e,T_c,T_d,V_f_2),hAPP(c_Product__Type_Oapfst(T_f,T_e,T_d,V_g_2),V_x_2))) # label(fact_apfst__compose) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1082 (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)) -> (c_Groups_Oone__class_Oone(T_a) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_u),V_v) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__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]. 5.89/5.93 1083 (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]. 5.89/5.93 1084 (all V_w all T_a (class_Int_Onumber__ring(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Groups_Oone__class_Oone(T_a)),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w)) = hAPP(c_Int_Onumber__class_Onumber__of(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)),V_w)))) # label(fact_add__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1085 (all V_n all V_xs all T_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)),V_n)) -> V_xs = hAPP(hAPP(c_List_Otake(T_a),V_n),V_xs))) # label(fact_take__all) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1086 (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]. 5.89/5.93 1087 (all V_xs all V_n all T_a hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),hAPP(c_List_Orotate(T_a,V_n),V_xs))) # label(fact_length__rotate) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1088 (all V_k2_2 all V_k1_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit0(V_k1_2)),c_Int_OBit1(V_k2_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k1_2),V_k2_2)))) # label(fact_less__int__code_I14_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1089 (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]. 5.89/5.93 1090 (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),V_xs_2)) & -hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2))) <-> hBOOL(hAPP(c_List_Odistinct(T_c),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_c,T_d,V_f_2),V_x_2),V_xs_2)))))) # label(fact_distinct__insort) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1091 (all V_w_2 all V_x_2 all V_z_2 all V_y_2 all T_c (class_Fields_Ofield(T_c) -> (c_Groups_Ozero__class_Ozero(T_c) != V_y_2 -> (V_z_2 != c_Groups_Ozero__class_Ozero(T_c) -> (hAPP(hAPP(c_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) <-> 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)))))) # label(fact_frac__eq__eq) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1092 (all 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)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z_2)))) # label(fact_int__one__le__iff__zero__less) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1093 (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]. 5.89/5.93 1094 (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]. 5.89/5.93 1095 (all V_w all V_z (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_w),V_z)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_z),V_w)))) # label(fact_zle__linear) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1096 (all V_x all V_y all T_a (class_Lattices_Osemilattice__inf(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_y),V_x)) -> V_y = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y)))) # label(fact_inf__absorb2) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1097 (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))))))) <-> V_x_2 != c_Groups_Ozero__class_Ozero(T_c) | c_Groups_Ozero__class_Ozero(T_c) != V_y_2))) # label(fact_sum__power2__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1098 (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]. 5.89/5.93 1099 (all V_k hAPP(c_Code__Numeral_Onat__of,hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),V_k)) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_k)) # label(fact_nat__of__number) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1100 (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]. 5.89/5.93 1101 (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]. 5.89/5.93 1102 (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]. 5.89/5.93 1103 (all V_x_2 all V_n_2 all T_c hAPP(c_List_Oset(T_c),c_List_Oreplicate(T_c,hAPP(c_Nat_OSuc,V_n_2),V_x_2)) = hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) # label(fact_set__replicate__Suc) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1104 (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]. 5.89/5.93 1105 (all V_l all V_k hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Int_OBit0(V_k)),c_Int_OBit0(V_l)) = c_Int_OBit0(hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_k),V_l))) # label(fact_add__Bit0__Bit0) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1106 (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]. 5.89/5.93 1107 (all V_k c_Quickcheck_Obeyond(V_k,c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)) = c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)) # label(fact_beyond__zero) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1108 (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]. 5.89/5.93 1109 (all V_v_2 all V_c_2 all V_b_2 all V_a_2 all T_c all T_d all T_e (hAPP(c_Fun_Ocomp(T_d,T_d,T_c,c_Fun_Oid(T_d)),V_c_2) = hAPP(c_Fun_Ocomp(T_e,T_d,T_c,V_a_2),V_b_2) -> hAPP(V_c_2,V_v_2) = hAPP(V_a_2,hAPP(V_b_2,V_v_2)))) # label(fact_o__eq__id__dest) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1110 (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]. 5.89/5.93 1111 (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 (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)) & c_Fun_Oinj__on(T_c,T_d,B_f,V_A_2))))))) # label(fact_inj__on__iff__card__le) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1112 (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]. 5.89/5.93 1113 (all T_a (class_Rings_Osemiring__1(T_a) -> c_Groups_Ozero__class_Ozero(T_a) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_of__nat__0) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1114 (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),V_w_2)) <-> c_Int_Oiszero(T_c,hAPP(c_Int_Onumber__class_Onumber__of(T_c),c_Int_OBit0(V_w_2)))))) # label(fact_iszero__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1115 (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]. 5.89/5.93 1116 (all V_b_2 all V_f_2 all T_d all T_c all V_a_2 all V_x_2 ((V_x_2 != V_a_2 -> hAPP(V_f_2,V_x_2) = hAPP(c_Fun_Ofun__upd(T_c,T_d,V_f_2,V_a_2,V_b_2),V_x_2)) & (V_a_2 = V_x_2 -> hAPP(c_Fun_Ofun__upd(T_c,T_d,V_f_2,V_a_2,V_b_2),V_x_2) = V_b_2))) # label(fact_fun__upd__def) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1117 (all V_n hAPP(c_Nat_OSuc,V_n) != V_n) # label(fact_n__not__Suc__n) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1118 (all V_B_2 all V_A_2 all V_C_2 all T_c hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_C_2),V_A_2)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_C_2),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))) # label(fact_Diff__Int__distrib) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1119 (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]. 5.89/5.93 1120 (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]. 5.89/5.93 1121 (all T_d all T_c all V_Pa_2 ((all B_a all B_b hBOOL(hAPP(V_Pa_2,hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),B_a),B_b)))) <-> (all B_x1 hBOOL(hAPP(V_Pa_2,B_x1))))) # label(fact_split__paired__All) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1122 (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]. 5.89/5.93 1123 (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]. 5.89/5.93 1124 (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]. 5.89/5.93 1125 (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]. 5.89/5.93 1126 (all V_a all T_a (class_Divides_Osemiring__div(T_a) -> (V_a != c_Groups_Ozero__class_Ozero(T_a) -> c_Groups_Oone__class_Oone(T_a) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_a),V_a)))) # label(fact_div__self) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1127 (all V_fun2_2 all V_com_2 all V_fun1_2 all V_fa_2 all T_c 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)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_triple_Osize_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1128 (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]. 5.89/5.93 1129 (all V_A_2 all V_a_2 all T_c c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_A_2)) # label(fact_empty__not__insert) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1130 (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_member(T_c),V_x_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))),V_B_2))) & (-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)))) & (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_B_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))) <-> 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]. 5.89/5.93 1131 (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]. 5.89/5.93 1132 (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]. 5.89/5.93 1133 (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_a_2 = V_b_2)) # label(fact_singleton__inject) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1134 (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]. 5.89/5.93 1135 (all V_r_2 all T_c (c_Relation_Osym(T_c,V_r_2) <-> c_Relation_Osym(T_c,hAPP(c_Relation_Oconverse(T_c,T_c),V_r_2)))) # label(fact_sym__converse) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1136 (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]. 5.89/5.93 1137 (all V_b_2 all V_a_2 all T_c (class_Rings_Olinordered__idom(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_a_2)),V_b_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_a_2),V_b_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(c_Groups_Oabs__class_Oabs(T_c),V_a_2)),V_b_2))))) # label(fact_abs__less__iff) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1138 (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]. 5.89/5.93 1139 (all V_z_2 all V_m_2 (hAPP(c_Int_Onat,V_z_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_z_2)) <-> V_z_2 = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m_2))) # label(fact_int__eq__iff) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1140 (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]. 5.89/5.93 1141 (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]. 5.89/5.93 1142 (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]. 5.89/5.93 1143 (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]. 5.89/5.93 1144 (all V_xs all V_n all T_a hAPP(hAPP(c_List_Odrop(T_a),V_n),hAPP(c_List_Otl(T_a),V_xs)) = hAPP(hAPP(c_List_Odrop(T_a),hAPP(c_Nat_OSuc,V_n)),V_xs)) # label(fact_drop__Suc) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1145 (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]. 5.89/5.93 1146 (all V_n_2 hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n_2))) = hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),V_n_2)) # label(fact_SetInterval_Otransfer__nat__int__set__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1147 (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]. 5.89/5.93 1148 (all V_k hAPP(c_Code__Numeral_Oof__nat,hAPP(c_Nat_OSuc,hAPP(c_Code__Numeral_Onat__of,V_k))) = c_Code__Numeral_OSuc__code__numeral(V_k)) # label(fact_Suc__code__numeral__def) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1149 (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]. 5.89/5.93 1150 (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]. 5.89/5.93 1151 (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]. 5.89/5.93 1152 (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__add__class_Osetsum(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__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)))) # label(fact_transfer__int__nat__sum__prod2_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1153 (all V_z_2 all T_c (class_Rings_Olinordered__idom(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),hAPP(c_Int_Oring__1__class_Oof__int(T_c),V_z_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z_2))))) # label(fact_of__int__0__less__iff) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1154 (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_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_I38_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1155 (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]. 5.89/5.93 1156 (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]. 5.89/5.93 1157 (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]. 5.89/5.93 1158 (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),hAPP(V_f_2,V_x_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_A_2))) # label(fact_insert__image) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1159 (all V_m_2 all V_n_2 (V_m_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) & V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n_2)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m_2)))))) # label(fact_int__zle__neg) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1160 (all V_c all V_b all V_v all T_b (class_Rings_Osemiring(T_b) & class_Int_Onumber(T_b) -> 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)) = 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)))) # label(fact_right__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1161 (all V_j_2 all V_i_2 all V_k_2 hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i_2),V_k_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_j_2),V_k_2)) = hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Nat_Onat,hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),V_k_2)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_i_2),V_j_2))) # label(fact_image__add__atLeastLessThan) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1162 (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]. 5.89/5.93 1163 (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]. 5.89/5.93 1164 (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]. 5.89/5.93 1165 (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),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(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__iff1) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1166 (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]. 5.89/5.93 1167 (all V_r_2 all V_y_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_x_2),V_y_2)),hAPP(c_Relation_Oconverse(tc_List_Olist(T_c),tc_List_Olist(T_c)),c_List_Olistrel1(T_c,V_r_2)))) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_x_2),V_y_2)),c_List_Olistrel1(T_c,hAPP(c_Relation_Oconverse(T_c,T_c),V_r_2)))))) # label(fact_in__listrel1__converse) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1168 (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) -> (V_x_2 = V_w_2 | V_y_2 = V_z_2 <-> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_w_2),V_y_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_x_2),V_z_2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_w_2),V_z_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_x_2),V_y_2))))) # label(fact_crossproduct__eq) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1169 (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 -> hAPP(c_Product__Type_Ofst(tc_Nat_Onat,tc_Nat_Onat),V_qr_2) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) & (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != V_n_2 -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Product__Type_Osnd(tc_Nat_Onat,tc_Nat_Onat),V_qr_2))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Product__Type_Osnd(tc_Nat_Onat,tc_Nat_Onat),V_qr_2)),V_n_2))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Product__Type_Osnd(tc_Nat_Onat,tc_Nat_Onat),V_qr_2)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2),hAPP(c_Product__Type_Osnd(tc_Nat_Onat,tc_Nat_Onat),V_qr_2))))) & V_m_2 = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Product__Type_Ofst(tc_Nat_Onat,tc_Nat_Onat),V_qr_2)),V_n_2)),hAPP(c_Product__Type_Osnd(tc_Nat_Onat,tc_Nat_Onat),V_qr_2)))) # label(fact_divmod__nat__rel__def) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1170 (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]. 5.89/5.93 1171 (all V_n_2 all V_m_2 (hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m_2),V_n_2) = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) <-> 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) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n_2 & V_m_2 = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_one__is__add) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1172 (all V_B_2 all V_A_2 all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_c,c_Lattices_Osemilattice__inf__class_Oinf(T_c)),V_B_2),V_A_2) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_c),V_B_2),c_Complete__Lattice_OInf__class_OInf(T_c,V_A_2))))) # label(fact_inf__Inf__fold__inf) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1173 (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]. 5.89/5.93 1174 (all V_c all V_b all V_a all T_a (class_Orderings_Oorder(T_a) -> (V_a = V_b -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_c),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_c),V_a)))))) # label(fact_xt1_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1175 (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]. 5.89/5.93 1176 (all V_n_2 all V_m_2 all T_c c_List_Orotate(T_c,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m_2),V_n_2)) = 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))) # label(fact_rotate__add) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1177 (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)),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]. 5.89/5.93 1178 (all V_k c_Int_OBit1(V_k) != c_Int_OPls) # label(fact_rel__simps_I46_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1179 (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]. 5.89/5.93 1180 (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)) -> 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),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)))))) # label(fact_min__max_OInf__fin_Oinsert__remove) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1181 (all V_x V_x = hAPP(c_Code__Numeral_Oof__nat,hAPP(c_Code__Numeral_Onat__of,V_x))) # label(fact_nat__of__inverse) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1182 (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]. 5.89/5.93 1183 (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]. 5.89/5.93 1184 (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]. 5.89/5.93 1185 (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]. 5.89/5.93 1186 (all T_a all V_xs (V_xs != c_List_Olist_ONil(T_a) -> c_List_Olast(T_a,V_xs) = hAPP(c_List_Ohd(T_a),hAPP(c_List_Orev(T_a),V_xs)))) # label(fact_hd__rev) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1187 (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]. 5.89/5.93 1188 (all V_R_2 all V_n_2 all T_c hAPP(hAPP(c_Relation_Orel__comp(T_c,T_c,T_c),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_c,T_c),tc_HOL_Obool),V_n_2),V_R_2)),V_R_2) = hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_c,T_c),tc_HOL_Obool),hAPP(c_Nat_OSuc,V_n_2)),V_R_2)) # label(fact_relpow_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1189 (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]. 5.89/5.93 1190 (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]. 5.89/5.93 1191 (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]. 5.89/5.93 1192 (all V_B_2 all V_A_2 all T_d all T_c hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Relation_ODomain(T_c,T_d),V_A_2)),hAPP(c_Relation_ODomain(T_c,T_d),V_B_2)) = hAPP(c_Relation_ODomain(T_c,T_d),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),V_A_2),V_B_2))) # label(fact_Domain__Un__eq) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1193 (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]. 5.89/5.93 1194 (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]. 5.89/5.93 1195 (all V_n_2 (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != V_n_2 <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n_2)))) # label(fact_neq0__conv) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1196 (all V_Pa_2 all V_Qa_2 (hBOOL(hAPP(hAPP(c_fdisj,V_Pa_2),V_Qa_2)) | -hBOOL(V_Qa_2))) # label(help_c__fdisj__2) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1197 (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_2) = hAPP(c_Int_Onat,V_z_H_2) <-> V_z_2 = V_z_H_2)))) # label(fact_eq__nat__nat__iff) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1198 (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]. 5.89/5.93 1199 (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_Oabs__class_Oabs(T_a),V_a) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)))) # label(fact_abs__of__nonpos) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1200 (all V_b all V_a all T_a (class_Rings_Odivision__ring(T_a) -> hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_b)) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),V_b))) # label(fact_minus__divide__left) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1201 (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]. 5.89/5.93 1202 (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]. 5.89/5.93 1203 (all V_x_2 all V_r_2 all T_d all T_c (hBOOL(hAPP(c_Predicate_ODomainP(T_c,T_d,hAPP(hAPP(c_COMBC(T_c,tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_fun(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool)),T_c),c_COMBC(T_d,tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_prod(T_c,T_d)),tc_fun(T_d,tc_fun(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),tc_HOL_Obool)),T_c),hAPP(c_COMBB(tc_prod(T_c,T_d),tc_fun(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),tc_HOL_Obool),T_d),c_member(tc_prod(T_c,T_d)))),c_Product__Type_OPair(T_c,T_d)))),V_r_2)),V_x_2)) <-> hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_Relation_ODomain(T_c,T_d),V_r_2))))) # label(fact_DomainP__Domain__eq) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1204 (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(V_f_2,hAPP(hAPP(c_List_Onth(T_c),V_xs_2),V_n_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))) # label(fact_nth__map) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1205 (all V_a_2 all V_y_2 all V_A_2 all V_z_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofun__left__comm(T_c,T_d,V_f_2) -> (hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_c,T_d,V_f_2,V_z_2,V_A_2),V_y_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> (exists B_y_H (hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_c,T_d,V_f_2,V_z_2,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))),B_y_H)) & hAPP(hAPP(V_f_2,V_a_2),B_y_H) = V_y_2)))))) # label(fact_fun__left__comm_Ofold__graph__insertE__aux) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1206 (all V_l_2 (V_l_2 = c_Int_OMin <-> c_Int_OBit1(V_l_2) = c_Int_OMin)) # label(fact_rel__simps_I43_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1207 (all V_x_2 all V_f_2 all T_e all T_c all T_d hAPP(c_Product__Type_Osnd(T_d,T_c),hAPP(c_Product__Type_Oapsnd(T_e,T_c,T_d,V_f_2),V_x_2)) = hAPP(V_f_2,hAPP(c_Product__Type_Osnd(T_d,T_e),V_x_2))) # label(fact_snd__apsnd) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1208 (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]. 5.89/5.93 1209 (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_Olinorder__class_OMax(T_c,V_A_2)),V_A_2))))))) # label(fact_Max_Oclosed) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1210 (all T_c (class_Power_Opower(T_c) -> c_Power_Opower__class_Opower(T_c) = c_Power_Opower_Opower(T_c,c_Groups_Oone__class_Oone(T_c),c_Groups_Otimes__class_Otimes(T_c)))) # label(fact_power__power__power) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1211 (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]. 5.89/5.93 1212 (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]. 5.89/5.93 1213 (all V_Pa_2 all T_d all T_c 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)) = 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)))) # label(fact_Domain__Collect__split) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1214 (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]. 5.89/5.93 1215 (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]. 5.89/5.93 1216 (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]. 5.89/5.93 1217 (all V_xy all V_i all V_ys all V_xs all T_b all T_a hAPP(hAPP(c_List_Olist__update(tc_prod(T_a,T_b),hAPP(hAPP(c_List_Ozip(T_a,T_b),V_xs),V_ys)),V_i),V_xy) = hAPP(hAPP(c_List_Ozip(T_a,T_b),hAPP(hAPP(c_List_Olist__update(T_a,V_xs),V_i),hAPP(c_Product__Type_Ofst(T_a,T_b),V_xy))),hAPP(hAPP(c_List_Olist__update(T_b,V_ys),V_i),hAPP(c_Product__Type_Osnd(T_a,T_b),V_xy)))) # label(fact_update__zip) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1218 (all V_r_2 all V_q_2 all T_c hAPP(hAPP(c_HOL_OLet(tc_Int_Oint,T_c),c_Nitpick_Oint__lcm(hAPP(c_Nitpick_Odenom(T_c),V_q_2),hAPP(c_Nitpick_Odenom(T_c),V_r_2))),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_Int_Oint,T_c),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_fun(tc_Int_Oint,T_c),tc_Int_Oint),c_Nitpick_Ofrac(T_c)),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint),tc_Int_Oint),c_Groups_Oplus__class_Oplus(tc_Int_Oint)),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_Int_Oint,tc_Int_Oint),hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Nitpick_Onum(T_c),V_q_2))),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Int_Oint,tc_Int_Oint),c_Divides_Odiv__class_Odiv(tc_Int_Oint)),hAPP(c_Nitpick_Odenom(T_c),V_q_2))))),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_Int_Oint,tc_Int_Oint),hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Nitpick_Onum(T_c),V_r_2))),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Int_Oint,tc_Int_Oint),c_Divides_Odiv__class_Odiv(tc_Int_Oint)),hAPP(c_Nitpick_Odenom(T_c),V_r_2)))))),c_COMBI(tc_Int_Oint))) = c_Nitpick_Oplus__frac(T_c,V_q_2,V_r_2)) # label(fact_Nitpick_Oplus__frac__def) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1219 (all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> (hAPP(c_Set_Oimage(T_c,T_d,V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) = c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool)) -> c_Fun_Obij__betw(T_c,T_d,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool)))))) # label(fact_bijI) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1220 (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_linorder__neq__iff) # label(axiom) # label(non_clause). [assumption]. 5.89/5.93 1221 (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]. 5.89/5.94 1222 (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]. 5.89/5.94 1223 (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]. 5.89/5.94 1224 (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]. 5.89/5.94 1225 (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]. 5.89/5.94 1226 (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]. 5.89/5.94 1227 (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]. 5.89/5.94 1228 (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]. 5.89/5.94 1229 (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]. 5.89/5.94 1230 (all V_m_2 all V_n_2 hAPP(c_List_Oset(tc_Nat_Onat),c_List_Oupt(V_n_2,hAPP(c_Nat_OSuc,V_m_2))) = c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_n_2,V_m_2)) # label(fact_atLeastAtMost__upt) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1231 (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_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)) & (V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> hAPP(c_Set_Oimage(T_c,T_d,hAPP(c_COMBK(T_d,T_c),V_c_2)),V_A_2) = c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool))))) # label(fact_image__constant__conv) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1232 (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]. 5.89/5.94 1233 (all V_nat_H_1 hAPP(c_Nat_OSuc,V_nat_H_1) != c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_nat_Osimps_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1234 (all V_x_2 all T_d all T_c (class_Orderings_Obot(T_c) -> c_Orderings_Obot__class_Obot(T_c) = hAPP(c_Orderings_Obot__class_Obot(tc_fun(T_d,T_c)),V_x_2))) # label(fact_bot__fun__def) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1235 (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)),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)),c_Transitive__Closure_Ortrancl(T_c,V_R_2))))))) # label(fact_tranclD) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1236 (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]. 5.89/5.94 1237 (all V_w all V_v (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))) -> hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)) = c_Divides_OposDivAlg(hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))) -> hAPP(c_Divides_Oadjust(hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)),c_Divides_OposDivAlg(hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)))) = c_Divides_OposDivAlg(hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))))) # label(fact_posDivAlg__eqn__number__of) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1238 (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]. 5.89/5.94 1239 (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]. 5.89/5.94 1240 (all V_B_2 all V_A_2 all T_e all V_f_2 all T_d all T_c (c_Fun_Obij__betw(T_c,T_d,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool))) -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_e,tc_fun(T_d,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool),T_e),c_Set_Oimage(T_c,T_d,V_f_2)),V_B_2)) = hAPP(c_Set_Oimage(T_c,T_d,V_f_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_e,tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)))) # label(fact_bij__image__INT) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1241 (all V_ry all V_rx all V_lx all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_rx),V_ry)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_rx),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),V_ry)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1242 (all V_a all T_a (class_Lattices_Oab__semigroup__idem__mult(T_a) -> V_a = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_a))) # label(fact_times_Oidem) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1243 (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]. 5.89/5.94 1244 (all V_f_2 all V_B_2 all V_A_2 all T_c all T_d (class_Complete__Lattice_Ocomplete__lattice(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_d,hAPP(hAPP(c_COMBB(T_d,tc_fun(T_d,T_d),T_c),c_Lattices_Osemilattice__inf__class_Oinf(T_d)),V_f_2)),V_B_2),V_A_2) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_d),V_B_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,T_d),V_A_2),V_f_2))))) # label(fact_inf__INFI__fold__inf) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1245 (all V_y all V_x (V_x != V_y -> (-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]. 5.89/5.94 1246 (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]. 5.89/5.94 1247 (all V_l_2 hAPP(hAPP(c_List_Omap(tc_Int_Oint,tc_Nat_Onat),c_Int_Onat),V_l_2) = c_List_Oreturn__list(V_l_2)) # label(fact_return__list__def) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1248 (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]. 5.89/5.94 1249 (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]. 5.89/5.94 1250 (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]. 5.89/5.94 1251 (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]. 5.89/5.94 1252 (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]. 5.89/5.94 1253 (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]. 5.89/5.94 1254 (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_y_2 = V_x_2 <-> hAPP(V_Abs_2,V_x_2) = hAPP(V_Abs_2,V_y_2)))))) # label(fact_type__definition_OAbs__inject) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1255 (all T_c c_Fun_Oid(tc_List_Olist(T_c)) = hAPP(c_List_Omap(T_c,T_c),c_Fun_Oid(T_c))) # label(fact_List_Omap_Oid) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1256 (all V_B_2 all V_A_2 all T_d all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(T_c,tc_HOL_Obool)),V_A_2),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_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))) # label(fact_Compl__UN) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1257 (all V_a_2 all V_times_2 all V_one_2 all T_c V_one_2 = hAPP(hAPP(c_Power_Opower_Opower(T_c,V_one_2,V_times_2),V_a_2),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_power_Opower_Opower__0) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1258 (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) -> hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_x) = V_y)))) # label(fact_compl__unique) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1259 (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]. 5.89/5.94 1260 (all V_x_2 all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(c_Nat_OSuc,hAPP(c_Finite__Set_Ocard(T_c),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))))) = hAPP(c_Finite__Set_Ocard(T_c),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)))) # label(fact_card__insert) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1261 (all V_A_2 all V_f_2 all T_c all T_d hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBC(T_d,tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(T_c,tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),T_d),c_Set_Oinsert(T_c)),V_f_2)),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) = hAPP(c_Set_Oimage(T_d,T_c,V_f_2),V_A_2)) # label(fact_image__eq__UN) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1262 (all V_A_2 all T_c ((all B_x -hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2))) <-> V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) # label(fact_all__not__in__conv) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1263 (all V_g_2 all V_f_2 all T_f all T_e all T_d all T_c hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,T_e),hAPP(hAPP(c_COMBB(T_f,tc_fun(T_d,T_e),T_c),V_f_2),V_g_2)) = 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))) # label(fact_split__comp__eq) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1264 (all V_B_2 all V_A_2 all V_a_2 all T_c (-hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))) # label(fact_Int__insert__right__if0) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1265 (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]. 5.89/5.94 1266 (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]. 5.89/5.94 1267 (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]. 5.89/5.94 1268 (all V_b_2 all V_a_2 all T_c (class_Rings_Olinordered__ring__strict(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_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)),V_b_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_b_2)),c_Groups_Ozero__class_Ozero(T_c)))))) # label(fact_mult__le__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1269 (all V_a_2 all T_c (class_Rings_Olinordered__idom(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_a_2),c_Groups_Ozero__class_Ozero(T_c))) <-> c_Groups_Osgn__class_Osgn(T_c,V_a_2) = hAPP(c_Groups_Ouminus__class_Ouminus(T_c),c_Groups_Oone__class_Oone(T_c))))) # label(fact_sgn__1__neg) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1270 (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]. 5.89/5.94 1271 (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]. 5.89/5.94 1272 (all V_b c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b)) # label(fact_zdiv__zero) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1273 (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]. 5.89/5.94 1274 (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]. 5.89/5.94 1275 (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]. 5.89/5.94 1276 (all V_A_2 all T_d all V_I_2 all V_k_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_k_2),V_I_2)) -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_HOL_Obool)),V_I_2),V_A_2) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_d,tc_HOL_Obool)),hAPP(V_A_2,V_k_2)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_HOL_Obool)),V_I_2),V_A_2)))) # label(fact_INT__absorb) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1277 (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__sup__class_Osup(T_c)),V_f_2)),c_Orderings_Obot__class_Obot(T_c)),V_xs_2) = 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))) # label(fact_SUPR__set__fold) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1278 (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]. 5.89/5.94 1279 (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]. 5.89/5.94 1280 (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]. 5.89/5.94 1281 (all V_a_2 all T_c (class_Groups_Olinordered__ab__group__add(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_a_2)),V_a_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_a_2))))) # label(fact_neg__less__nonneg) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1282 (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]. 5.89/5.94 1283 (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]. 5.89/5.94 1284 (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_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_set__eq__subset) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1285 (all V_b all V_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a)) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_b))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_a)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_b),V_a))) # label(fact_pos__zdiv__mult__2) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1286 (all V_t_2 all V_B_2 all V_D_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_D_2)) -> (all B_x ((all B_xa (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xa),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),V_D_2))) -> (all B_xb (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xb),V_B_2)) -> B_x != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B_xb),B_xa))))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B_x),V_t_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B_x),V_D_2)),V_t_2))))))) # label(fact_bset_I6_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1287 (all V_n_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),hAPP(c_Nat_OSuc,V_n_2))) -> c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_m_2,hAPP(c_Nat_OSuc,V_n_2)) = hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_n_2)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_m_2,V_n_2)))) # label(fact_atLeastAtMostSuc__conv) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1288 (all V_r_2 all T_c hAPP(hAPP(c_Relation_Orel__comp(T_c,T_c,T_c),c_Transitive__Closure_Ortrancl(T_c,V_r_2)),V_r_2) = c_Transitive__Closure_Otrancl(T_c,V_r_2)) # label(fact_trancl__unfold__right) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1289 (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]. 5.89/5.94 1290 (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]. 5.89/5.94 1291 (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]. 5.89/5.94 1292 (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]. 5.89/5.94 1293 (all V_XS_2 all V_A_2 all T_c hAPP(c_Set_OCollect(tc_List_Olist(T_c)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool,tc_List_Olist(T_c)),c_HOL_OEx(T_c)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(tc_List_Olist(T_c),tc_HOL_Obool)),tc_fun(T_c,tc_HOL_Obool),tc_List_Olist(T_c)),hAPP(c_COMBB(tc_fun(tc_List_Olist(T_c),tc_HOL_Obool),tc_HOL_Obool,T_c),c_HOL_OEx(tc_List_Olist(T_c)))),hAPP(hAPP(c_COMBC(tc_List_Olist(T_c),tc_fun(T_c,tc_fun(tc_List_Olist(T_c),tc_HOL_Obool)),tc_fun(T_c,tc_fun(tc_List_Olist(T_c),tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(tc_fun(tc_List_Olist(T_c),tc_HOL_Obool),tc_fun(tc_List_Olist(T_c),tc_HOL_Obool))),tc_fun(tc_fun(T_c,tc_fun(tc_List_Olist(T_c),tc_HOL_Obool)),tc_fun(T_c,tc_fun(tc_List_Olist(T_c),tc_HOL_Obool))),tc_List_Olist(T_c)),c_COMBS(T_c,tc_fun(tc_List_Olist(T_c),tc_HOL_Obool),tc_fun(tc_List_Olist(T_c),tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(tc_List_Olist(T_c),tc_fun(tc_HOL_Obool,tc_HOL_Obool))),tc_fun(T_c,tc_fun(tc_fun(tc_List_Olist(T_c),tc_HOL_Obool),tc_fun(tc_List_Olist(T_c),tc_HOL_Obool))),tc_List_Olist(T_c)),hAPP(c_COMBB(tc_fun(tc_List_Olist(T_c),tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_List_Olist(T_c),tc_HOL_Obool),tc_fun(tc_List_Olist(T_c),tc_HOL_Obool)),T_c),c_COMBS(tc_List_Olist(T_c),tc_HOL_Obool,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(tc_List_Olist(T_c),tc_HOL_Obool)),tc_fun(T_c,tc_fun(tc_List_Olist(T_c),tc_fun(tc_HOL_Obool,tc_HOL_Obool))),tc_List_Olist(T_c)),hAPP(c_COMBB(tc_fun(tc_List_Olist(T_c),tc_HOL_Obool),tc_fun(tc_List_Olist(T_c),tc_fun(tc_HOL_Obool,tc_HOL_Obool)),T_c),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_List_Olist(T_c)),c_fconj))),hAPP(hAPP(c_COMBC(tc_List_Olist(T_c),tc_fun(T_c,tc_fun(tc_List_Olist(T_c),tc_List_Olist(T_c))),tc_fun(T_c,tc_fun(tc_List_Olist(T_c),tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(tc_List_Olist(T_c),tc_List_Olist(T_c)),tc_fun(tc_List_Olist(T_c),tc_HOL_Obool)),tc_fun(tc_fun(T_c,tc_fun(tc_List_Olist(T_c),tc_List_Olist(T_c))),tc_fun(T_c,tc_fun(tc_List_Olist(T_c),tc_HOL_Obool))),tc_List_Olist(T_c)),c_COMBB(tc_fun(tc_List_Olist(T_c),tc_List_Olist(T_c)),tc_fun(tc_List_Olist(T_c),tc_HOL_Obool),T_c)),hAPP(hAPP(c_COMBB(tc_fun(tc_List_Olist(T_c),tc_HOL_Obool),tc_fun(tc_fun(tc_List_Olist(T_c),tc_List_Olist(T_c)),tc_fun(tc_List_Olist(T_c),tc_HOL_Obool)),tc_List_Olist(T_c)),c_COMBB(tc_List_Olist(T_c),tc_HOL_Obool,tc_List_Olist(T_c))),c_fequal))),c_List_Olist_OCons(T_c)))))),hAPP(hAPP(c_COMBC(T_c,tc_fun(tc_List_Olist(T_c),tc_HOL_Obool),tc_fun(tc_List_Olist(T_c),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_fun(tc_fun(tc_List_Olist(T_c),tc_HOL_Obool),tc_fun(tc_List_Olist(T_c),tc_HOL_Obool)),T_c),c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_List_Olist(T_c))),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_c),c_fconj),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_A_2)))),hAPP(hAPP(c_COMBC(tc_List_Olist(T_c),tc_fun(tc_List_Olist(T_c),tc_HOL_Obool),tc_HOL_Obool),c_member(tc_List_Olist(T_c))),V_XS_2)))))) = c_List_Oset__Cons(T_c,V_A_2,V_XS_2)) # label(fact_set__Cons__def) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1294 (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)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_c,V_m_2),V_u_2)))) # label(fact_ivl__disj__int_I9_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1295 (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]. 5.89/5.94 1296 (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]. 5.89/5.94 1297 (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]. 5.89/5.94 1298 (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]. 5.89/5.94 1299 (all V_n_2 hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_n_2)) = hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Nat_Onat,c_Nat_OSuc),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),V_n_2)))) # label(fact_lessThan__Suc__eq__insert__0) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1300 (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]. 5.89/5.94 1301 (all V_zs_2 all V_ys_2 all V_xs_2 all V_x_2 all T_c (hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2) = hAPP(hAPP(c_List_Oappend(T_c),V_ys_2),V_zs_2) <-> (exists B_ys_H (hAPP(hAPP(c_List_Oappend(T_c),B_ys_H),V_zs_2) = V_xs_2 & hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),B_ys_H) = V_ys_2)) | c_List_Olist_ONil(T_c) = V_ys_2 & V_zs_2 = hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2))) # label(fact_Cons__eq__append__conv) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1302 (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]. 5.89/5.94 1303 (all V_C_2 all V_B_2 all V_A_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_C_2))),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_C_2),V_A_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_C_2))),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_C_2),V_A_2))) # label(fact_Un__Int__crazy) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1304 (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]. 5.89/5.94 1305 (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]. 5.89/5.94 1306 (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]. 5.89/5.94 1307 (all V_b_2 all V_a_2 all V_c_2 all T_c (class_Rings_Olinordered__ring__strict(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_c_2),c_Groups_Ozero__class_Ozero(T_c))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_c_2),V_a_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_c_2),V_b_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_b_2),V_a_2)))))) # label(fact_mult__le__cancel__left__neg) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1308 (all V_n hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),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))) # label(fact_of__int__int__eq) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1309 (all V_x all V_m all V_n all T_a c_List_Oreplicate(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),V_m),V_x) = hAPP(hAPP(c_List_Oappend(T_a),c_List_Oreplicate(T_a,V_n,V_x)),c_List_Oreplicate(T_a,V_m,V_x))) # label(fact_replicate__add) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1310 (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]. 5.89/5.94 1311 (all V_x_2 all T_c (class_Rings_Oring__1__no__zero__divisors(T_c) -> (c_Groups_Oone__class_Oone(T_c) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_x_2),V_x_2) <-> hAPP(c_Groups_Ouminus__class_Ouminus(T_c),c_Groups_Oone__class_Oone(T_c)) = V_x_2 | V_x_2 = c_Groups_Oone__class_Oone(T_c)))) # label(fact_square__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1312 (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_y_2)) <-> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_c),V_x_2),V_y_2) = V_x_2))) # label(fact_le__iff__inf) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1313 (all V_F_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofolding(T_c,T_d,V_f_2,V_F_2) -> c_Fun_Oid(T_d) = hAPP(V_F_2,c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))) # label(fact_folding_Oempty) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1314 (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(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__eq(T_c),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]. 5.89/5.94 1315 (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]. 5.89/5.94 1316 (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]. 5.89/5.94 1317 (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]. 5.89/5.94 1318 (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]. 5.89/5.94 1319 (all V_xs_2 all T_c (V_xs_2 = c_List_Oremdups(T_c,V_xs_2) <-> hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),c_List_Oremdups(T_c,V_xs_2)))) # label(fact_length__remdups__eq) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1320 (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]. 5.89/5.94 1321 (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)) -> (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(c_Finite__Set_Ofold1(T_c,c_Groups_Otimes__class_Otimes(T_c)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(c_Finite__Set_Ofold1(T_c,c_Groups_Otimes__class_Otimes(T_c)),V_A_2)),hAPP(c_Finite__Set_Ofold1(T_c,c_Groups_Otimes__class_Otimes(T_c)),V_B_2))))))))) # label(fact_fold1__Un) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1322 (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]. 5.89/5.94 1323 (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]. 5.89/5.94 1324 (all V_a all T_a (class_Divides_Osemiring__div(T_a) -> V_a = c_Divides_Odiv__class_Omod(T_a,V_a,c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_mod__by__0) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1325 (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]. 5.89/5.94 1326 (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]. 5.89/5.94 1327 (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]. 5.89/5.94 1328 (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]. 5.89/5.94 1329 (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]. 5.89/5.94 1330 (all V_z all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_y),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),V_z)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_y),V_z)))) # label(fact_min__max_Oinf__left__commute) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1331 (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]. 5.89/5.94 1332 (all V_x all T_a c_List_Oremove1(T_a,V_x,c_List_Olist_ONil(T_a)) = c_List_Olist_ONil(T_a)) # label(fact_remove1_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1333 (all V_t_2 all V_A_2 all V_D_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_D_2)) -> (all B_x ((all B_xa (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xa),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),V_D_2))) -> (all B_xb (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xb),V_A_2)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B_xb),B_xa) != B_x)))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_t_2),B_x)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_t_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B_x),V_D_2)))))))) # label(fact_aset_I8_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1334 (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_equation__minus__iff) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1335 (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),V_xs_2) = c_List_OdropWhile(T_c,V_Pa_2,hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2))) & (hBOOL(hAPP(V_Pa_2,V_x_2)) -> c_List_OdropWhile(T_c,V_Pa_2,hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2)) = c_List_OdropWhile(T_c,V_Pa_2,V_xs_2)))) # label(fact_dropWhile_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1336 (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]. 5.89/5.94 1337 (all V_i_2 all V_xq_2 all V_f_2 all T_c all T_d c_Lazy__Sequence_Omap(T_d,T_c,V_f_2,hAPP(V_xq_2,V_i_2)) = c_New__DSequence_Opos__map(T_d,T_c,V_f_2,V_xq_2,V_i_2)) # label(fact_New__DSequence_Opos__map__def) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1338 (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]. 5.89/5.94 1339 (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]. 5.89/5.94 1340 (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_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_A_2)) # label(fact_insert__Diff) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1341 (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]. 5.89/5.94 1342 (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]. 5.89/5.94 1343 (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]. 5.89/5.94 1344 (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]. 5.89/5.94 1345 (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]. 5.89/5.94 1346 (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]. 5.89/5.94 1347 (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]. 5.89/5.94 1348 (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]. 5.89/5.94 1349 (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]. 5.89/5.94 1350 (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]. 5.89/5.94 1351 (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),V_y),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_x),V_z)))) # label(fact_min__max_Osup__left__commute) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1352 (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]. 5.89/5.94 1353 (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)))) -> V_x_2 = c_Big__Operators_Olinorder__class_OMax(T_c,V_A_2)) & (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) -> c_Big__Operators_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))))))))))) # label(fact_Max_Oremove) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1354 (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]. 5.89/5.94 1355 (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_x_2 = V_y_2)))) # label(fact_fun__left__comm_Ofold__graph__determ) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1356 (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]. 5.89/5.94 1357 (all V_f_2 all V_m_2 all V_n_2 all T_c 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)) = 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)) # label(fact_funpow__mult) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1358 (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]. 5.89/5.94 1359 (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]. 5.89/5.94 1360 (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]. 5.89/5.94 1361 (all V_a_2 all V_r_2 all T_c (hBOOL(hAPP(c_Wellfounded_Oaccp(T_c,V_r_2),V_a_2)) <-> (all B_x (hBOOL(hAPP(hAPP(V_r_2,B_x),V_a_2)) -> hBOOL(hAPP(c_Wellfounded_Oaccp(T_c,V_r_2),B_x)))))) # label(fact_accp_Osimps) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1362 (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]. 5.89/5.94 1363 (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]. 5.89/5.94 1364 (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]. 5.89/5.94 1365 (all V_A_2 all V_I_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(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(T_c,tc_HOL_Obool)),V_I_2),V_A_2))) <-> (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))))))) # label(fact_INT__subset__iff) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1366 (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]. 5.89/5.94 1367 (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]. 5.89/5.94 1368 (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]. 5.89/5.94 1369 (all V_r_2 all T_c (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))) <-> c_Predicate_Osymp(T_c,V_r_2))) # label(fact_symp__def) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1370 (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]. 5.89/5.94 1371 (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]. 5.89/5.94 1372 (all V_x all T_a hAPP(c_Fun_Oid(T_a),V_x) = V_x) # label(fact_id__apply) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1373 (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]. 5.89/5.94 1374 (all V_y_2 all V_n_2 all V_x_2 all V_m_2 all T_c ((V_m_2 != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> V_y_2 = V_x_2) & V_m_2 = V_n_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]. 5.89/5.94 1375 (all V_code__numeral_H_1 c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != c_Code__Numeral_OSuc__code__numeral(V_code__numeral_H_1)) # label(fact_code__numeral_Osimps_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1376 (all V_n all T_a all V_xs (V_xs != c_List_Olist_ONil(T_a) -> 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]. 5.89/5.94 1377 (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(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) = 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))))) # label(fact_folding__image_Odistrib) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1378 (all V_b all V_a all T_b all T_a hAPP(c_Product__Type_Ofst(T_a,T_b),hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_a),V_b)) = V_a) # label(fact_fst__conv) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1379 (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]. 5.89/5.94 1380 (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]. 5.89/5.94 1381 (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_fun__Compl__def) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1382 (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]. 5.89/5.94 1383 (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]. 5.89/5.94 1384 (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),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__eq(tc_Nat_Onat),V_i_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j_2),V_k_2)))))) # label(fact_le__diff__conv2) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1385 (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_2 = V_a_H_2 & V_b_2 = V_b_H_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]. 5.89/5.94 1386 (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_y_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_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_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]. 5.89/5.94 1387 (all V_x all V_ys all V_xs all T_a all V_n ((-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))) -> hAPP(hAPP(c_List_Olist__update(T_a,hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys)),V_n),V_x) = hAPP(hAPP(c_List_Oappend(T_a),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))) & (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)))) # label(fact_list__update__append) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1388 (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]. 5.89/5.94 1389 (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]. 5.89/5.94 1390 (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]. 5.89/5.94 1391 (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),V_v)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H)),V_k)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_v),V_v_H))),V_k)) & (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]. 5.89/5.94 1392 (all V_xs_2 all T_c all V_x_2 all V_Pa_2 ((-hBOOL(hAPP(V_Pa_2,V_x_2)) -> c_List_OtakeWhile(T_c,V_Pa_2,hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2)) = c_List_Olist_ONil(T_c)) & (hBOOL(hAPP(V_Pa_2,V_x_2)) -> c_List_OtakeWhile(T_c,V_Pa_2,hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2)) = hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),c_List_OtakeWhile(T_c,V_Pa_2,V_xs_2))))) # label(fact_takeWhile_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1393 (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]. 5.89/5.94 1394 (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 hAPP(c_Int_ORep__Integ,B_x) != V_y_2))) # label(fact_Rep__Integ__cases) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1395 (all V_b all V_a all T_a (class_Orderings_Oorder(T_a) -> (V_a != V_b -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_a),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),V_b)))))) # label(fact_order__neq__le__trans) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1396 (all V_b_2 all V_a_2 all T_c (class_Groups_Oordered__ab__group__add(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_a_2),V_b_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_c),V_a_2),V_b_2)),c_Groups_Ozero__class_Ozero(T_c)))))) # label(fact_less__iff__diff__less__0) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1397 (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]. 5.89/5.94 1398 (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]. 5.89/5.94 1399 (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]. 5.89/5.94 1400 (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]. 5.89/5.94 1401 (all V_k all V_n all V_m hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n)),V_k) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_k)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n),V_k))) # label(fact_add__mult__distrib) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1402 (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]. 5.89/5.94 1403 (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]. 5.89/5.94 1404 (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]. 5.89/5.94 1405 (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),hAPP(hAPP(c_List_Oappend(T_d),V_xs_2),V_ys_2)) = 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))) # label(fact_map__append) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1406 (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(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)) = 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))))))) # label(fact_fold__image__Un__one) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1407 (all V_m all V_n hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Code__Numeral_Ocode__numeral),V_n),V_m) = 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)))) # label(fact_minus__code__numeral__def) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1408 (all V_f_2 all V_F_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_F_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_F_2)) -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(V_f_2,B_x))) <-> 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]. 5.89/5.94 1409 (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]. 5.89/5.94 1410 (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(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),c_List_Oremove1(T_c,V_x_2,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))))) # label(fact_length__remove1) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1411 (all V_n_2 all V_m_2 (V_m_2 = V_n_2 | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),hAPP(c_Nat_OSuc,V_n_2))))) # label(fact_less__Suc__eq) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1412 (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]. 5.89/5.94 1413 (all V_xs_2 all V_x_2 all T_c hAPP(hAPP(c_HOL_OLet(tc_List_Olist(tc_List_Olist(T_c)),tc_List_Olist(tc_List_Olist(T_c))),c_Enum_Osublists(T_c,V_xs_2)),hAPP(hAPP(c_COMBS(tc_List_Olist(tc_List_Olist(T_c)),tc_List_Olist(tc_List_Olist(T_c)),tc_List_Olist(tc_List_Olist(T_c))),hAPP(hAPP(c_COMBB(tc_List_Olist(tc_List_Olist(T_c)),tc_fun(tc_List_Olist(tc_List_Olist(T_c)),tc_List_Olist(tc_List_Olist(T_c))),tc_List_Olist(tc_List_Olist(T_c))),c_List_Oappend(tc_List_Olist(T_c))),hAPP(c_List_Omap(tc_List_Olist(T_c),tc_List_Olist(T_c)),hAPP(c_List_Olist_OCons(T_c),V_x_2)))),c_COMBI(tc_List_Olist(tc_List_Olist(T_c))))) = c_Enum_Osublists(T_c,hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2))) # label(fact_sublists_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.94 1414 (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]. 5.89/5.95 1415 (all V_B_2 all V_t_2 all V_D_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_D_2)) -> (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),V_t_2),V_B_2)) -> (all B_x ((all B_xa (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xa),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),V_D_2))) -> (all B_xb (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xb),V_B_2)) -> B_x != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B_xb),B_xa))))) -> (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]. 5.89/5.95 1416 (all V_xs all V_i all T_a hAPP(c_List_Orev(T_a),hAPP(hAPP(c_List_Odrop(T_a),V_i),V_xs)) = hAPP(hAPP(c_List_Otake(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)),V_i)),hAPP(c_List_Orev(T_a),V_xs))) # label(fact_rev__drop) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1417 (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]. 5.89/5.95 1418 (all V_A_2 all V_B_2 all V_x_2 all T_c ((hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_B_2)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)),V_B_2) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) & (-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_B_2)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)),V_B_2) = hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))))) # label(fact_insert__Diff__if) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1419 (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(tc_Int_Oint),V_r_1),V_b_1)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_r_1))) -> (V_b_1 != c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a_1),V_b_1) = V_y)))) # label(fact_divmod__int__rel__div__eq) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1420 (all V_n hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_n)),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = V_n) # label(fact_diff__Suc__1) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1421 (all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> 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)))) = V_x_2)) # label(fact_Max_Osingleton) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1422 (all V_c_2 all T_e all T_d all T_c all V_q_2 all V_p_2 (V_p_2 = V_q_2 -> hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,T_e),V_c_2),V_p_2) = hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,T_e),V_c_2),V_q_2))) # label(fact_split__weak__cong) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1423 (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]. 5.89/5.95 1424 (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]. 5.89/5.95 1425 (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]. 5.89/5.95 1426 (all V_n all V_z (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z)) -> hAPP(c_Int_Onat,hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_z),V_n)) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),hAPP(c_Int_Onat,V_z)),V_n))) # label(fact_nat__power__eq) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1427 (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]. 5.89/5.95 1428 (all V_a_2 all V_b_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a_2),V_b_2)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_a_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))))) # label(fact_pos__imp__zdiv__neg__iff) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1429 (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]. 5.89/5.95 1430 (all V_x_2 all V_n_2 all V_f_2 all T_c all T_d c_List_Oreplicate(T_c,V_n_2,hAPP(V_f_2,V_x_2)) = hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),c_List_Oreplicate(T_d,V_n_2,V_x_2))) # label(fact_map__replicate) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1431 (all V_g_2 all V_x_2 all V_y_2 all T_e all V_F_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofolding(T_c,T_d,V_f_2,V_F_2) -> hAPP(c_Fun_Ocomp(T_d,T_d,T_e,hAPP(V_f_2,V_x_2)),hAPP(c_Fun_Ocomp(T_d,T_d,T_e,hAPP(V_f_2,V_y_2)),V_g_2)) = hAPP(c_Fun_Ocomp(T_d,T_d,T_e,hAPP(V_f_2,V_y_2)),hAPP(c_Fun_Ocomp(T_d,T_d,T_e,hAPP(V_f_2,V_x_2)),V_g_2)))) # label(fact_folding_Ocommute__left__comp) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1432 (all V_fun2_2 all V_com_2 all V_fun1_2 all V_f1_2 all T_c all T_d c_Hoare__Mirabelle_Otriple_Otriple__rec(T_d,T_c,V_f1_2,hAPP(hAPP(hAPP(c_Hoare__Mirabelle_Otriple_Otriple(T_d),V_fun1_2),V_com_2),V_fun2_2)) = hAPP(hAPP(hAPP(V_f1_2,V_fun1_2),V_com_2),V_fun2_2)) # label(fact_triple_Orecs) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1433 (all V_z_2 all T_c (class_Rings_Olinordered__idom(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),c_Groups_Ozero__class_Ozero(T_c)),hAPP(c_Int_Oring__1__class_Oof__int(T_c),V_z_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z_2))))) # label(fact_of__int__0__le__iff) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1434 (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]. 5.89/5.95 1435 (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]. 5.89/5.95 1436 (all V_Pa_2 ((exists B_x1 hBOOL(hAPP(V_Pa_2,B_x1))) <-> (exists B_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B_x)) & hBOOL(hAPP(V_Pa_2,hAPP(c_Int_Onat,B_x))))))) # label(fact_ex__nat) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1437 (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]. 5.89/5.95 1438 (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]. 5.89/5.95 1439 (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]. 5.89/5.95 1440 (all V_A_2 all T_c (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)) <-> c_Equiv__Relations_Oequiv(T_c,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)),V_A_2))) # label(fact_equivp__equiv) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1441 (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)),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__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_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]. 5.89/5.95 1442 (all T_c hAPP(c_Relation_Oconverse(T_c,T_c),c_Relation_OId(T_c)) = c_Relation_OId(T_c)) # label(fact_converse__Id) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1443 (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]. 5.89/5.95 1444 (all V_r_2 all T_c ((all B_x all B_y (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),B_x),B_y)),V_r_2)) -> (all B_z (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),B_y),B_z)),V_r_2)) -> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),B_x),B_z)),V_r_2)))))) <-> c_Relation_Otrans(T_c,V_r_2))) # label(fact_trans__def) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1445 (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]. 5.89/5.95 1446 (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)) -> (hAPP(V_f_2,V_x_2) = V_b_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]. 5.89/5.95 1447 (all V_I_2 all V_xs_2 all T_c hAPP(c_List_Oset(T_c),c_List_Osublist(T_c,V_xs_2,V_I_2)) = hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool,T_c),c_HOL_OEx(tc_Nat_Onat)),hAPP(hAPP(c_COMBC(T_c,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),T_c),c_COMBS(tc_Nat_Onat,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),T_c),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Nat_Onat),c_fconj)),hAPP(hAPP(c_COMBC(T_c,tc_fun(tc_Nat_Onat,T_c),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(tc_Nat_Onat,T_c),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),T_c),c_COMBB(T_c,tc_HOL_Obool,tc_Nat_Onat)),c_fequal)),hAPP(c_List_Onth(T_c),V_xs_2))))),hAPP(hAPP(c_COMBS(tc_Nat_Onat,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Nat_Onat),c_fconj),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_HOL_Obool),c_Orderings_Oord__class_Oless(tc_Nat_Onat)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2)))),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool),c_member(tc_Nat_Onat)),V_I_2)))))) # label(fact_set__sublist) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1448 (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]. 5.89/5.95 1449 (all V_xs all T_a all V_ys (c_List_Olist_ONil(T_a) != V_ys -> c_List_Olast(T_a,V_ys) = c_List_Olast(T_a,hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys)))) # label(fact_last__appendR) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1450 (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]. 5.89/5.95 1451 (all V_B_2 all V_A_2 all T_c all T_d hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(T_c,tc_HOL_Obool)),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_d,tc_HOL_Obool),V_A_2)),V_B_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBC(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_c,tc_HOL_Obool)),c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(T_c,tc_HOL_Obool))),V_B_2))) # label(fact_INT__extend__simps_I8_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1452 (all V_n_2 all V_m_2 (c_Nat__Transfer_Ois__nat(V_m_2) -> (c_Nat__Transfer_Ois__nat(V_n_2) -> hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(c_Int_Onat,V_m_2),hAPP(c_Int_Onat,V_n_2))) = c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,V_m_2,V_n_2)))) # label(fact_SetInterval_Otransfer__int__nat__set__functions) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1453 (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]. 5.89/5.95 1454 (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]. 5.89/5.95 1455 (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]. 5.89/5.95 1456 (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]. 5.89/5.95 1457 (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]. 5.89/5.95 1458 (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]. 5.89/5.95 1459 (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]. 5.89/5.95 1460 (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]. 5.89/5.95 1461 (all V_m all V_n hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),V_m)),V_m) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),V_n),V_m)) # label(fact_nat__minus__add__max) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1462 (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]. 5.89/5.95 1463 (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]. 5.89/5.95 1464 (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_a),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]. 5.89/5.95 1465 (all V_R_2 all T_c all T_d hAPP(c_Relation_OImage(T_d,T_c,V_R_2),c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) # label(fact_Image__empty) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1466 (all V_A_2 all V_y_2 all V_x_2 all T_c hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_y_2),V_A_2)) = hAPP(hAPP(c_Set_Oinsert(T_c),V_y_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2))) # label(fact_insert__commute) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1467 (all V_size_2 all V_nrandom_2 all V_x_2 all T_c hAPP(c_Product__Type_OPair(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_HOL_Obool,tc_Option_Ooption(tc_Lazy__Sequence_Olazy__sequence(T_c)))),tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral)),hAPP(c_DSequence_Osingle(T_c),V_x_2)) = hAPP(hAPP(hAPP(c_Random__Sequence_Osingle(T_c),V_x_2),V_nrandom_2),V_size_2)) # label(fact_Random__Sequence_Osingle__def) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1468 (all V_A_2 all V_f_2 all T_c hAPP(c_Finite__Set_Ofold1(T_c,V_f_2),V_A_2) = c_HOL_OThe(T_c,c_Finite__Set_Ofold1Set(T_c,V_f_2,V_A_2))) # label(fact_fold1__def) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1469 (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]. 5.89/5.95 1470 (all T_1 all T_2 (class_Lazy__Sequence_Osmall__lazy(T_1) & class_Lazy__Sequence_Osmall__lazy(T_2) -> class_Lazy__Sequence_Osmall__lazy(tc_prod(T_2,T_1)))) # label(arity_prod__Lazy__Sequence_Osmall__lazy) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1471 (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]. 5.89/5.95 1472 (all V_x_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> c_Big__Operators_Olattice_OSup__fin(T_c,c_Orderings_Oord__class_Omax(T_c),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)) = hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_c,c_Orderings_Oord__class_Omax(T_c)),V_x_2),V_A_2))))) # label(fact_min__max_OSup__fin_Oeq__fold_H) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1473 (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]. 5.89/5.95 1474 (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]. 5.89/5.95 1475 (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]. 5.89/5.95 1476 (all V_c all V_b all V_a hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_c)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_b),V_c))),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_c)),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_b,V_c))),V_c)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_a),V_b)),V_c)) # label(fact_zdiv__zadd1__eq) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1477 (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]. 5.89/5.95 1478 (all V_z all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),hAPP(hAPP(c_Orderings_Oord__class_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_Oinf__sup__distrib1) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1479 (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]. 5.89/5.95 1480 (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]. 5.89/5.95 1481 (all V_x_2 all V_A_2 all T_d all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> hAPP(c_Complete__Lattice_OInf__class_OInf(tc_fun(T_d,T_c),V_A_2),V_x_2) = c_Complete__Lattice_OInf__class_OInf(T_c,hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_d,T_c),tc_HOL_Obool),tc_HOL_Obool,T_c),c_HOL_OEx(tc_fun(T_d,T_c))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_d,T_c),tc_HOL_Obool),tc_fun(tc_fun(T_d,T_c),tc_HOL_Obool),T_c),hAPP(c_COMBS(tc_fun(T_d,T_c),tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_fun(T_d,T_c)),c_fconj),hAPP(hAPP(c_COMBC(tc_fun(T_d,T_c),tc_fun(tc_fun(T_d,T_c),tc_HOL_Obool),tc_HOL_Obool),c_member(tc_fun(T_d,T_c))),V_A_2)))),hAPP(hAPP(c_COMBC(T_c,tc_fun(tc_fun(T_d,T_c),T_c),tc_fun(tc_fun(T_d,T_c),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(tc_fun(T_d,T_c),T_c),tc_fun(tc_fun(T_d,T_c),tc_HOL_Obool)),T_c),c_COMBB(T_c,tc_HOL_Obool,tc_fun(T_d,T_c))),c_fequal)),hAPP(hAPP(c_COMBC(tc_fun(T_d,T_c),T_d,T_c),c_COMBI(tc_fun(T_d,T_c))),V_x_2)))))))) # label(fact_Inf__apply) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1482 (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]. 5.89/5.95 1483 (all V_i_2 all V_Pa_2 (hBOOL(hAPP(V_Pa_2,hAPP(c_Int_Onat,V_i_2))) <-> (all B_n (V_i_2 = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),B_n) -> 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]. 5.89/5.95 1484 (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]. 5.89/5.95 1485 (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]. 5.89/5.95 1486 (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]. 5.89/5.95 1487 (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_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_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))) # label(fact_INT__simps_I9_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1488 (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]. 5.89/5.95 1489 (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]. 5.89/5.95 1490 (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)) -> (V_S_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> c_Big__Operators_Olinorder__class_OMin(T_c,hAPP(c_Set_Oimage(T_c,T_c,c_Groups_Ouminus__class_Ouminus(T_c)),V_S_2)) = hAPP(c_Groups_Ouminus__class_Ouminus(T_c),c_Big__Operators_Olinorder__class_OMax(T_c,V_S_2)))))) # label(fact_minus__Max__eq__Min) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1491 (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]. 5.89/5.95 1492 (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]. 5.89/5.95 1493 (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]. 5.89/5.95 1494 (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]. 5.89/5.95 1495 (all V_r_2 all T_c c_List_Olistrel(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_listrel__rtrancl__eq__rtrancl__listrel1) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1496 (all V_ry all V_rx all V_ly all V_lx all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_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))) = 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_I14_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1497 (all V_xs all T_a (class_Groups_Ocomm__monoid__add(T_a) -> hAPP(c_List_Omonoid__add__class_Olistsum(T_a),V_xs) = hAPP(c_List_Omonoid__add__class_Olistsum(T_a),hAPP(c_List_Orev(T_a),V_xs)))) # label(fact_listsum__rev) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1498 (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]. 5.89/5.95 1499 (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]. 5.89/5.95 1500 (all V_g_2 all V_x_2 all V_f_2 all T_c all T_d all T_e (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)) <-> 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))) # label(fact_apsnd__eq__conv) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1501 (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(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),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2))))) # label(fact_fun__left__comm_Ofold__insert__remove) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1502 (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]. 5.89/5.95 1503 (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]. 5.89/5.95 1504 (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]. 5.89/5.95 1505 (all V_x all T_a all V_xs (V_xs != c_List_Olist_ONil(T_a) -> c_List_Olast(T_a,hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)) = c_List_Olast(T_a,V_xs))) # label(fact_last__ConsR) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1506 (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),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_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]. 5.89/5.95 1507 (all V_A_2 all V_x_2 all T_c 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_x_2),V_A_2))) # label(fact_insert__absorb2) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1508 (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_Ozero__class_Ozero(T_c) <-> c_Groups_Ozero__class_Ozero(T_c) = V_a_2))) # label(fact_sgn__0__0) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1509 (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]. 5.89/5.95 1510 (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]. 5.89/5.95 1511 (all V_v all T_a (class_Int_Onumber__ring(T_a) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))) -> hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)) = c_Groups_Ozero__class_Ozero(T_a)) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))) -> hAPP(c_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__lemma) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1512 (all V_x all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_x),V_x) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1513 (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)),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)) = 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))) # label(fact_Union__Un__distrib) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1514 (all T_a c_Nitpick_OAbs__Frac(T_a,hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Groups_Oone__class_Oone(tc_Int_Oint))) = c_Nitpick_Ozero__frac(T_a)) # label(fact_Nitpick_Ozero__frac__def) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1515 (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]. 5.89/5.95 1516 (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]. 5.89/5.95 1517 (all V_u_2 all V_l_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) = c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Int_Oint,V_l_2,V_u_2)) # label(fact_atLeastPlusOneAtMost__greaterThanAtMost__int) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1518 (all V_C_2 all V_B_2 all V_A_2 all T_c (hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_C_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)),V_C_2) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_C_2),V_A_2)))) # label(fact_Un__Int__assoc__eq) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1519 (all V_n all V_a all V_b all T_a (class_Fields_Ofield(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) != V_b -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_b)),V_n) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_b),V_n))))) # label(fact_nonzero__power__divide) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1520 (all V_B_2 all V_A_2 all T_d all T_c all V_C_2 ((V_C_2 = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool),T_c),hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_d,tc_HOL_Obool)),V_A_2)),V_B_2)) = c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool))) & (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_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool),T_c),hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_d,tc_HOL_Obool)),V_A_2)),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_d,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),V_B_2))))) # label(fact_INT__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1521 (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]. 5.89/5.95 1522 (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]. 5.89/5.95 1523 (all V_B_2 all V_A_2 all T_d all T_c all V_C_2 ((V_C_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool),T_c),hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_d,tc_HOL_Obool)),V_A_2)),V_B_2)) = 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))) & (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = V_C_2 -> 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]. 5.89/5.95 1524 (all V_m all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n),V_m)) -> c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n),V_n) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n))) # label(fact_le__mod__geq) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1525 (all V_n_2 all V_m_2 (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_n_2)),V_m_2)))) # label(fact_not__less__eq__eq) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1526 (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]. 5.89/5.95 1527 (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]. 5.89/5.95 1528 (all V_v ((-hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))) -> hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)) = 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))) -> hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)) = c_Groups_Ozero__class_Ozero(tc_Int_Oint)))) # label(fact_int__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1529 (all V_xs_2 all V_n_2 all T_c (hAPP(hAPP(c_List_Odrop(T_c),V_n_2),V_xs_2) = c_List_Olist_ONil(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)))) # label(fact_drop__eq__Nil) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1530 (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__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)) = 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))) # label(fact_Un__Int__distrib) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1531 (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]. 5.89/5.95 1532 (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_OMin(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),V_x_2),c_Big__Operators_Olinorder__class_OMin(T_c,V_A_2))))))) # label(fact_Min_Oinsert) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1533 (all V_yq all V_xq all V_x all T_a c_Lazy__Sequence_Olazy__sequence_OInsert(T_a,V_x,c_Lazy__Sequence_Oappend(T_a,V_xq,V_yq)) = c_Lazy__Sequence_Oappend(T_a,c_Lazy__Sequence_Olazy__sequence_OInsert(T_a,V_x,V_xq),V_yq)) # label(fact__01) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1534 (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))) -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(c_Int_Onat,V_z))) # label(fact_nat__le__0) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1535 (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]. 5.89/5.95 1536 (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 -> c_Nitpick_Onat__gcd(V_y_2,c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_x_2,V_y_2)) = c_Nitpick_Onat__gcd(V_x_2,V_y_2)) & (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_y_2 -> c_Nitpick_Onat__gcd(V_x_2,V_y_2) = V_x_2))) # label(fact_nat__gcd_Opsimps) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1537 (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]. 5.89/5.95 1538 (all V_x all T_a (class_Lattices_Oboolean__algebra(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_x)),V_x) = c_Orderings_Obot__class_Obot(T_a))) # label(fact_compl__inf__bot) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1539 (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),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)))) = 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)))) # label(fact_sublist__shift__lemma__Suc) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1540 (all V_c_2 all V_b_2 all V_a_2 all T_c (class_Fields_Olinordered__field__inverse__zero(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_a_2),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_b_2),V_c_2))) <-> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_c_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__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)))))))) # label(fact_le__divide__eq) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1541 (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]. 5.89/5.95 1542 (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]. 5.89/5.95 1543 (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]. 5.89/5.95 1544 (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__eq(T_c),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_c_2)))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),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__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]. 5.89/5.95 1545 (all V_B_2 all V_A_2 all T_c all T_d hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Relation_ORange(T_d,T_c),V_A_2)),hAPP(c_Relation_ORange(T_d,T_c),V_B_2)) = hAPP(c_Relation_ORange(T_d,T_c),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_d,T_c),tc_HOL_Obool)),V_A_2),V_B_2))) # label(fact_Range__Un__eq) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1546 (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)),c_List_Olinorder__class_Osort__key(T_c,T_d,V_f_2,V_xs_2)) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2))) # label(fact_length__sort) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1547 (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]. 5.89/5.95 1548 (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]. 5.89/5.95 1549 (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__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]. 5.89/5.95 1550 (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]. 5.89/5.95 1551 (all V_A_2 all V_f_2 all T_d all T_c 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)) = 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))) # label(fact_vimage__Union) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1552 (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]. 5.89/5.95 1553 (all V_i ((-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_i),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),V_i) = V_i) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_i),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_i) = hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),V_i)))) # label(fact_zabs__def) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1554 (all V_A_2 all V_a_2 all T_c hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_A_2) = hAPP(hAPP(c_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__single) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1555 (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]. 5.89/5.95 1556 (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]. 5.89/5.95 1557 (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)) & c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_A_2)) # label(fact_card__gt__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1558 (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]. 5.89/5.95 1559 (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_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)),V_a_2))))) # label(fact_minus__less__iff) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1560 (all V_x all V_i all V_xs all T_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)),V_i)) -> hAPP(hAPP(c_List_Olist__update(T_a,V_xs),V_i),V_x) = V_xs)) # label(fact_list__update__beyond) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1561 (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]. 5.89/5.95 1562 (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]. 5.89/5.95 1563 (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]. 5.89/5.95 1564 (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]. 5.89/5.95 1565 (all V_M_2 all V_B_2 all V_A_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_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)) = 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)) # label(fact_INT__Un) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1566 (all V_ys all V_xs all V_n all T_b all T_a hAPP(hAPP(c_List_Otake(tc_prod(T_a,T_b)),V_n),hAPP(hAPP(c_List_Ozip(T_a,T_b),V_xs),V_ys)) = hAPP(hAPP(c_List_Ozip(T_a,T_b),hAPP(hAPP(c_List_Otake(T_a),V_n),V_xs)),hAPP(hAPP(c_List_Otake(T_b),V_n),V_ys))) # label(fact_take__zip) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1567 (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]. 5.89/5.95 1568 (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]. 5.89/5.95 1569 (all V_a all T_a (class_Groups_Ogroup__add(T_a) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a)) = V_a)) # label(fact_diff__0__right) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1570 (all V_R_2 all T_c V_R_2 = 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))) # label(fact_eq__comp__r) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1571 (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]. 5.89/5.95 1572 (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]. 5.89/5.95 1573 (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]. 5.89/5.95 1574 (all V_m all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n)) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),V_n) = hAPP(c_Nat_OSuc,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)),V_n))))) # label(fact_div__geq) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1575 (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]. 5.89/5.95 1576 (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]. 5.89/5.95 1577 (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]. 5.89/5.95 1578 (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]. 5.89/5.95 1579 (all V_m all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n)) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n),V_m)),V_n) = V_m)) # label(fact_div__mult__self1__is__m) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1580 (all V_t_2 all V_D_2 all V_d_2 all T_c (class_Rings_Ocomm__ring(T_c) & class_Rings_Odvd(T_c) -> (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_c),V_d_2),V_D_2)) -> (all B_x all B_k (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_c),V_d_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),B_x),V_t_2))) <-> hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_c),V_d_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_c),B_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),B_k),V_D_2))),V_t_2)))))))) # label(fact_inf__period_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1581 (all V_y_2 all V_x_2 all T_c (class_Lattices_Oboolean__algebra(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_y_2),V_x_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_x_2)),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_y_2)))))) # label(fact_compl__le__compl__iff) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1582 (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(V_F_2,V_A_2) = hAPP(c_Finite__Set_Ofold1(T_c,V_f_2),V_A_2)))) # label(fact_semilattice__big_OF__eq) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1583 (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]. 5.89/5.95 1584 (all V_S_2 all V_a_2 all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) & class_Orderings_Olinorder(T_c) -> ((exists B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_S_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_a_2),B_x)))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_a_2),c_Complete__Lattice_OSup__class_OSup(T_c,V_S_2)))))) # label(fact_less__Sup__iff) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1585 (all T_c (class_Lattices_Osemilattice__inf(T_c) & class_Orderings_Olinorder(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]. 5.89/5.95 1586 (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]. 5.89/5.95 1587 (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]. 5.89/5.95 1588 (all V_ys_2 all V_xs_2 all T_c (c_List_Oconcat(T_c,V_ys_2) = c_List_Oconcat(T_c,V_xs_2) -> (hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_List_Olist(T_c))),V_xs_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_List_Olist(T_c))),V_ys_2) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),B_x),hAPP(c_List_Oset(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_List_Ozip(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_xs_2),V_ys_2)))) -> hBOOL(hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(tc_List_Olist(T_c),tc_List_Olist(T_c),tc_HOL_Obool),hAPP(hAPP(c_COMBC(tc_List_Olist(T_c),tc_fun(tc_List_Olist(T_c),tc_Nat_Onat),tc_fun(tc_List_Olist(T_c),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_fun(tc_List_Olist(T_c),tc_Nat_Onat),tc_fun(tc_List_Olist(T_c),tc_HOL_Obool)),tc_List_Olist(T_c)),c_COMBB(tc_Nat_Onat,tc_HOL_Obool,tc_List_Olist(T_c))),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_List_Olist(T_c)),c_fequal),c_Nat_Osize__class_Osize(tc_List_Olist(T_c))))),c_Nat_Osize__class_Osize(tc_List_Olist(T_c)))),B_x)))) -> V_ys_2 = V_xs_2)))) # label(fact_concat__injective) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1589 (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]. 5.89/5.95 1590 (all V_A_2 all V_B_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_A_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2) = V_A_2)) # label(fact_Un__absorb2) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1591 (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]. 5.89/5.95 1592 (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]. 5.89/5.95 1593 (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]. 5.89/5.95 1594 (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]. 5.89/5.95 1595 (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]. 5.89/5.95 1596 (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]. 5.89/5.95 1597 (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]. 5.89/5.95 1598 (all V_s_2 all V_r_2 all V_b_H_2 all V_a_H_2 all V_b_2 all V_a_2 all T_d all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_prod(T_c,T_d),tc_prod(T_c,T_d))),hAPP(hAPP(c_Product__Type_OPair(tc_prod(T_c,T_d),tc_prod(T_c,T_d)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_a_2),V_b_2)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_a_H_2),V_b_H_2))),c_Wellfounded_Olex__prod(T_c,T_d,V_r_2,V_s_2))) <-> 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)) | V_a_2 = V_a_H_2 & hBOOL(hAPP(hAPP(c_member(tc_prod(T_d,T_d)),hAPP(hAPP(c_Product__Type_OPair(T_d,T_d),V_b_2),V_b_H_2)),V_s_2)))) # label(fact_in__lex__prod) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1599 (all V_a_2 all T_c (class_Rings_Olinordered__idom(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_Power_Opower__class_Opower(T_c),V_a_2),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))))))) # label(fact_zero__less__power2) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1600 (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]. 5.89/5.95 1601 (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]. 5.89/5.95 1602 (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]. 5.89/5.95 1603 (all V_xss_2 all T_c (c_List_Oconcat(T_c,V_xss_2) = c_List_Olist_ONil(T_c) <-> (all B_x (hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_c)),B_x),hAPP(c_List_Oset(tc_List_Olist(T_c)),V_xss_2))) -> c_List_Olist_ONil(T_c) = B_x)))) # label(fact_Nil__eq__concat__conv) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1604 (all V_z2 all V_z1 all V_w 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)) = 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))) # label(fact_zdiff__zmult__distrib2) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1605 (all V_xs_2 all V_f_2 all T_c all T_d (c_List_Olist_ONil(T_d) = V_xs_2 <-> hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_xs_2) = c_List_Olist_ONil(T_c))) # label(fact_map__is__Nil__conv) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1606 (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]. 5.89/5.95 1607 (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]. 5.89/5.95 1608 (all V_k_2 all V_A_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),V_A_2),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_k_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_k_2),hAPP(c_Finite__Set_Ocard(tc_Nat_Onat),V_A_2))))) -> hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_k_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_k_2),hAPP(c_Finite__Set_Ocard(tc_Nat_Onat),V_A_2))) = V_A_2)) # label(fact_subset__card__intvl__is__intvl) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1609 (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]. 5.89/5.95 1610 (all V_x_2 all T_c (c_List_Oremdups(T_c,V_x_2) = c_List_Olist_ONil(T_c) <-> c_List_Olist_ONil(T_c) = V_x_2)) # label(fact_remdups__eq__nil__right__iff) # label(axiom) # label(non_clause). [assumption]. 5.89/5.95 1611 (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]. 5.89/5.95 1612 (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]. 5.89/5.95 1613 (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]. 5.89/5.96 1614 (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]. 5.89/5.96 1615 (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]. 5.89/5.96 1616 (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_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))) = hAPP(hAPP(c_List_Onth(tc_List_Olist(T_c)),V_xs_2),V_i_2)))) # label(fact_transpose__column) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1617 (all V_n all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) | V_n = V_m -> 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]. 5.89/5.96 1618 (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]. 5.89/5.96 1619 (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]. 5.89/5.96 1620 (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) <-> V_z_2 = c_Groups_Ozero__class_Ozero(tc_Int_Oint)))) # label(fact_of__int__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1621 (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_OgreaterThanAtMost(T_c,V_a_2,V_b_2)),c_SetInterval_Oord__class_OgreaterThanAtMost(T_c,V_c_2,V_d_2)) = 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)))) # label(fact_Int__greaterThanAtMost) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1622 (all V_x all V_i all T_a hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),c_List_Oreplicate(T_a,V_i,V_x)) = 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)))) # label(fact_replicate__append__same) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1623 (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_A_2 = V_B_2)))) # label(fact_insert__ident) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1624 (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_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(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]. 5.89/5.96 1625 (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]. 5.89/5.96 1626 (all V_Pa_2 all T_c all T_d (class_Enum_Oenum(T_d) & class_Enum_Oenum(T_c) -> (hBOOL(hAPP(c_Enum_Oenum__class_Oenum__all(T_c),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_HOL_Obool,T_c),c_Enum_Oenum__class_Oenum__all(T_d)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_prod(T_c,T_d)),tc_fun(T_d,tc_HOL_Obool),T_c),hAPP(c_COMBB(tc_prod(T_c,T_d),tc_HOL_Obool,T_d),V_Pa_2)),c_Product__Type_OPair(T_c,T_d))))) <-> hBOOL(hAPP(c_Enum_Oenum__class_Oenum__all(tc_prod(T_c,T_d)),V_Pa_2))))) # label(fact_enum__all__prod__def) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1627 (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]. 5.89/5.96 1628 (all V_n_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n_2)))) # label(fact_zero__less__int__conv) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1629 (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]. 5.89/5.96 1630 (all V_Qa_2 all V_Pa_2 (hBOOL(V_Qa_2) | hBOOL(V_Pa_2) | -hBOOL(hAPP(hAPP(c_fdisj,V_Pa_2),V_Qa_2)))) # label(help_c__fdisj__3) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1631 (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]. 5.89/5.96 1632 (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),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),V_q)),V_x) = 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_I28_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1633 (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]. 5.89/5.96 1634 (all V_n_2 all 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) <-> V_n_2 = c_Groups_Oone__class_Oone(tc_Int_Oint) & V_m_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]. 5.89/5.96 1635 (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]. 5.89/5.96 1636 (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_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)) = 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))) # label(fact_Int__UN__distrib) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1637 (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]. 5.89/5.96 1638 (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]. 5.89/5.96 1639 (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]. 5.89/5.96 1640 (all V_b_2 all V_a_2 all T_c (class_Fields_Olinordered__field__inverse__zero(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_a_2),V_b_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),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))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_b_2),c_Groups_Ozero__class_Ozero(T_c)))))) # label(fact_zero__less__divide__iff) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1641 (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]. 5.89/5.96 1642 (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]. 5.89/5.96 1643 (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]. 5.89/5.96 1644 (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]. 5.89/5.96 1645 (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]. 5.89/5.96 1646 (all V_xs all T_a hAPP(hAPP(c_List_Odrop(T_a),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_xs) = V_xs) # label(fact_drop__0) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1647 (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]. 5.89/5.96 1648 (all V_size_2 all V_nrandom_2 all T_c c_Random__Sequence_Oempty(T_c,V_nrandom_2,V_size_2) = hAPP(c_Product__Type_OPair(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_HOL_Obool,tc_Option_Ooption(tc_Lazy__Sequence_Olazy__sequence(T_c)))),tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral)),c_DSequence_Oempty(T_c))) # label(fact_Random__Sequence_Oempty__def) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1649 (all V_T_2 all V_S_2 all V_R_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_R_2),V_S_2)),hAPP(hAPP(c_Relation_Orel__comp(T_c,T_e,T_d),V_R_2),V_T_2)) = 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))) # label(fact_rel__comp__distrib) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1650 (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]. 5.89/5.96 1651 (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]. 5.89/5.96 1652 (all V_xs all V_x all V_n all T_a hAPP(hAPP(c_List_Oappend(T_a),c_List_Oreplicate(T_a,V_n,V_x)),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)) = hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),hAPP(hAPP(c_List_Oappend(T_a),c_List_Oreplicate(T_a,V_n,V_x)),V_xs))) # label(fact_replicate__app__Cons__same) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1653 (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]. 5.89/5.96 1654 (all V_b all V_a all T_a (class_Divides_Oring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a),V_b) = c_Divides_Odiv__class_Omod(T_a,hAPP(c_Groups_Ouminus__class_Ouminus(T_a),c_Divides_Odiv__class_Omod(T_a,V_a,V_b)),V_b))) # label(fact_mod__minus__eq) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1655 (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 (V_k_2 = hAPP(V_f_2,B_i) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B_i),V_n_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),B_i))))))))) # label(fact_nat__intermed__int__val) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1656 (all T_a c_Nitpick_OAbs__Frac(T_a,hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),c_Groups_Oone__class_Oone(tc_Int_Oint))) = c_Nitpick_Oone__frac(T_a)) # label(fact_Nitpick_Oone__frac__def) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1657 (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]. 5.89/5.96 1658 (all V_B_2 all V_A_2 all T_e all V_f_2 all T_d all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_e,tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_e),c_Set_Ovimage(T_c,T_d,V_f_2)),V_B_2)) = hAPP(c_Set_Ovimage(T_c,T_d,V_f_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_e,tc_fun(T_d,tc_HOL_Obool)),V_A_2),V_B_2))) # label(fact_vimage__INT) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1659 (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]. 5.89/5.96 1660 (all V_Pa_2 all T_c all T_d hAPP(c_Relation_ORange(T_d,T_c),hAPP(c_Set_OCollect(tc_prod(T_d,T_c)),hAPP(c_Product__Type_Oprod_Oprod__case(T_d,T_c,tc_HOL_Obool),V_Pa_2))) = hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_HOL_Obool,T_c),c_HOL_OEx(T_d)),hAPP(c_COMBC(T_d,T_c,tc_HOL_Obool),V_Pa_2)))) # label(fact_Range__Collect__split) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1661 (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]. 5.89/5.96 1662 (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]. 5.89/5.96 1663 (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(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_Oreflp(T_c,V_r_2))) # label(fact_reflp__def) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1664 (all V_f_2 all V_r_2 all T_d all T_c ((all B_x (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),B_x),V_r_2)) -> hBOOL(hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_c,tc_HOL_Obool),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,T_d),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(tc_fun(T_c,T_d),tc_fun(T_c,tc_HOL_Obool)),T_c),c_COMBB(T_d,tc_HOL_Obool,T_c)),hAPP(hAPP(c_COMBB(T_d,tc_fun(T_d,tc_HOL_Obool),T_c),c_fequal),V_f_2))),V_f_2)),B_x)))) <-> c_Equiv__Relations_Ocongruent(T_c,T_d,V_r_2,V_f_2))) # label(fact_congruent__def) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1665 (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]. 5.89/5.96 1666 (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]. 5.89/5.96 1667 (all V_B_2 all V_A_2 all T_c (class_Lattices_Olattice(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> (V_B_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> 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]. 5.89/5.96 1668 (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_OatLeastAtMost(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_OatLeastAtMost(T_c,V_l_2,V_u_2))))) # label(fact_ivl__disj__un_I22_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1669 (all V_n all V_m c_Divides_Odiv__class_Omod(tc_Int_Oint,V_m,V_n) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_m),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_m),V_n)),V_n))) # label(fact_zmod__zdiv__equality_H) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1670 (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]. 5.89/5.96 1671 (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]. 5.89/5.96 1672 (all V_x_2 all V_A_2 all V_F_2 all V_g_2 all V_z_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofolding__image__simple__idem(T_c,T_d,V_f_2,V_z_2,V_g_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_d),V_x_2),V_A_2)) -> hAPP(V_F_2,V_A_2) = hAPP(hAPP(V_f_2,hAPP(V_g_2,V_x_2)),hAPP(V_F_2,V_A_2)))))) # label(fact_folding__image__simple__idem_Oin__idem) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1673 (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_ORange(T_d,T_c),V_r_2) <-> 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]. 5.89/5.96 1674 (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]. 5.89/5.96 1675 (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]. 5.89/5.96 1676 (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]. 5.89/5.96 1677 (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]. 5.89/5.96 1678 (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]. 5.89/5.96 1679 (all V_x_2 all V_r_2 all T_d all T_c (hBOOL(hAPP(hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool),T_c),hAPP(c_COMBC(T_d,tc_fun(T_d,tc_HOL_Obool),tc_HOL_Obool),c_member(T_d))),V_r_2)),V_x_2)) <-> hBOOL(hAPP(hAPP(c_member(T_d),V_x_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))),V_r_2))))) # label(fact_SUP__UN__eq) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1680 (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]. 5.89/5.96 1681 (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]. 5.89/5.96 1682 (all V_x all T_a all 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)) & (V_xs != c_List_Olist_ONil(T_a) -> c_List_Obutlast(T_a,hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)) = hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),c_List_Obutlast(T_a,V_xs))))) # label(fact_butlast_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1683 (all V_n_2 all 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 <-> hAPP(hAPP(c_Groups_Otimes__class_Otimes(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]. 5.89/5.96 1684 (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]. 5.89/5.96 1685 (all V_m all V_n hAPP(hAPP(c_Product__Type_OPair(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),V_n),V_m)),c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,V_n,V_m)) = c_Code__Numeral_Odiv__mod__code__numeral(V_n,V_m)) # label(fact_div__mod__code__numeral__def) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1686 (all T_c c_Relation_Otrans(T_c,c_Relation_OId(T_c))) # label(fact_trans__Id) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1687 (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]. 5.89/5.96 1688 (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]. 5.89/5.96 1689 (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]. 5.89/5.96 1690 (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]. 5.89/5.96 1691 (all V_ys_2 all V_xs_2 all T_c (V_ys_2 = c_List_Olist_ONil(T_c) & V_xs_2 = c_List_Olist_ONil(T_c) <-> hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_ys_2) = c_List_Olist_ONil(T_c))) # label(fact_Nil__is__append__conv) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1692 (all V_S_2 all T_c all T_d 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)) = 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))) # label(fact_Range__Union) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1693 (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]. 5.89/5.96 1694 (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]. 5.89/5.96 1695 (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]. 5.89/5.96 1696 (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]. 5.89/5.96 1697 (all T_1 all T_2 (class_Enum_Oenum(T_1) & class_Enum_Oenum(T_2) -> class_Enum_Oenum(tc_sum(T_2,T_1)))) # label(arity_sum__Enum_Oenum) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1698 (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]. 5.89/5.96 1699 (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_Un__absorb1) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1700 (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]. 5.89/5.96 1701 (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]. 5.89/5.96 1702 (all V_n all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) -> c_Divides_Odivmod__nat(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))) # label(fact_divmod__nat__base) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1703 (all V_x_2 all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> hAPP(c_Nat_OSuc,hAPP(c_Finite__Set_Ocard(T_c),V_A_2)) = hAPP(c_Finite__Set_Ocard(T_c),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2))) & (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_Finite__Set_Ocard(T_c),V_A_2)))) # label(fact_card__insert__if) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1704 (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_x_2) = hAPP(V_f_2,V_y_2)))))) # label(fact_inj__on__iff) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1705 (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]. 5.89/5.96 1706 (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]. 5.89/5.96 1707 (all V_l all V_k c_Int_OBit1(V_l) != c_Int_OBit0(V_k)) # label(fact_rel__simps_I49_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1708 (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]. 5.89/5.96 1709 (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]. 5.89/5.96 1710 (all V_n hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = V_n) # label(fact_nat__mult__1__right) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1711 (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]. 5.89/5.96 1712 (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]. 5.89/5.96 1713 (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]. 5.89/5.96 1714 (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),V_a_2),V_b_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),V_a_2),V_c_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),V_b_2),V_c_2)))))) # label(fact_add__le__cancel__right) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1715 (all V_A_2 all T_c 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)))) = c_Complete__Lattice_OInf__class_OInf(tc_fun(T_c,tc_HOL_Obool),V_A_2)) # label(fact_Inter__eq) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1716 (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]. 5.89/5.96 1717 (all V_b all V_a all V_c all T_a (class_Divides_Osemiring__div(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) = V_c -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b)) = c_Groups_Ozero__class_Ozero(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__if) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1718 (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]. 5.89/5.96 1719 (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]. 5.89/5.96 1720 (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]. 5.89/5.96 1721 (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(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_c,T_d)),V_f_2),V_g_2)) <-> hBOOL(hAPP(c_Enum_Oenum__class_Oenum__ex(T_c),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_c),c_fNot),hAPP(hAPP(c_COMBS(T_c,T_d,tc_HOL_Obool),hAPP(hAPP(c_COMBB(T_d,tc_fun(T_d,tc_HOL_Obool),T_c),c_fequal),V_f_2)),V_g_2)))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,T_d)),V_f_2),V_g_2))))) # label(fact_order__fun_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1722 (all V_x_2 all V_A_2 all V_F_2 all V_f_2 all T_c (c_Finite__Set_Ofolding__one(T_c,V_f_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> V_x_2 = hAPP(V_F_2,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2))) & (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,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]. 5.89/5.96 1723 (all V_m_2 all V_q_2 all V_n_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n_2)) -> (V_q_2 = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m_2),V_n_2) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n_2),hAPP(c_Nat_OSuc,V_q_2)))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n_2),V_q_2)),V_m_2))))) # label(fact_split__div__lemma) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1724 (all V_f_2 all V_S_2 all T_c all T_d (class_Groups_Ocomm__monoid__add(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(T_c,tc_HOL_Obool)),V_S_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),B_x),V_S_2)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),B_x)))) -> ((all B_T1 (hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),B_T1),V_S_2)) -> (all B_T2 (hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),B_T2),V_S_2)) -> (B_T1 != B_T2 -> (all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),B_T1)) -> (hBOOL(hAPP(hAPP(c_member(T_c),B_x),B_T2)) -> 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),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]. 5.89/5.96 1725 (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]. 5.89/5.96 1726 (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]. 5.89/5.96 1727 (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]. 5.89/5.96 1728 (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]. 5.89/5.96 1729 (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]. 5.89/5.96 1730 (all V_n all T_a c_List_Olist_ONil(T_a) = hAPP(hAPP(c_List_Otake(T_a),V_n),c_List_Olist_ONil(T_a))) # label(fact_take__Nil) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1731 (all V_u_2 all V_l_2 hAPP(c_Finite__Set_Ocard(tc_Nat_Onat),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_l_2,V_u_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_u_2)),V_l_2)) # label(fact_card__atLeastAtMost) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1732 (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]. 5.89/5.96 1733 (all V_b_2 all V_a_2 all T_c (class_Groups_Ogroup__add(T_c) -> (c_Groups_Ozero__class_Ozero(T_c) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),V_a_2),V_b_2) <-> hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_b_2) = V_a_2))) # label(fact_eq__neg__iff__add__eq__0) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1734 (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]. 5.89/5.96 1735 (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]. 5.89/5.96 1736 (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]. 5.89/5.96 1737 (all V_B_2 all V_A_2 all T_c hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)),V_B_2) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) # label(fact_Diff__idemp) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1738 (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]. 5.89/5.96 1739 (all V_a_2 all V_f_2 all T_c 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)))) = V_a_2) # label(fact_fold1__singleton) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1740 (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]. 5.89/5.96 1741 (all V_B_2 all V_A_2 all T_d all T_c all V_C_2 ((c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_C_2 -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_d,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),V_A_2)),V_B_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool)),T_c),c_Groups_Ominus__class_Ominus(tc_fun(T_d,tc_HOL_Obool))),V_A_2)),V_B_2))) & (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = V_C_2 -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_d,tc_HOL_Obool)),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)))) # label(fact_INT__extend__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1742 (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]. 5.89/5.96 1743 (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]. 5.89/5.96 1744 (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]. 5.89/5.96 1745 (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]. 5.89/5.96 1746 (all V_ys all T_a all V_xs ((c_List_Olist_ONil(T_a) = V_xs -> 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]. 5.89/5.96 1747 (all V_y_2 all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_y_2),V_x_2)) <-> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_x_2),V_y_2))))) # label(fact_linorder__not__less) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1748 (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]. 5.89/5.96 1749 (all V_C_2 all V_B_2 all V_A_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_C_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)),V_C_2)) # label(fact_Int__assoc) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1750 (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]. 5.89/5.96 1751 (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))) -> hAPP(c_Divides_Oadjust(hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)),c_Divides_OposDivAlg(c_Groups_Oone__class_Oone(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)))) = c_Divides_OposDivAlg(c_Groups_Oone__class_Oone(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w))))) # label(fact_posDivAlg__eqn__1__number__of) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1752 (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]. 5.89/5.96 1753 (all V_C_2 all V_J_2 all V_I_2 all T_d all T_c hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_I_2),V_J_2)),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_I_2),V_C_2)),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),V_J_2),V_C_2))) # label(fact_Sigma__Int__distrib1) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1754 (all V_xs all T_a all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))) -> hAPP(hAPP(c_List_Oappend(T_a),hAPP(hAPP(c_List_Otake(T_a),V_n),V_xs)),hAPP(hAPP(c_List_Olist_OCons(T_a),hAPP(c_List_Ohd(T_a),hAPP(hAPP(c_List_Odrop(T_a),V_n),V_xs))),c_List_Olist_ONil(T_a))) = hAPP(hAPP(c_List_Otake(T_a),hAPP(c_Nat_OSuc,V_n)),V_xs))) # label(fact_take__hd__drop) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1755 (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]. 5.89/5.96 1756 (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]. 5.89/5.96 1757 (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]. 5.89/5.96 1758 (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]. 5.89/5.96 1759 (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]. 5.89/5.96 1760 (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)) -> 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__pos__pos__trivial) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1761 (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),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_v),V_w))),V_z) = 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)))) # label(fact_add__number__of__left) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1762 (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]. 5.89/5.96 1763 (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]. 5.89/5.96 1764 (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]. 5.89/5.96 1765 (all V_xs_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_List_Omember(T_c),V_xs_2),V_x_2)) <-> hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2))))) # label(fact_in__set__member) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1766 (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]. 5.89/5.96 1767 (all V_xs_2 all V_n_2 all T_c (c_List_Olist_ONil(T_c) = V_xs_2 | V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) <-> hAPP(hAPP(c_List_Otake(T_c),V_n_2),V_xs_2) = c_List_Olist_ONil(T_c))) # label(fact_take__eq__Nil) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1768 (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]. 5.89/5.96 1769 (all V_n_2 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(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2)),V_n_2)) = hAPP(hAPP(c_HOL_OLet(tc_Int_Oint,T_c),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_Opred(V_v_2))),hAPP(hAPP(c_COMBS(tc_Int_Oint,T_c,T_c),hAPP(hAPP(c_COMBC(tc_Int_Oint,T_c,tc_fun(T_c,T_c)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(T_c,tc_fun(T_c,T_c)),tc_Int_Oint),c_If(T_c)),c_Nat__Numeral_Oneg)),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))))) # label(fact_nat__case__add__eq__if) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1770 (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]. 5.89/5.96 1771 (all V_n all V_b all V_a all T_a (class_Fields_Ofield__inverse__zero(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_b)),V_n) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_b),V_n)))) # label(fact_power__divide) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1772 (all V_r_2 all T_d all V_a_2 all T_c ((exists B_y hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_d)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_a_2),B_y)),V_r_2))) <-> hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),hAPP(c_Relation_ODomain(T_c,T_d),V_r_2))))) # label(fact_Domain__iff) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1773 (all V_b_2 all V_a_2 all T_c (class_Rings_Oring__no__zero__divisors(T_c) -> (V_b_2 = c_Groups_Ozero__class_Ozero(T_c) | c_Groups_Ozero__class_Ozero(T_c) = V_a_2 <-> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_b_2) = c_Groups_Ozero__class_Ozero(T_c)))) # label(fact_mult__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1774 (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]. 5.89/5.96 1775 (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]. 5.89/5.96 1776 (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]. 5.89/5.96 1777 (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]. 5.89/5.96 1778 (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,c_Fun_Ofun__upd(T_c,T_d,V_f_2,V_x_2,V_y_2),V_x_2,V_z_2) = c_Fun_Ofun__upd(T_c,T_d,V_f_2,V_x_2,V_z_2)) # label(fact_fun__upd__upd) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1779 (all V_w all T_a (class_Int_Onumber__ring(T_a) -> hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OBit0(V_w)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Groups_Ozero__class_Ozero(T_a)),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w))),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w)))) # label(fact_number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1780 (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]. 5.89/5.96 1781 (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]. 5.89/5.96 1782 (all V_d_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_OatMost(T_c),V_d_2)) = 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)))) # label(fact_Int__atLeastAtMostL1) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1783 (all V_z all V_x all V_y all T_a (class_Fields_Ofield__inverse__zero(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) != V_y -> hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_x),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_z),V_y))),V_y) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),V_y)),V_z)))) # label(fact_add__frac__num) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1784 (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]. 5.89/5.96 1785 (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_x_2 = V_y_2 <-> hAPP(V_f_2,V_y_2) = hAPP(V_f_2,V_x_2)))) # label(fact_inj__eq) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1786 (all V_b_2 all V_a_2 all V_g_2 all V_f_2 all T_d all T_f all T_c all T_e hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),hAPP(V_f_2,V_a_2)),hAPP(V_g_2,V_b_2)) = hAPP(c_Product__Type_Omap__pair(T_e,T_c,T_f,T_d,V_f_2,V_g_2),hAPP(hAPP(c_Product__Type_OPair(T_e,T_f),V_a_2),V_b_2))) # label(fact_map__pair__simp) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1787 (all V_N_2 all V_h_2 all T_c (class_Orderings_Olinorder(T_c) -> ((all B_x all B_y hAPP(V_h_2,hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),B_x),B_y)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),hAPP(V_h_2,B_x)),hAPP(V_h_2,B_y))) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_N_2)) -> (V_N_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> 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]. 5.89/5.96 1788 (all V_code__numeral hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Code__Numeral_Ocode__numeral_Ocode__numeral__size(V_code__numeral)),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) = c_Code__Numeral_Ocode__numeral_Ocode__numeral__size(c_Code__Numeral_OSuc__code__numeral(V_code__numeral))) # label(fact_code__numeral_Osize_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1789 (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]. 5.89/5.96 1790 (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]. 5.89/5.96 1791 (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]. 5.89/5.96 1792 (all V_ys all T_a all V_xs (V_xs != c_List_Olist_ONil(T_a) -> 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]. 5.89/5.96 1793 (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]. 5.89/5.96 1794 (all V_d_2 all T_c all V_Pa_2 (((exists B_x ((all B_y (hBOOL(hAPP(V_Pa_2,B_y)) -> B_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)) & (-(exists B_x (hBOOL(hAPP(V_Pa_2,B_x)) & (all B_y (hBOOL(hAPP(V_Pa_2,B_y)) -> B_x = B_y)))) -> V_d_2 = c_FunDef_OTHE__default(T_c,V_d_2,V_Pa_2)))) # label(fact_THE__default__def) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1795 (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]. 5.89/5.96 1796 (all V_A_2 all T_d all V_I_2 all V_k_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_k_2),V_I_2)) -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_HOL_Obool)),V_I_2),V_A_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_d,tc_HOL_Obool)),hAPP(V_A_2,V_k_2)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_HOL_Obool)),V_I_2),V_A_2)))) # label(fact_UN__absorb) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1797 (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__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)) = V_A_2)) # label(fact_Un__empty) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 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__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]. 5.89/5.96 1799 (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]. 5.89/5.96 1800 (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]. 5.89/5.96 1801 (all V_n_2 all V_m_2 (hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m_2),V_n_2) <-> V_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) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n_2 & V_m_2 = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_add__is__1) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1802 (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]. 5.89/5.96 1803 (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]. 5.89/5.96 1804 (all V_n_2 all V_m_2 (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_m_2),V_n_2)),c_Transitive__Closure_Otrancl(tc_Nat_Onat,c_Wellfounded_Opred__nat))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)))) # label(fact_less__eq) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1805 (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_d,T_e,V_f_H_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_f_H_2),V_f_2),V_A_2)))) # label(fact_comp__inj__on__iff) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1806 (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(c_Finite__Set_Ofold1(T_c,V_f_2),V_A_2) = hAPP(V_F_2,V_A_2)))) # label(fact_folding__one_Oeq__fold) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1807 (all V_y_2 all V_x_2 all T_c (class_Lattices_Oboolean__algebra(T_c) -> (V_y_2 = V_x_2 <-> hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_x_2) = hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_y_2)))) # label(fact_compl__eq__compl__iff) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1808 (all V_n_2 all V_m_2 (V_m_2 = V_n_2 | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2)))) # label(fact_le__eq__less__or__eq) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1809 (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]. 5.89/5.96 1810 (all V_r_2 all V_x_2 all T_c 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))) = hAPP(hAPP(c_Equiv__Relations_Oquotient(T_c),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))),V_r_2)) # label(fact_singleton__quotient) # label(axiom) # label(non_clause). [assumption]. 5.89/5.96 1811 (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]. 5.89/5.96 1812 (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]. 5.89/5.97 1813 (all V_com2_2 all V_com1_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_OSemi(V_com1_2,V_com2_2)) # label(fact_com_Osimps_I47_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1814 (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]. 5.89/5.97 1815 (all V_n_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m_2)),V_n_2)) <-> hBOOL(c_Nat_Onat_Onat__case(tc_HOL_Obool,c_fFalse,hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2)))) # label(fact_less__eq__nat_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1816 (all V_N_2 all V_h_2 all T_c (class_Lattices_Olattice(T_c) -> ((all B_x all B_y hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_c),hAPP(V_h_2,B_x)),hAPP(V_h_2,B_y)) = hAPP(V_h_2,hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_c),B_x),B_y))) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_N_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_N_2 -> hAPP(V_h_2,c_Big__Operators_Olattice__class_OInf__fin(T_c,V_N_2)) = c_Big__Operators_Olattice__class_OInf__fin(T_c,hAPP(c_Set_Oimage(T_c,T_c,V_h_2),V_N_2))))))) # label(fact_Inf__fin_Ohom__commute) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1817 (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]. 5.89/5.97 1818 (all V_b all V_a all V_c all T_a (class_Groups_Oordered__ab__semigroup__add__imp__le(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__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]. 5.89/5.97 1819 (all V_A_2 all T_c V_A_2 = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)))) # label(fact_Int__UNIV__right) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1820 (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]. 5.89/5.97 1821 (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]. 5.89/5.97 1822 (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]. 5.89/5.97 1823 (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]. 5.89/5.97 1824 (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]. 5.89/5.97 1825 (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]. 5.89/5.97 1826 (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]. 5.89/5.97 1827 (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]. 5.89/5.97 1828 (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]. 5.89/5.97 1829 (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]. 5.89/5.97 1830 (all V_b all V_a all T_a (class_Groups_Ogroup__add(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_b)),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)))) # label(fact_minus__add) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1831 (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]. 5.89/5.97 1832 (all V_xs_2 all T_c hAPP(hAPP(c_List_Omap(T_c,T_c),c_COMBI(T_c)),V_xs_2) = V_xs_2) # label(fact_map__ident) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1833 (all V_B_2 all V_A_2 all T_e all V_r_2 all T_c all T_d hAPP(c_Relation_OImage(T_d,T_c,V_r_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_e,tc_fun(T_d,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_e,tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_e),c_Relation_OImage(T_d,T_c,V_r_2)),V_B_2))) # label(fact_Image__UN) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1834 (all V_Pa_2 all V_A_2 all T_d all T_c 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))) = 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)))) # label(fact_Collect__ball__eq) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1835 (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]. 5.89/5.97 1836 (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]. 5.89/5.97 1837 (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]. 5.89/5.97 1838 (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) -> (hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2) = V_B_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_c,T_e),V_g_2),V_A_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_d,T_e),V_h_2),V_B_2)))))) # label(fact_setprod__reindex__cong) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1839 (all V_a5_2 all V_a1_2 all V_a2_2 all V_a4_2 all V_a3_2 all T_d all T_c (c_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)) <-> (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) & c_Nitpick_Ofold__graph_H(T_c,T_d,V_a3_2,V_a4_2,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_a2_2),hAPP(hAPP(c_Set_Oinsert(T_c),B_x),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))),B_y) & hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_a2_2)))) | 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)))) # label(fact_fold__graph_H_Oequations_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1840 (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]. 5.89/5.97 1841 (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]. 5.89/5.97 1842 (all V_n all V_a all T_a (class_Rings_Oring__1(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_n)) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_n)))) # label(fact_power__minus__even) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1843 (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]. 5.89/5.97 1844 (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]. 5.89/5.97 1845 (all V_A_2 all V_f_2 all T_d all T_c (class_Groups_Ocomm__monoid__mult(T_c) & class_Rings_Osemiring__1(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]. 5.89/5.97 1846 (all V_xs_2 all T_c all V_x_2 all V_Pa_2 (-hBOOL(hAPP(V_Pa_2,V_x_2)) -> c_List_Oremove1(T_c,V_x_2,hAPP(c_List_Ofilter(T_c,V_Pa_2),V_xs_2)) = hAPP(c_List_Ofilter(T_c,V_Pa_2),V_xs_2))) # label(fact_remove1__filter__not) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1847 (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]. 5.89/5.97 1848 (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) -> 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) -> 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)) = V_a_2))) # label(fact_divide__eq__eq__number__of1) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1849 (all T_1 all T_2 (class_Finite__Set_Ofinite(T_1) & class_Finite__Set_Ofinite(T_2) -> class_Finite__Set_Ofinite(tc_fun(T_2,T_1)))) # label(arity_fun__Finite__Set_Ofinite) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1850 (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]. 5.89/5.97 1851 (all V_f_2 all V_A_2 all T_c all T_d (class_Complete__Lattice_Ocomplete__lattice(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,T_d),V_A_2),V_f_2) = hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_d,hAPP(hAPP(c_COMBB(T_d,tc_fun(T_d,T_d),T_c),c_Lattices_Osemilattice__inf__class_Oinf(T_d)),V_f_2)),c_Orderings_Otop__class_Otop(T_d)),V_A_2)))) # label(fact_INFI__fold__inf) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1852 (all V_ys_2 all T_c all V_xs_2 (c_List_Olist_ONil(T_c) = V_ys_2 <-> V_xs_2 = hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_ys_2))) # label(fact_self__append__conv) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1853 (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]. 5.89/5.97 1854 (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 -> 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]. 5.89/5.97 1855 (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]. 5.89/5.97 1856 (all V_y_2 all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> (V_x_2 = V_y_2 <-> hAPP(c_SetInterval_Oord__class_OatLeast(T_c),V_y_2) = hAPP(c_SetInterval_Oord__class_OatLeast(T_c),V_x_2)))) # label(fact_atLeast__eq__iff) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1857 (all V_a all T_a (class_Rings_Odivision__ring__inverse__zero(T_a) -> (V_a = c_Groups_Ozero__class_Ozero(T_a) -> hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_a) = c_Groups_Ozero__class_Ozero(T_a)) & (V_a != c_Groups_Ozero__class_Ozero(T_a) -> c_Groups_Oone__class_Oone(T_a) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_a)))) # label(fact_divide__self__if) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1858 (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]. 5.89/5.97 1859 (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]. 5.89/5.97 1860 (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]. 5.89/5.97 1861 (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]. 5.89/5.97 1862 (all V_a_2 all T_c (class_Groups_Ogroup__add(T_c) -> (c_Groups_Ozero__class_Ozero(T_c) = hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_a_2) <-> c_Groups_Ozero__class_Ozero(T_c) = V_a_2))) # label(fact_neg__equal__0__iff__equal) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1863 (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]. 5.89/5.97 1864 (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]. 5.89/5.97 1865 (all V_i_1 all T_a hAPP(c_New__DSequence_Opos__empty(T_a),V_i_1) = c_Lazy__Sequence_Oempty(T_a)) # label(fact_New__DSequence_Opos__empty__def) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1866 (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)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(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_C_2)),V_B_2)) # label(fact_Diff__Int2) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1867 (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]. 5.89/5.97 1868 (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_I14_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1869 (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]. 5.89/5.97 1870 (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]. 5.89/5.97 1871 (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]. 5.89/5.97 1872 (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))))) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_t_2),V_D_2) = V_t_2)))) # label(fact_bset_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1873 (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]. 5.89/5.97 1874 (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]. 5.89/5.97 1875 (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]. 5.89/5.97 1876 (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)))) <-> c_Groups_Ozero__class_Ozero(T_c) != V_x_2 | V_y_2 != c_Groups_Ozero__class_Ozero(T_c)))) # label(fact_sum__squares__gt__zero__iff) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1877 (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)) -> ((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]. 5.89/5.97 1878 (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]. 5.89/5.97 1879 (all V_i all V_j all V_k (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_k),V_j)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j),V_k)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),V_j)),V_k))) # label(fact_diff__add__assoc) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1880 (all V_b all V_a hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a),V_b)) = c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_b))) # label(fact_zmod__zminus2) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1881 (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]. 5.89/5.97 1882 (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),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_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))) & (-hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_f_2),V_A_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_f_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))))))))) # label(fact_setprod__diff1) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1883 (all V_xs_2 all T_c hAPP(hAPP(c_List_Ozip(T_c,T_c),V_xs_2),V_xs_2) = hAPP(hAPP(c_List_Omap(T_c,tc_prod(T_c,T_c)),hAPP(hAPP(c_COMBS(T_c,T_c,tc_prod(T_c,T_c)),c_Product__Type_OPair(T_c,T_c)),c_COMBI(T_c))),V_xs_2)) # label(fact_zip__same__conv__map) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1884 (all V_c all V_a all V_b all T_a (class_Lattices_Osemilattice__inf(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_b),V_c)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_b),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),V_c)))) # label(fact_inf_Oleft__commute) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1885 (all V_xs_2 all V_f_2 all T_c all V_x_2 all V_Pa_2 all T_d (class_Orderings_Olinorder(T_d) -> (-hBOOL(hAPP(V_Pa_2,V_x_2)) -> hAPP(c_List_Ofilter(T_c,V_Pa_2),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_c,T_d,V_f_2),V_x_2),V_xs_2)) = hAPP(c_List_Ofilter(T_c,V_Pa_2),V_xs_2)))) # label(fact_filter__insort__triv) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1886 (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]. 5.89/5.97 1887 (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),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_x_2)),V_y_2))))) # label(fact_minus__dvd__iff) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1888 (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]. 5.89/5.97 1889 (all V_a_2 all T_c (class_Groups_Oordered__ab__group__add(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_a_2),c_Groups_Ozero__class_Ozero(T_c))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),c_Groups_Ozero__class_Ozero(T_c)),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_a_2)))))) # label(fact_neg__0__le__iff__le) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1890 (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]. 5.89/5.97 1891 (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 -> (hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_c_2) = V_b_2 <-> hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_b_2),V_c_2) = V_a_2)))) # label(fact_nonzero__eq__divide__eq) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1892 (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]. 5.89/5.97 1893 (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]. 5.89/5.97 1894 (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_b),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_a),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_Oleft__commute) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1895 (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_x != B_xa)))))) # label(fact_disjoint__iff__not__equal) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1896 (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]. 5.89/5.97 1897 (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]. 5.89/5.97 1898 (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]. 5.89/5.97 1899 (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]. 5.89/5.97 1900 (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]. 5.89/5.97 1901 (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]. 5.89/5.97 1902 (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]. 5.89/5.97 1903 (all V_y_2 all V_x_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_y_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_x_2),V_y_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onat,V_x_2)),hAPP(c_Int_Onat,V_y_2))))))) # label(fact_transfer__nat__int__relations_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1904 (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]. 5.89/5.97 1905 (all V_xs all V_n all T_a hAPP(c_List_Orotate1(T_a),hAPP(c_List_Orotate(T_a,V_n),V_xs)) = hAPP(c_List_Orotate(T_a,V_n),hAPP(c_List_Orotate1(T_a),V_xs))) # label(fact_rotate1__rotate__swap) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1906 (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]. 5.89/5.97 1907 (all V_c_2 all T_d all V_A_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> hAPP(c_Set_Oimage(T_c,T_d,hAPP(c_COMBK(T_d,T_c),V_c_2)),V_A_2) = hAPP(hAPP(c_Set_Oinsert(T_d),V_c_2),c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool))))) # label(fact_image__constant) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1908 (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]. 5.89/5.97 1909 (all V_f_2 all V_S_2 all V_T_2 all T_c all T_d (class_Groups_Ocomm__monoid__mult(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_T_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_S_2),V_T_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_T_2),V_S_2))) -> hAPP(V_f_2,B_x) = c_Groups_Oone__class_Oone(T_d))) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_f_2),V_T_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_f_2),V_S_2)))))) # label(fact_setprod__mono__one__left) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1910 (all V_k_2 all T_c (class_Orderings_Olinorder(T_c) -> hAPP(c_SetInterval_Oord__class_OatMost(T_c),V_k_2) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OgreaterThan(T_c),V_k_2)))) # label(fact_Compl__greaterThan) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1911 (all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> c_Big__Operators_Olinorder__class_OMin(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) = V_x_2)) # label(fact_Min_Osingleton) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1912 (all V_f_2 all T_d all T_c all V_S_2 (c_Orderings_Obot__class_Obot(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool)) != V_S_2 -> ((all B_A (hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),B_A),V_S_2)) -> c_Fun_Oinj__on(T_c,T_d,V_f_2,B_A))) -> c_Fun_Oinj__on(T_c,T_d,V_f_2,c_Complete__Lattice_OInf__class_OInf(tc_fun(T_c,tc_HOL_Obool),V_S_2))))) # label(fact_inj__on__Inter) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1913 (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]. 5.89/5.97 1914 (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]. 5.89/5.97 1915 (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]. 5.89/5.97 1916 (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 (V_p_2 = hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),B_a),B_b) -> 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]. 5.89/5.97 1917 (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]. 5.89/5.97 1918 (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]. 5.89/5.97 1919 (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]. 5.89/5.97 1920 (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]. 5.89/5.97 1921 (all V_xs_2 all T_c (hBOOL(hAPP(c_List_Odistinct(T_c),V_xs_2)) -> 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_distinct__card) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1922 (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]. 5.89/5.97 1923 (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]. 5.89/5.97 1924 (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]. 5.89/5.97 1925 (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]. 5.89/5.97 1926 (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]. 5.89/5.97 1927 (all V_B_2 all V_A_2 all T_d all V_C_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_C_2)) -> (hAPP(hAPP(c_Product__Type_OSigma(T_d,T_c),V_B_2),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_d),V_C_2)) = hAPP(hAPP(c_Product__Type_OSigma(T_d,T_c),V_A_2),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_d),V_C_2)) <-> V_B_2 = V_A_2))) # label(fact_Times__eq__cancel2) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1928 (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]. 5.89/5.97 1929 (all V_x_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_c,c_Orderings_Oord__class_Omin(T_c)),V_x_2),V_A_2) = c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)))))) # label(fact_min__max_OInf__fin_Oeq__fold_H) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1930 (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]. 5.89/5.97 1931 (all V_xs_2 all V_x_2 all T_c (-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2))) -> V_xs_2 = c_List_Oremove1(T_c,V_x_2,V_xs_2))) # label(fact_remove1__idem) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1932 (all V_v_H_2 all 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) <-> (-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))) -> c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v_2)) & (-hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v_H_2))) -> V_v_H_2 = V_v_2)) & (hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v_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)))))) # label(fact_eq__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1933 (all V_x_2 all V_Pa_2 all V_xs_2 all V_f_2 all T_d all T_c (class_Orderings_Olinorder(T_c) -> (c_List_Olinorder__class_Osorted(T_c,hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_xs_2)) -> (hBOOL(hAPP(V_Pa_2,V_x_2)) -> hAPP(c_List_Ofilter(T_d,V_Pa_2),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_d,T_c,V_f_2),V_x_2),V_xs_2)) = hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_d,T_c,V_f_2),V_x_2),hAPP(c_List_Ofilter(T_d,V_Pa_2),V_xs_2)))))) # label(fact_filter__insort) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1934 (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]. 5.89/5.97 1935 (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]. 5.89/5.97 1936 (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]. 5.89/5.97 1937 (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]. 5.89/5.97 1938 (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]. 5.89/5.97 1939 (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]. 5.89/5.97 1940 (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]. 5.89/5.97 1941 (all V_xs all V_x all T_a all V_n ((V_n = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> V_x = hAPP(hAPP(c_List_Onth(T_a),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)),V_n)) & (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != V_n -> hAPP(hAPP(c_List_Onth(T_a),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)),V_n) = hAPP(hAPP(c_List_Onth(T_a),V_xs),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),c_Groups_Oone__class_Oone(tc_Nat_Onat)))))) # label(fact_nth__Cons_H) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1942 (all V_r_2 all V_x_2 all T_c all V_y_2 ((exists B_b exists B_z hAPP(hAPP(c_List_Olist_OCons(T_c),B_b),B_z) = V_y_2) -> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_x_2),hAPP(hAPP(c_List_Oappend(T_c),V_x_2),V_y_2))),c_List_Olexord(T_c,V_r_2))))) # label(fact_lexord__append__rightI) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1943 (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]. 5.89/5.97 1944 (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]. 5.89/5.97 1945 (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]. 5.89/5.97 1946 (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]. 5.89/5.97 1947 (all V_k_2 all V_n_2 all V_Pa_2 (hBOOL(hAPP(V_Pa_2,c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_n_2,V_k_2))) <-> (V_k_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> hBOOL(hAPP(V_Pa_2,V_n_2))) & (V_k_2 != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> (all B_i all B_j (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_j),V_k_2)) -> (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)))))))) # label(fact_split__mod) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1948 (all V_n all V_m hAPP(c_Product__Type_Ofst(tc_Nat_Onat,tc_Nat_Onat),c_Divides_Odivmod__nat(V_m,V_n)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),V_n)) # label(fact_div__nat__def) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1949 (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]. 5.89/5.97 1950 (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]. 5.89/5.97 1951 (all V_seed_2 all V_size_2 all V_nrandom_2 all V_f_2 all V_R_2 all T_c all T_d c_New__Random__Sequence_Opos__decr__bind(T_d,T_c,V_R_2,V_f_2,V_nrandom_2,V_size_2,V_seed_2) = c_New__DSequence_Opos__decr__bind(T_d,T_c,hAPP(hAPP(hAPP(V_R_2,V_nrandom_2),V_size_2),V_seed_2),hAPP(hAPP(c_COMBC(T_d,tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Lazy__Sequence_Olazy__sequence(T_c))),hAPP(hAPP(c_COMBC(T_d,tc_Code__Numeral_Ocode__numeral,tc_fun(tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Lazy__Sequence_Olazy__sequence(T_c)))),hAPP(hAPP(c_COMBC(T_d,tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Lazy__Sequence_Olazy__sequence(T_c))))),V_f_2),V_nrandom_2)),V_size_2)),V_seed_2))) # label(fact_New__Random__Sequence_Opos__decr__bind__def) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1952 (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_x_2 & V_y_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),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]. 5.89/5.97 1953 (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]. 5.89/5.97 1954 (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) <-> 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__eq__add__iff2) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1955 (all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> hAPP(c_Groups_Oabs__class_Oabs(T_a),c_Groups_Ozero__class_Ozero(T_a)) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_abs__zero) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1956 (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]. 5.89/5.97 1957 (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]. 5.89/5.97 1958 (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_Set_Oinsert(T_c),V_x_2),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))))))))) # label(fact_sorted__list__of__set__insert) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1959 (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]. 5.89/5.97 1960 (all V_m hAPP(c_Nat_OSuc,V_m) != c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_Zero__neq__Suc) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1961 (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]. 5.89/5.97 1962 (all V_A_2 all V_f_2 all T_d all V_B_2 all T_c ((exists B_AA (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_d,tc_HOL_Obool)),B_AA),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_subset__image__iff) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1963 (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]. 5.89/5.97 1964 (all V_xs all T_a all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))) -> hAPP(hAPP(c_List_Oappend(T_a),hAPP(hAPP(c_List_Otake(T_a),V_i),V_xs)),hAPP(hAPP(c_List_Olist_OCons(T_a),hAPP(hAPP(c_List_Onth(T_a),V_xs),V_i)),c_List_Olist_ONil(T_a))) = hAPP(hAPP(c_List_Otake(T_a),hAPP(c_Nat_OSuc,V_i)),V_xs))) # label(fact_take__Suc__conv__app__nth) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1965 (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]. 5.89/5.97 1966 (all V_B_2 all V_f_2 all T_c all T_d (class_Groups_Ocomm__monoid__add(T_d) -> (c_Fun_Oinj__on(T_c,T_d,V_f_2,V_B_2) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_d),c_Fun_Oid(T_d)),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_B_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_f_2),V_B_2)))) # label(fact_setsum__reindex__id) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1967 (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]. 5.89/5.97 1968 (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]. 5.89/5.97 1969 (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)) <-> (exists B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(c_List_Oset(T_c),c_List_Oreplicate(T_c,V_n_2,V_a_2)))) & hBOOL(hAPP(V_Pa_2,B_x)))))) # label(fact_Bex__set__replicate) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1970 (all V_br all V_bq all V_b all V_ar all V_aq all V_c all V_a (hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a,V_c),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_aq),V_ar))) -> (hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_b,V_c),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_bq),V_br))) -> (c_Groups_Ozero__class_Ozero(tc_Int_Oint) != V_c -> hBOOL(hAPP(c_Divides_Odivmod__int__rel(hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_a),V_b),V_c),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_aq),V_bq)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_ar),V_br)),V_c))),c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_ar),V_br),V_c)))))))) # label(fact_zadd1__lemma) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1971 (all V_n_2 (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n_2 <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))))) # label(fact_less__Suc0) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1972 (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(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))))))) # label(fact_abs__power2) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1973 (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]. 5.89/5.97 1974 (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]. 5.89/5.97 1975 (all V_c all V_b all V_a all T_a (class_Rings_Odivision__ring(T_a) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_c)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_b),V_c)) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_b)),V_c))) # label(fact_diff__divide__distrib) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1976 (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]. 5.89/5.97 1977 (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]. 5.89/5.97 1978 (all V_a all T_a (class_Lattices_Osemilattice__inf(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),V_a) = V_a)) # label(fact_inf_Oidem) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1979 (all V_n_2 all V_R_2 all T_c hAPP(hAPP(c_Relation_Orel__comp(T_c,T_c,T_c),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_c,T_c),tc_HOL_Obool),V_n_2),V_R_2)),V_R_2) = hAPP(hAPP(c_Relation_Orel__comp(T_c,T_c,T_c),V_R_2),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_c,T_c),tc_HOL_Obool),V_n_2),V_R_2))) # label(fact_rel__pow__commute) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1980 (all V_m_2 all V_n_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2),V_m_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_n_2)),V_m_2)))) # label(fact_less__eq__Suc__le) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1981 (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]. 5.89/5.97 1982 (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]. 5.89/5.97 1983 (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]. 5.89/5.97 1984 (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]. 5.89/5.97 1985 (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]. 5.89/5.97 1986 (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]. 5.89/5.97 1987 (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]. 5.89/5.97 1988 (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]. 5.89/5.97 1989 (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]. 5.89/5.97 1990 (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_Nat_Onat,T_c),V_f_2),V_A_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,T_c),hAPP(hAPP(c_COMBB(tc_Nat_Onat,T_c,tc_Int_Oint),V_f_2),c_Int_Onat)),hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),V_A_2)))) # label(fact_transfer__nat__int__sum__prod_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1991 (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]. 5.89/5.97 1992 (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_f,T_d,T_c,hAPP(c_Fun_Ocomp(T_e,T_d,T_f,V_f_2),V_g_2)),V_h_2) = 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))) # label(fact_o__assoc) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1993 (all V_l_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit1(V_k_2)),c_Int_OBit0(V_l_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),V_l_2)))) # label(fact_rel__simps_I16_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1994 (all V_r_2 all T_c c_Transitive__Closure_Ortrancl(T_c,V_r_2) = c_Transitive__Closure_Ortrancl(T_c,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_r_2),c_Relation_OId(T_c)))) # label(fact_rtrancl__r__diff__Id) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1995 (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]. 5.89/5.97 1996 (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_c),hAPP(c_Set_Ovimage(T_c,T_d,V_h_2),V_F_2))) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),V_F_2))))) # label(fact_finite__vimage__iff) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1997 (all V_x all V_n all T_a c_List_Oreplicate(T_a,V_n,V_x) = hAPP(c_List_Orev(T_a),c_List_Oreplicate(T_a,V_n,V_x))) # label(fact_rev__replicate) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1998 (all 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)) = 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)))) # label(fact_one__div__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 1999 (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_a_2,hAPP(V_b_2,V_v_2)) = hAPP(V_c_2,V_v_2))) # label(fact_o__eq__dest__lhs) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 2000 (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]. 5.89/5.97 2001 (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]. 5.89/5.97 2002 (all V_y_2 all V_xs_2 all T_c (class_Orderings_Olinorder(T_c) -> (c_List_Olinorder__class_Osorted(T_c,hAPP(c_List_Orev(T_c),V_xs_2)) -> (c_List_Olist_ONil(T_c) = V_xs_2 -> V_y_2 = c_List_Ofoldr(T_c,T_c,c_Orderings_Oord__class_Omax(T_c),V_xs_2,V_y_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))))) # label(fact_foldr__max__sorted) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 2003 (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]. 5.89/5.97 2004 (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]. 5.89/5.97 2005 (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]. 5.89/5.97 2006 (all V_n_2 all V_m_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) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),V_m_2),hAPP(c_Nat_OSuc,V_n_2))) # label(fact_min__Suc2) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 2007 (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]. 5.89/5.97 2008 (all V_A_2 all V_a_2 all T_c (V_a_2 = c_List_Olist_ONil(T_c) | (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(tc_List_Olist(T_c)),B_l),c_List_Olists(T_c,V_A_2))) & hBOOL(hAPP(hAPP(c_member(T_c),B_a),V_A_2)))) <-> hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_c)),V_a_2),c_List_Olists(T_c,V_A_2))))) # label(fact_lists_Osimps) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 2009 (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]. 5.89/5.97 2010 (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]. 5.89/5.97 2011 (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]. 5.89/5.97 2012 (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]. 5.89/5.97 2013 (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_Olinorder__class_OMax(T_c,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))))))) # label(fact_Max_Osubset__idem) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 2014 (all V_w all T_a (class_Int_Onumber__ring(T_a) & class_Int_Oring__char__0(T_a) -> -c_Int_Oiszero(T_a,hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OBit1(V_w))))) # label(fact_iszero__number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 2015 (all V_k_2 all V_j_2 all V_i_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i_2),V_j_2)) -> (hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_k_2),V_i_2) = V_j_2 <-> V_k_2 = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j_2),V_i_2)))) # label(fact_le__imp__diff__is__add) # label(axiom) # label(non_clause). [assumption]. 5.89/5.97 2016 (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]. 5.89/5.98 2017 (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]. 5.89/5.98 2018 (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)) -> hAPP(V_B_2,B_x) = 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(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__UNIV__conv_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2019 (all V_k all V_j all V_i hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_i),V_j)),V_k) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_i),V_k)),V_j)) # label(fact_diff__commute) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2020 (all V_n all V_a all T_a (class_Rings_Olinordered__idom(T_a) -> hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)) = hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),V_n)))) # label(fact_abs__power__minus) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2021 (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]. 5.89/5.98 2022 (all V_v ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_v),c_Int_OPls)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = c_Groups_Oone__class_Oone(tc_Nat_Onat)) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_v),c_Int_OPls)) -> hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_Osucc(V_v)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),c_Groups_Oone__class_Oone(tc_Nat_Onat))))) # label(fact_nat__number__of__add__1) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2023 (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]. 5.89/5.98 2024 (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]. 5.89/5.98 2025 (all V_y_2 all T_c (class_Rings_Olinordered__idom(T_c) & class_Int_Onumber__ring(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),c_Groups_Ozero__class_Ozero(T_c)),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_y_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OPls),V_y_2))))) # label(fact_le__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2026 (all V_x c_Int_Osucc(c_Int_Opred(V_x)) = V_x) # label(fact_succ__pred) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2027 (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]. 5.89/5.98 2028 (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,V_xs_2,c_List_Ofoldr(T_d,T_c,V_f_2,V_ys_2,V_a_2)) = 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)) # label(fact_foldr__append) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2029 (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]. 5.89/5.98 2030 (all V_A_H_2 all T_d all T_c all V_A_2 (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_A_2 -> ((exists B_g V_A_2 = hAPP(c_Set_Oimage(T_d,T_c,B_g),V_A_H_2)) <-> (exists B_f (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]. 5.89/5.98 2031 (all V_x_2 all V_A_2 all T_c (hBOOL(hAPP(c_Predicate_OPowp(T_c,hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_A_2)),V_x_2)) <-> hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),V_x_2),hAPP(c_Set_OPow(T_c),V_A_2))))) # label(fact_Powp__Pow__eq) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2032 (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]. 5.89/5.98 2033 (all V_x_2 all T_c (class_Groups_Oordered__ab__group__add(T_c) -> hAPP(c_Set_Oimage(T_c,T_c,c_Groups_Ouminus__class_Ouminus(T_c)),hAPP(c_SetInterval_Oord__class_OatLeast(T_c),V_x_2)) = hAPP(c_SetInterval_Oord__class_OatMost(T_c),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_x_2)))) # label(fact_image__uminus__atLeast) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2034 (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]. 5.89/5.98 2035 (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]. 5.89/5.98 2036 (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]. 5.89/5.98 2037 (all V_a_2 all T_c (class_Groups_Oordered__ab__group__add__abs(T_c) -> (hAPP(c_Groups_Oabs__class_Oabs(T_c),V_a_2) = c_Groups_Ozero__class_Ozero(T_c) <-> V_a_2 = c_Groups_Ozero__class_Ozero(T_c)))) # label(fact_abs__eq__0) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2038 (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]. 5.89/5.98 2039 (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]. 5.89/5.98 2040 (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]. 5.89/5.98 2041 (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]. 5.89/5.98 2042 (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]. 5.89/5.98 2043 (all V_B_2 all V_A_2 all T_c ((all B_X (hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),B_X),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),B_X),V_B_2)))) -> (c_Orderings_Obot__class_Obot(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool)) != V_A_2 -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),c_Complete__Lattice_OInf__class_OInf(tc_fun(T_c,tc_HOL_Obool),V_A_2)),V_B_2))))) # label(fact_Inter__subset) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2044 (all V_A_2 all V_f_2 all T_d all T_c hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Set_Ovimage(T_c,T_d,V_f_2),V_A_2)) = hAPP(c_Set_Ovimage(T_c,T_d,V_f_2),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_d,tc_HOL_Obool)),V_A_2))) # label(fact_vimage__Compl) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2045 (all V_A_2 all T_c (-hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(c_Finite__Set_Ocard(T_c),V_A_2) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_card__infinite) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2046 (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]. 5.89/5.98 2047 (all V_code__numeral_H_2 all V_code__numeral_2 (V_code__numeral_H_2 = V_code__numeral_2 <-> c_Code__Numeral_OSuc__code__numeral(V_code__numeral_H_2) = c_Code__Numeral_OSuc__code__numeral(V_code__numeral_2))) # label(fact_code__numeral_Oinject) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2048 (all V_B_2 all V_A_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),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(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__cartesian__product) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2049 (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]. 5.89/5.98 2050 (all V_n all V_m hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),V_n)),V_n)) # label(fact_div__mod__equality_H) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2051 (all V_t_2 all V_A_2 all V_D_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_D_2)) -> (all B_x ((all B_xa (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xa),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),V_D_2))) -> (all B_xb (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xb),V_A_2)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B_xb),B_xa) != B_x)))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_t_2),B_x)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_t_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B_x),V_D_2)))))))) # label(fact_aset_I7_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2052 (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]. 5.89/5.98 2053 (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]. 5.89/5.98 2054 (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]. 5.89/5.98 2055 (all V_c all V_r all V_q all V_b all V_a (hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a,V_b),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_q),V_r))) -> (c_Groups_Ozero__class_Ozero(tc_Int_Oint) != V_b -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_c)) -> hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),V_c)),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_q),V_c)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_q,V_c))),V_r)))))))) # label(fact_zmult2__lemma) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2056 (all V_A_2 all T_c (V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))))) # label(fact_subset__empty) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2057 (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_d,tc_Nat_Onat),V_k_2),V_t_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)),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))))) # label(fact_setsum__multicount__gen) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2058 (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]. 5.89/5.98 2059 (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]. 5.89/5.98 2060 (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]. 5.89/5.98 2061 (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),c_Divides_Odiv__class_Omod(T_c,V_m_2,V_n_2))) <-> hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_c),V_k_2),V_m_2)))))) # label(fact_dvd__mod__iff) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2062 (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]. 5.89/5.98 2063 (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]. 5.89/5.98 2064 (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]. 5.89/5.98 2065 (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]. 5.89/5.98 2066 (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 -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),V_k),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)) = c_Random_Oinc__shift(V_v,V_k)))) # label(fact_inc__shift__def) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2067 (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]. 5.89/5.98 2068 (all V_q all T_a all T_b (class_Rings_Oring__1(T_b) & class_Rings_Oinverse(T_b) -> hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_b),hAPP(c_Int_Oring__1__class_Oof__int(T_b),hAPP(c_Nitpick_Onum(T_a),V_q))),hAPP(c_Int_Oring__1__class_Oof__int(T_b),hAPP(c_Nitpick_Odenom(T_a),V_q))) = c_Nitpick_Oof__frac(T_a,T_b,V_q))) # label(fact_Nitpick_Oof__frac__def) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2069 (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]. 5.89/5.98 2070 (all V_B_2 all T_d all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),V_B_2)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_c),V_A_2)),hAPP(c_Finite__Set_Ocard(T_d),V_B_2)) = hAPP(c_Finite__Set_Ocard(tc_sum(T_c,T_d)),c_Sum__Type_OPlus(T_c,T_d,V_A_2,V_B_2))))) # label(fact_card__Plus) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2071 (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]. 5.89/5.98 2072 (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]. 5.89/5.98 2073 (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]. 5.89/5.98 2074 (all V_xs_2 all V_x_2 all V_Pa_2 all T_c (c_List_Olist__all(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,V_Pa_2,V_xs_2))) # label(fact_list__all__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2075 (all V_code__numeral hAPP(c_Nat_Osize__class_Osize(tc_Code__Numeral_Ocode__numeral),c_Code__Numeral_OSuc__code__numeral(V_code__numeral)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_Code__Numeral_Ocode__numeral),V_code__numeral)),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_code__numeral_Osize_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2076 (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]. 5.89/5.98 2077 (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]. 5.89/5.98 2078 (all V_x_2 all T_c (class_Lattices_Olattice(T_c) -> V_x_2 = c_Big__Operators_Olattice__class_OSup__fin(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))))) # label(fact_Sup__fin_Osingleton) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2079 (all V_B_2 all V_A_2 all T_c (hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),V_B_2))))) # label(fact_disjoint__eq__subset__Compl) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2080 (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]. 5.89/5.98 2081 (all V_Pa_2 hAPP(c_Set_OCollect(tc_Nat_Onat),V_Pa_2) = hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),hAPP(c_Set_OCollect(tc_Int_Oint),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Int_Oint),c_fconj),hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_HOL_Obool,tc_Int_Oint),V_Pa_2),c_Int_Onat))))) # label(fact_Nat__Transfer_Otransfer__nat__int__set__functions_I5_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2082 (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]. 5.89/5.98 2083 (all V_u_2 all V_m_2 all V_l_2 all T_c (class_Orderings_Oorder(T_c) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_c,V_l_2),V_m_2)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_c,V_m_2),V_u_2)) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) # label(fact_ivl__disj__int_I11_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2084 (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))) -> hAPP(V_g_2,B_x) = V_z_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(hAPP(V_f_2,hAPP(V_F_2,V_A_2)),hAPP(V_F_2,V_B_2))))))) # label(fact_folding__image__simple_Ounion__inter__neutral) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2085 (all T_c all V_ys_2 all V_xs_2 (V_ys_2 = V_xs_2 <-> c_List_Olist__all2(T_c,T_c,c_fequal,V_xs_2,V_ys_2))) # label(fact_list__all2__eq) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2086 (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]. 5.89/5.98 2087 (all V_B_2 all V_A_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),V_A_2)),V_B_2) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))) # label(fact_Compl__Diff__eq) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2088 (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]. 5.89/5.98 2089 (all V_a all T_a (class_Groups_Ogroup__add(T_a) -> c_Groups_Ozero__class_Ozero(T_a) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_a))) # label(fact_diff__self) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2090 (all V_xs_2 all V_n_2 all V_ys_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_List_Olist(T_c)),V_ys_2),hAPP(c_List_Oset(tc_List_Olist(T_c)),c_Enum_On__lists(T_c,V_n_2,V_xs_2)))) -> hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_ys_2) = V_n_2)) # label(fact_length__n__lists__elem) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2091 (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_Otranspose(T_c,c_List_Otranspose(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))) # label(fact_transpose__transpose) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2092 (all V_a all T_a (class_Rings_Omult__zero(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_mult__zero__left) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2093 (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]. 5.89/5.98 2094 (all V_xs_2 all V_x_2 all V_ys_2 all V_Pa_2 all T_c (hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2) = hAPP(c_List_Ofilter(T_c,V_Pa_2),V_ys_2) <-> (exists B_us exists B_vs (hBOOL(hAPP(V_Pa_2,V_x_2)) & V_xs_2 = hAPP(c_List_Ofilter(T_c,V_Pa_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(hAPP(c_List_Oappend(T_c),B_us),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),B_vs)) = V_ys_2)))) # label(fact_filter__eq__Cons__iff) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2095 (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]. 5.89/5.98 2096 (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]. 5.89/5.98 2097 (all V_c all V_b all V_a hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_a),V_c)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_b),V_c))),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_a,V_c)),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_b,V_c))),V_c)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_a),V_b)),V_c)) # label(fact_div__add1__eq) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2098 (all V_k_2 all V_R_2 all V_T_2 all T_d all V_S_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_S_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),V_T_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_d),B_x),V_T_2)) -> hAPP(c_Finite__Set_Ocard(T_c),hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBS(T_c,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_c),c_fconj),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_S_2))),hAPP(hAPP(c_COMBC(T_c,T_d,tc_HOL_Obool),V_R_2),B_x)))) = V_k_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)),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]. 5.89/5.98 2099 (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]. 5.89/5.98 2100 (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)),c_Groups_Oone__class_Oone(T_a)) = hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) # label(fact_one__add__one__is__two) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2101 (all V_a all T_a (class_Rings_Olinordered__idom(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> c_Groups_Oone__class_Oone(T_a) = c_Groups_Osgn__class_Osgn(T_a,V_a)))) # label(fact_sgn__pos) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2102 (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]. 5.89/5.98 2103 (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]. 5.89/5.98 2104 (all V_i_2 all V_f_2 all V_s_2 all T_c all T_d (class_Groups_Oordered__ab__group__add(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_s_2)) -> ((all B_i (hBOOL(hAPP(hAPP(c_member(T_c),B_i),V_s_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_d),c_Groups_Ozero__class_Ozero(T_d)),hAPP(V_f_2,B_i))))) -> (hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_f_2),V_s_2) = c_Groups_Ozero__class_Ozero(T_d) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_i_2),V_s_2)) -> c_Groups_Ozero__class_Ozero(T_d) = hAPP(V_f_2,V_i_2))))))) # label(fact_setsum__nonneg__0) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2105 (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)))) -> (B_x != V_t_2 -> V_t_2 != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B_x),V_D_2))))))) # label(fact_aset_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2106 (all V_B_2 all V_A_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_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_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_d,tc_HOL_Obool))),V_A_2)),V_B_2))) # label(fact_Sigma__Int__distrib2) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2107 (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]. 5.89/5.98 2108 (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]. 5.89/5.98 2109 (all V_B_2 all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_d,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2)),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_B_2)) = hAPP(c_Set_Oimage(T_c,T_d,V_f_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)))) # label(fact_image__set__diff) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2110 (all V_A_2 all V_f_2 all T_d all T_c (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)) = 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]. 5.89/5.98 2111 (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]. 5.89/5.98 2112 (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]. 5.89/5.98 2113 (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]. 5.89/5.98 2114 (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]. 5.89/5.98 2115 (all V_w all V_v (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)))) -> c_Divides_OnegDivAlg(hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)) = hAPP(c_Divides_Oadjust(hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)),c_Divides_OnegDivAlg(hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)))) -> 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))))) # label(fact_negDivAlg__eqn__number__of) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2116 (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_arith__simps_I30_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2117 (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]. 5.89/5.98 2118 (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]. 5.89/5.98 2119 (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]. 5.89/5.98 2120 (all V_u_2 hAPP(c_Finite__Set_Ocard(tc_Int_Oint),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_u_2)) = hAPP(c_Int_Onat,V_u_2)) # label(fact_card__atLeastZeroLessThan__int) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2121 (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]. 5.89/5.98 2122 (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]. 5.89/5.98 2123 (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]. 5.89/5.98 2124 (all V_Q all V_P all T_b all T_a hAPP(hAPP(c_COMBK(T_a,T_b),V_P),V_Q) = V_P) # label(help_c__COMBK__1) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2125 (all 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)) = hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_k_2))) # label(fact_atMost__Suc) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2126 (all V_f_2 all T_d all T_c all V_xs_2 (V_xs_2 != c_List_Olist_ONil(T_c) -> hAPP(V_f_2,hAPP(c_List_Ohd(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)))) # label(fact_hd__map) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2127 (all V_x all T_a (class_Lattices_Olattice(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_x) = V_x)) # label(fact_Sup__fin_Oidem) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2128 (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]. 5.89/5.98 2129 (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]. 5.89/5.98 2130 (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]. 5.89/5.98 2131 (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]. 5.89/5.98 2132 (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]. 5.89/5.98 2133 (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]. 5.89/5.98 2134 (all V_A_2 all T_c ((exists B_x hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2))) <-> c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_A_2)) # label(fact_ex__in__conv) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2135 (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]. 5.89/5.98 2136 (all V_c all V_b all V_a c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_a),V_b),V_c) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_a),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_b,V_c)),V_c)) # label(fact_zmod__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2137 (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]. 5.89/5.98 2138 (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]. 5.89/5.98 2139 (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]. 5.89/5.98 2140 (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]. 5.89/5.98 2141 (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]. 5.89/5.98 2142 (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]. 5.89/5.98 2143 (all V_n all V_a all T_a (class_Rings_Oring__1(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),c_Groups_Oone__class_Oone(T_a))),V_n)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),V_n))) # label(fact_power__minus) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2144 (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]. 5.89/5.98 2145 (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]. 5.89/5.98 2146 (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]. 5.89/5.98 2147 (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]. 5.89/5.98 2148 (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_d,c_Groups_Otimes__class_Otimes(T_e)),V_g_2),c_Groups_Oone__class_Oone(T_e)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_HOL_Obool)),V_I_2),V_A_2)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_e,T_c,c_Groups_Otimes__class_Otimes(T_e)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),T_e,T_c),hAPP(hAPP(c_Finite__Set_Ofold__image(T_e,T_d,c_Groups_Otimes__class_Otimes(T_e)),V_g_2),c_Groups_Oone__class_Oone(T_e))),V_A_2)),c_Groups_Oone__class_Oone(T_e)),V_I_2)))))) # label(fact_fold__image__UN__disjoint) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2149 (all V_r_2 all V_q_2 all V_b_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)) = 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)))) # label(fact_adjust__eq) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2150 (all V_x all T_a (class_Lattices_Oboolean__algebra(T_a) -> hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_x)) = V_x)) # label(fact_double__compl) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2151 (all V_Pa_2 all V_f_2 all T_c all T_d 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)) = c_New__Random__Sequence_Opos__map(T_d,T_c,V_f_2,V_Pa_2)) # label(fact_pos__map__def) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2152 (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]. 5.89/5.98 2153 (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_Oappend(tc_Int_Oint,hAPP(c_Lazy__Sequence_Osingle(tc_Int_Oint),V_i_2),c_Lazy__Sequence_Osmall__lazy_H(V_d_2,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_i_2),c_Groups_Oone__class_Oone(tc_Int_Oint)))) = c_Lazy__Sequence_Osmall__lazy_H(V_d_2,V_i_2)) & (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)))) # label(fact_small__lazy_H_Opsimps) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2154 (all V_x_2 all T_c c_HOL_OThe(T_c,hAPP(c_fequal,V_x_2)) = V_x_2) # label(fact_the__sym__eq__trivial) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2155 (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]. 5.89/5.98 2156 (all V_R_2 all T_c ((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))) <-> c_Equiv__Relations_Oequivp(T_c,V_R_2))) # label(fact_equivp__def) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2157 (all V_B_2 all V_A_2 all T_c all T_d c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,tc_HOL_Obool),hAPP(c_Set_Oimage(T_d,tc_fun(T_c,tc_HOL_Obool),V_B_2),V_A_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) # label(fact_UNION__eq__Union__image) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2158 (all V_i ((V_i = c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> c_Groups_Osgn__class_Osgn(tc_Int_Oint,V_i) = c_Groups_Ozero__class_Ozero(tc_Int_Oint)) & (c_Groups_Ozero__class_Ozero(tc_Int_Oint) != V_i -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_i)) -> hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)) = c_Groups_Osgn__class_Osgn(tc_Int_Oint,V_i)) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_i)) -> c_Groups_Osgn__class_Osgn(tc_Int_Oint,V_i) = c_Groups_Oone__class_Oone(tc_Int_Oint))))) # label(fact_zsgn__def) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2159 (all V_ys_2 all T_d all V_xs_2 all T_c (hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_d)),V_ys_2) -> V_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]. 5.89/5.98 2160 (all V_Pa_2 all T_c (class_Enum_Oenum(T_c) -> (hBOOL(hAPP(c_Enum_Oenum__class_Oenum__ex(T_c),V_Pa_2)) <-> (exists B_x1 hBOOL(hAPP(V_Pa_2,B_x1)))))) # label(fact_enum__ex) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2161 (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]. 5.89/5.98 2162 (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]. 5.89/5.98 2163 (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))) & hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)))) # label(fact_mem__Sigma__iff) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2164 (all T_1 class_Nat_Osize(tc_List_Olist(T_1))) # label(arity_List__Olist__Nat_Osize) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2165 (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]. 5.89/5.98 2166 (all V_zs_2 all V_y_2 all V_x_2 all T_c (class_Groups_Osemigroup__add(T_c) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),V_x_2),hAPP(c_List_Ofoldl(T_c,T_c,c_Groups_Oplus__class_Oplus(T_c),V_y_2),V_zs_2)) = hAPP(c_List_Ofoldl(T_c,T_c,c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),V_x_2),V_y_2)),V_zs_2))) # label(fact_foldl__assoc) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2167 (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]. 5.89/5.98 2168 (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]. 5.89/5.98 2169 (all V_B_2 all V_A_2 (c_Nat__Transfer_Onat__set(V_A_2) -> (c_Nat__Transfer_Onat__set(V_B_2) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_Int_Oint,tc_HOL_Obool)),V_A_2),V_B_2) = hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),hAPP(hAPP(c_Lattices_Osemilattice__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)))))) # label(fact_Nat__Transfer_Otransfer__int__nat__set__functions_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2170 (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]. 5.89/5.98 2171 (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_Orotate1(T_c),V_xs_2))) # label(fact_rotate1__is__Nil__conv) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2172 (all V_xs_2 all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> (c_List_Olinorder__class_Osorted(T_c,V_xs_2) & (all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(c_List_Oset(T_c),V_xs_2))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_x_2),B_x)))) <-> c_List_Olinorder__class_Osorted(T_c,hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2))))) # label(fact_sorted__Cons) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2173 (all V_c all V_b all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c)),V_b) = c_Divides_Odiv__class_Omod(T_a,V_a,V_b))) # label(fact_mod__mult__self2) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2174 (all V_lazy__sequence all V_a all T_a hAPP(c_Nat_Osize__class_Osize(tc_Lazy__Sequence_Olazy__sequence(T_a)),c_Lazy__Sequence_Olazy__sequence_OInsert(T_a,V_a,V_lazy__sequence)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_Lazy__Sequence_Olazy__sequence(T_a)),V_lazy__sequence)),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_lazy__sequence_Osize_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2175 (all V_B_2 all V_A_2 all V_f_2 all T_e all T_d all T_c hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Set_Ovimage(T_c,T_d,hAPP(c_Fun_Ocomp(tc_prod(T_d,T_e),T_d,T_c,c_Product__Type_Ofst(T_d,T_e)),V_f_2)),V_A_2)),hAPP(c_Set_Ovimage(T_c,T_e,hAPP(c_Fun_Ocomp(tc_prod(T_d,T_e),T_e,T_c,c_Product__Type_Osnd(T_d,T_e)),V_f_2)),V_B_2)) = hAPP(c_Set_Ovimage(T_c,tc_prod(T_d,T_e),V_f_2),hAPP(hAPP(c_Product__Type_OSigma(T_d,T_e),V_A_2),hAPP(c_COMBK(tc_fun(T_e,tc_HOL_Obool),T_d),V_B_2)))) # label(fact_vimage__Times) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2176 (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]. 5.89/5.98 2177 (all V_w_2 all V_v_2 all T_c (class_Int_Onumber(T_c) & class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_v_2)),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2))) <-> -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2)),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_v_2)))))) # label(fact_le__number__of__eq__not__less) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2178 (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]. 5.89/5.98 2179 (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_OBit1(c_Int_OPls))))) <-> c_Groups_Oone__class_Oone(T_c) = hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_x_2)))) # label(fact_eq__special_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2180 (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]. 5.89/5.98 2181 (all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_y),V_x)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y)))) # label(fact_linorder__linear) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2182 (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]. 5.89/5.98 2183 (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]. 5.89/5.98 2184 (all V_k_2 all T_c (class_Orderings_Olinorder(T_c) -> hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(T_c),V_k_2)) = hAPP(c_SetInterval_Oord__class_OatLeast(T_c),V_k_2))) # label(fact_Compl__lessThan) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2185 (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),V_b_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_a_2),V_b_2))) -> c_Divides_OnegDivAlg(V_a_2,V_b_2) = hAPP(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))))) # label(fact_negDivAlg_Opsimps) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2186 (all V_Y_2 (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),V_Y_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)) != V_Y_2 -> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool),tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool))),V_Y_2)),c_FunDef_Omax__strict))))) # label(fact_smax__emptyI) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2187 (all V_f_2 all V_B_2 all T_d all V_A_2 all T_c all T_e (class_Groups_Ocomm__monoid__add(T_e) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),hAPP(V_B_2,B_x))))) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_e),hAPP(hAPP(c_COMBS(T_c,tc_fun(T_d,tc_HOL_Obool),T_e),hAPP(hAPP(c_COMBB(tc_fun(T_d,T_e),tc_fun(tc_fun(T_d,tc_HOL_Obool),T_e),T_c),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_e)),V_f_2)),V_B_2)),V_A_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_prod(T_c,T_d),T_e),hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,T_e),V_f_2)),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),V_A_2),V_B_2)))))) # label(fact_setsum__Sigma) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2188 (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]. 5.89/5.98 2189 (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]. 5.89/5.98 2190 (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]. 5.89/5.98 2191 (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(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(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__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]. 5.89/5.98 2192 (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]. 5.89/5.98 2193 (all V_b all V_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b)) -> hAPP(c_Product__Type_Ofst(tc_Int_Oint,tc_Int_Oint),c_Divides_OposDivAlg(V_a,V_b)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b)))) # label(fact_div__pos__pos) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2194 (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]. 5.89/5.98 2195 (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_fun__diff__def) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2196 (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_Otransp(T_c,V_R_2) & c_Predicate_Osymp(T_c,V_R_2))) # label(fact_equivp__reflp__symp__transp) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2197 (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]. 5.89/5.98 2198 (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]. 5.89/5.98 2199 (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]. 5.89/5.98 2200 (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_Ovimage(T_c,T_d,V_f_2),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2)) = V_A_2)) # label(fact_inj__vimage__image__eq) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2201 (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]. 5.89/5.98 2202 (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(c_Set_Ovimage(T_d,T_c,V_f_2),V_A_2)) = 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))))) # label(fact_image__vimage__eq) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2203 (all V_k_2 all V_i_2 (c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_i_2),V_k_2) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_i_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),V_i_2)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_i_2),V_k_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_i_2)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = V_k_2)) # label(fact_zdiv__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.89/5.98 2204 (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)) -> 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,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) = V_x_2))))) # label(fact_Inf__fin_Oremove) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2205 (all V_b_2 all V_a_2 all T_c (class_Groups_Oordered__ab__group__add(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_a_2)),V_b_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_b_2)),V_a_2))))) # label(fact_minus__le__iff) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2206 (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]. 5.89/5.99 2207 (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]. 5.89/5.99 2208 (all V_A_2 all V_I_2 all T_d all V_B_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_B_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(T_c,tc_HOL_Obool)),V_I_2),V_A_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(T_c,tc_HOL_Obool)),V_I_2),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_d),hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_B_2)),V_A_2))) # label(fact_Un__INT__distrib) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2209 (all V_xs_2 all V_x_2 all T_c ((-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2))) -> c_List_Oinsert(T_c,V_x_2,V_xs_2) = hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2)) & (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2))) -> c_List_Oinsert(T_c,V_x_2,V_xs_2) = V_xs_2))) # label(fact_List_Oinsert__def) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2210 (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]. 5.89/5.99 2211 (all V_c all V_b all V_a all T_a (class_Lattices_Osemilattice__inf(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),V_b)),V_c) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_b),V_c)))) # label(fact_inf_Oassoc) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2212 (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]. 5.89/5.99 2213 (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(hAPP(V_f_2,hAPP(V_h_2,B_x)),hAPP(V_h_2,B_y)) = hAPP(V_h_2,hAPP(hAPP(V_f_2,B_x),B_y))) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_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]. 5.89/5.99 2214 (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]. 5.89/5.99 2215 (all T_d all T_c hAPP(c_Product__Type_Oprod_Oprod__case(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_fun(tc_fun(T_d,T_c),tc_prod(tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool),tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool)))),hAPP(hAPP(c_COMBC(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_fun(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_fun(tc_fun(T_d,T_c),tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool))),tc_fun(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_fun(tc_fun(T_d,T_c),tc_prod(tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool),tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool))))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(tc_fun(T_d,T_c),tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool)),tc_fun(tc_fun(T_d,T_c),tc_prod(tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool),tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool)))),tc_fun(tc_fun(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_fun(tc_fun(T_d,T_c),tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool))),tc_fun(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_fun(tc_fun(T_d,T_c),tc_prod(tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool),tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool))))),tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),c_COMBB(tc_fun(tc_fun(T_d,T_c),tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool)),tc_fun(tc_fun(T_d,T_c),tc_prod(tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool),tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool))),tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_d,T_c),tc_fun(tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool),tc_prod(tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool),tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool)))),tc_fun(tc_fun(tc_fun(T_d,T_c),tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool)),tc_fun(tc_fun(T_d,T_c),tc_prod(tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool),tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool)))),tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),c_COMBS(tc_fun(T_d,T_c),tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool),tc_prod(tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool),tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool)))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_d,T_c),tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool)),tc_fun(tc_fun(T_d,T_c),tc_fun(tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool),tc_prod(tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool),tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool)))),tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),hAPP(c_COMBB(tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool),tc_fun(tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool),tc_prod(tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool),tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool))),tc_fun(T_d,T_c)),c_Product__Type_OPair(tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool),tc_fun(tc_prod(T_d,T_d),tc_HOL_Obool)))),c_Relation_Oinv__image(T_c,T_d))))),c_Relation_Oinv__image(T_c,T_d))) = c_FunDef_Orp__inv__image(T_c,T_d)) # label(fact_rp__inv__image__def) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2216 (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_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_inj__on__iff__eq__card) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2217 (all V_Pa_2 all T_d all T_c c_Hilbert__Choice_OEps(tc_prod(T_c,T_d),hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,tc_HOL_Obool),V_Pa_2)) = 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)))) # label(fact_Eps__split) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2218 (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_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = 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__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]. 5.89/5.99 2219 (all V_A_2 all V_f_2 all T_c all T_d (c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)) = hAPP(c_Set_Oimage(T_d,T_c,V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool))) -> 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)) = V_A_2)) # label(fact_image__surj__f__inv__f) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2220 (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]. 5.89/5.99 2221 (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_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2222 (all V_n_2 all V_m_2 all V_k_2 (hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_n_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_m_2) <-> V_n_2 = V_m_2 | V_k_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_mult__cancel1) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2223 (all V_Qa_2 all V_Pa_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(c_Set_OCollect(T_c),V_Pa_2))) & hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(c_Set_OCollect(T_c),V_Qa_2))) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBS(T_c,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_c),c_fdisj),V_Pa_2)),V_Qa_2)))))) # label(fact_finite__Collect__disjI) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2224 (all V_n_2 all V_m_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_k_2)) -> (hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_n_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),V_m_2) <-> V_m_2 = V_n_2))) # label(fact_nat__mult__eq__cancel1) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2225 (all V_l all V_k hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),c_Int_OBit1(V_k)),c_Int_OBit1(V_l)) = c_Int_OBit0(hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_k),V_l))) # label(fact_diff__bin__simps_I10_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2226 (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]. 5.89/5.99 2227 (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]. 5.89/5.99 2228 (all V_m all V_x c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_x),V_m) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_x,V_m)),V_m)) # label(fact_zminus__zmod) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2229 (all V_xs_2 all V_c_2 all V_f_2 all T_d all T_c (class_Rings_Osemiring__0(T_c) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(c_List_Omonoid__add__class_Olistsum(T_c),hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_xs_2))),V_c_2) = hAPP(c_List_Omonoid__add__class_Olistsum(T_c),hAPP(hAPP(c_List_Omap(T_d,T_c),hAPP(hAPP(c_COMBC(T_d,T_c,T_c),hAPP(hAPP(c_COMBB(T_c,tc_fun(T_c,T_c),T_d),c_Groups_Otimes__class_Otimes(T_c)),V_f_2)),V_c_2)),V_xs_2)))) # label(fact_listsum__mult__const) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2230 (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]. 5.89/5.99 2231 (all V_r_2 all T_c hAPP(c_Relation_ORange(T_c,T_c),V_r_2) = hAPP(c_Relation_ORange(T_c,T_c),c_Transitive__Closure_Otrancl(T_c,V_r_2))) # label(fact_trancl__range) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2232 (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]. 5.89/5.99 2233 (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]. 5.89/5.99 2234 (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]. 5.89/5.99 2235 (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]. 5.89/5.99 2236 (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_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))) <-> 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)))) # label(fact_conversep__converse__eq) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2237 (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]. 5.89/5.99 2238 (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(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)))) = hAPP(c_List_Omonoid__add__class_Olistsum(T_d),hAPP(hAPP(c_List_Omap(T_c,T_d),V_f_2),V_xs_2))))) # label(fact_listsum__map__remove1) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2239 (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]. 5.89/5.99 2240 (all V_a_2 all V_b_2 (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_b_2),V_a_2)) -> c_SetInterval_Oord_OatLeastLessThan(tc_Nat_Onat,c_Rings_Odvd__class_Odvd(tc_Nat_Onat),hAPP(hAPP(c_COMBS(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),tc_Nat_Onat),c_COMBS(tc_Nat_Onat,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_Nat_Onat),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Nat_Onat),c_fconj)),c_Rings_Odvd__class_Odvd(tc_Nat_Onat)))),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_Nat_Onat),hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Nat_Onat),c_fNot)),hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_HOL_Obool),c_Rings_Odvd__class_Odvd(tc_Nat_Onat)))),V_a_2,V_b_2) = c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) # label(fact_dvd_OatLeastLessThan__empty) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2241 (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_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_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 <-> V_zs_2 = hAPP(hAPP(c_List_Ozip(T_c,T_d),V_xs_2),V_ys_2)))) # label(fact_zip__eq__conv) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2242 (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]. 5.89/5.99 2243 (all V_r_2 all T_c c_Transitive__Closure_Otrancl(T_c,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_r_2),c_Relation_OId(T_c))) = c_Transitive__Closure_Ortrancl(T_c,V_r_2)) # label(fact_trancl__reflcl) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2244 (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]. 5.89/5.99 2245 (all T_d all T_c hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,tc_prod(T_c,T_d)),c_Product__Type_OPair(T_c,T_d)) = c_Fun_Oid(tc_prod(T_c,T_d))) # label(fact_split__Pair) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2246 (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_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]. 5.89/5.99 2247 (all T_a all V_xs (V_xs != c_List_Olist_ONil(T_a) -> c_List_Olast(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))))) # label(fact_last__conv__nth) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2248 (all V_n (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != V_n -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n)))) # label(fact_gr0I) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2249 (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__map) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2250 (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]. 5.89/5.99 2251 (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]. 5.89/5.99 2252 (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]. 5.89/5.99 2253 (all V_b all V_a all T_a (class_Divides_Osemiring__div(T_a) -> V_a = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Divides_Odiv__class_Omod(T_a,V_a,V_b)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_a),V_b)),V_b)))) # label(fact_semiring__div__class_Omod__div__equality_H) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2254 (all V_x all T_a (class_Rings_Ocomm__semiring__1(T_a) -> V_x = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_x),c_Groups_Oone__class_Oone(tc_Nat_Onat)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2255 (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]. 5.89/5.99 2256 (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]. 5.89/5.99 2257 (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]. 5.89/5.99 2258 (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]. 5.89/5.99 2259 (all V_x_2 all T_c V_x_2 = c_Hilbert__Choice_OEps(T_c,hAPP(c_fequal,V_x_2))) # label(fact_some__sym__eq__trivial) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2260 (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]. 5.89/5.99 2261 (all V_a all V_b (V_b != c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> (c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b) = c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_b)) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b))) & (c_Groups_Ozero__class_Ozero(tc_Int_Oint) != c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_b)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b))),c_Groups_Oone__class_Oone(tc_Int_Oint))))) # label(fact_zdiv__zminus2__eq__if) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2262 (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]. 5.89/5.99 2263 (all V_xs_2 all V_Qa_2 all V_Pa_2 all T_c hAPP(c_List_Ofilter(T_c,hAPP(hAPP(c_COMBS(T_c,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_c),c_fconj),V_Qa_2)),V_Pa_2)),V_xs_2) = hAPP(c_List_Ofilter(T_c,V_Pa_2),hAPP(c_List_Ofilter(T_c,V_Qa_2),V_xs_2))) # label(fact_filter__filter) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2264 (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]. 5.89/5.99 2265 (all V_b_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> (hBOOL(hAPP(hAPP(c_member(T_d),V_b_2),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))))) <-> (exists B_x (V_b_2 = hAPP(V_f_2,B_x) & (all B_y (V_b_2 = hAPP(V_f_2,B_y) -> B_x = B_y))))))) # label(fact_range__ex1__eq) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2266 (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]. 5.89/5.99 2267 (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)) -> (hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> c_Big__Operators_Olattice__class_OInf__fin(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)) = V_x_2) & (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) -> c_Big__Operators_Olattice__class_OInf__fin(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_c),V_x_2),c_Big__Operators_Olattice__class_OInf__fin(T_c,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))))))))) # label(fact_Inf__fin_Oinsert__remove) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2268 (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]. 5.89/5.99 2269 (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_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) = 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))))) # label(fact_setsum__setsum__restrict) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2270 (all V_n_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m_2)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n_2))))) # label(fact_zle__int) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2271 (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_Set_Oimage(tc_Nat_Onat,tc_Nat_Onat,hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Groups_Ominus__class_Ominus(tc_Nat_Onat)),V_c_2)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_x_2),V_y_2)) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_x_2),V_c_2)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_y_2),V_c_2))) & (-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(c_Set_Oimage(tc_Nat_Onat,tc_Nat_Onat,hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Groups_Ominus__class_Ominus(tc_Nat_Onat)),V_c_2)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_x_2),V_y_2)) = hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_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]. 5.89/5.99 2272 (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_x) = hAPP(V_f_2,B_xa) -> B_x = B_xa))))))) # label(fact_inj__on__def) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2273 (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]. 5.89/5.99 2274 (all V_xs all T_a V_xs = hAPP(hAPP(c_List_Oappend(T_a),V_xs),c_List_Olist_ONil(T_a))) # label(fact_append__Nil2) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2275 (all V_n_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n_2),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) <-> V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_le__0__eq) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2276 (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]. 5.89/5.99 2277 (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]. 5.89/5.99 2278 (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]. 5.89/5.99 2279 (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]. 5.89/5.99 2280 (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]. 5.89/5.99 2281 (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]. 5.89/5.99 2282 (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]. 5.89/5.99 2283 (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]. 5.89/5.99 2284 (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]. 5.89/5.99 2285 (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),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_v),V_v_H))),V_k)) & (hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v_H))) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),V_k) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H)),V_k)))))) # label(fact_nat__number__of__add__left) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2286 (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]. 5.89/5.99 2287 (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]. 5.89/5.99 2288 (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)) -> V_y_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__equality) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2289 (all V_n_2 all V_v_2 (hBOOL(hAPP(hAPP(c_HOL_OLet(tc_Int_Oint,tc_HOL_Obool),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_Opred(V_v_2))),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Int_Oint),c_fimplies),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Int_Oint),c_fNot),c_Nat__Numeral_Oneg))),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Nat_Onat,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_Int_Oint),c_Orderings_Oord__class_Oless(tc_Nat_Onat)),c_Int_Onat)),V_n_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))))) # label(fact_less__number__of__Suc) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2290 (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]. 5.89/5.99 2291 (all V_x all T_a (class_Lattices_Oboolean__algebra(T_a) -> c_Orderings_Otop__class_Otop(T_a) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_x)),V_x))) # label(fact_compl__sup__top) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2292 (all V_m all T_a (class_Rings_Osemiring__1(T_a) -> hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),hAPP(c_Nat_OSuc,V_m)) = 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)))) # label(fact_of__nat__Suc) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2293 (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]. 5.89/5.99 2294 (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]. 5.89/5.99 2295 (all V_y_2 all V_x_2 all T_d all T_c c_HOL_OThe(tc_prod(T_c,T_d),hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,tc_HOL_Obool),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_fun(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool)),T_c),c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_d)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_c),c_fconj),hAPP(c_fequal,V_x_2)))),hAPP(c_fequal,V_y_2)))) = hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_x_2),V_y_2)) # label(fact_The__split__eq) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2296 (all V_n_2 all T_c all V_x_2 all V_Pa_2 ((hBOOL(hAPP(V_Pa_2,V_x_2)) -> hAPP(c_List_Ofilter(T_c,V_Pa_2),c_List_Oreplicate(T_c,V_n_2,V_x_2)) = c_List_Oreplicate(T_c,V_n_2,V_x_2)) & (-hBOOL(hAPP(V_Pa_2,V_x_2)) -> hAPP(c_List_Ofilter(T_c,V_Pa_2),c_List_Oreplicate(T_c,V_n_2,V_x_2)) = c_List_Olist_ONil(T_c)))) # label(fact_filter__replicate) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2297 (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]. 5.89/5.99 2298 (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 (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_d,tc_HOL_Obool)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_d,tc_HOL_Obool)),V_A_H_2),V_B_H_2) -> (c_Fun_Obij__betw(T_c,T_d,V_f_2,V_A_2,V_A_H_2) -> (c_Fun_Obij__betw(T_c,T_d,V_f_2,V_B_2,V_B_H_2) -> c_Fun_Obij__betw(T_c,T_d,V_f_2,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_d,tc_HOL_Obool)),V_A_H_2),V_B_H_2))))))) # label(fact_bij__betw__Disj__Un) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2299 (all V_A_2 all V_f_2 all T_c 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) = 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))) # label(fact_int__setprod) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2300 (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]. 5.89/5.99 2301 (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]. 5.89/5.99 2302 (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]. 5.89/5.99 2303 (all V_f2_2 all V_f1_2 all T_c V_f1_2 = hAPP(c_Nat_Onat_Onat__rec(T_c,V_f1_2,V_f2_2),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_nat__rec__0) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2304 (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]. 5.89/5.99 2305 (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]. 5.89/5.99 2306 (all V_m_2 all V_n_2 (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_m_2 & V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) <-> hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n_2)) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m_2))) # label(fact_negative__eq__positive) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2307 (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]. 5.89/5.99 2308 (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_h_2),V_z_2),V_A_2) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(T_d,T_c,c_Groups_Otimes__class_Otimes(T_d)),V_g_2),V_z_2),V_A_2))))) # label(fact_fold__image__cong) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2309 (all V_f_2 all V_y_2 all V_x_2 all T_c (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)),hAPP(c_Wellfounded_Omeasure(T_c),V_f_2))))) # label(fact_in__measure) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2310 (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]. 5.89/5.99 2311 (all V_a all V_b all T_a (class_Divides_Osemiring__div(T_a) -> (V_b != c_Groups_Ozero__class_Ozero(T_a) -> V_a = hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_a)),V_b)))) # label(fact_div__mult__self1__is__id) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2312 (all V_s_2 all V_r_2 all T_e all T_c all T_d c_Predicate_Opred__comp(T_c,T_e,T_d,c_Predicate_Oconversep(T_e,T_c,V_s_2),c_Predicate_Oconversep(T_d,T_e,V_r_2)) = c_Predicate_Oconversep(T_d,T_c,c_Predicate_Opred__comp(T_d,T_e,T_c,V_r_2,V_s_2))) # label(fact_converse__pred__comp) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2313 (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]. 5.89/5.99 2314 (all V_N_2 all V_h_2 all T_c (class_Lattices_Olattice(T_c) -> ((all B_x all B_y hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_c),hAPP(V_h_2,B_x)),hAPP(V_h_2,B_y)) = hAPP(V_h_2,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_c),B_x),B_y))) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_N_2)) -> (V_N_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> c_Big__Operators_Olattice__class_OSup__fin(T_c,hAPP(c_Set_Oimage(T_c,T_c,V_h_2),V_N_2)) = hAPP(V_h_2,c_Big__Operators_Olattice__class_OSup__fin(T_c,V_N_2))))))) # label(fact_Sup__fin_Ohom__commute) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2315 (all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> V_a = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),c_Groups_Oone__class_Oone(T_a)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I12_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2316 (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]. 5.89/5.99 2317 (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]. 5.89/5.99 2318 (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]. 5.89/5.99 2319 (all V_Ga_2 all V_F_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_Ga_2)) & hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_F_2)) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_F_2),V_Ga_2))))) # label(fact_finite__Un) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2320 (all V_b all V_a ((c_Groups_Ozero__class_Ozero(tc_Int_Oint) = c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b) -> 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) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b)),V_b) = c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_b))))) # label(fact_zmod__zminus2__eq__if) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2321 (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(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)) = hAPP(hAPP(c_List_Omap(T_c,T_d),V_f_2),hAPP(c_List_OremoveAll(T_c,V_x_2),V_xs_2)))) # label(fact_map__removeAll__inj) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2322 (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]. 5.89/5.99 2323 (all V_n_2 all V_m_2 all V_Pa_2 ((exists B_q (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n_2),B_q)),V_m_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n_2),hAPP(c_Nat_OSuc,B_q)))) & hBOOL(hAPP(V_Pa_2,B_q)))) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n_2 & hBOOL(hAPP(V_Pa_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) <-> hBOOL(hAPP(V_Pa_2,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m_2),V_n_2))))) # label(fact_split__div_H) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2324 (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]. 5.89/5.99 2325 (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]. 5.89/5.99 2326 (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]. 5.89/5.99 2327 (all V_b_2 all V_a_2 all V_c_2 all T_c (class_Rings_Olinordered__ring__strict(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_c_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_a_2),V_b_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_c_2),V_a_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_c_2),V_b_2))))))) # label(fact_mult__less__cancel__left__pos) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2328 (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)) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),B_x),V_A_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)),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]. 5.89/5.99 2329 (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]. 5.89/5.99 2330 (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]. 5.89/5.99 2331 (all V_z hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),V_z)),V_z) != c_Groups_Ozero__class_Ozero(tc_Int_Oint)) # label(fact_odd__nonzero) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2332 (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]. 5.89/5.99 2333 (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]. 5.89/5.99 2334 (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))) <-> hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_m_2),V_n_2)) | V_k_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_nat__mult__dvd__cancel__disj) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2335 (all V_k_2 all V_less_2 all V_i_2 all T_c (hBOOL(hAPP(hAPP(V_less_2,V_i_2),V_k_2)) <-> hBOOL(hAPP(hAPP(c_member(T_c),V_i_2),c_SetInterval_Oord_OlessThan(T_c,V_less_2,V_k_2))))) # label(fact_ord_OlessThan__iff) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2336 (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) = 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))))) # label(fact_INTER__def) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2337 (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(tc_Nat_Onat),V_m_2),V_n_2)) <-> 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)))))) # label(fact_of__nat__le__iff) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2338 (all V_xs_2 all T_c c_Random_Oselect(T_c,V_xs_2) = hAPP(hAPP(c_Product__Type_Oscomp(tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_Code__Numeral_Ocode__numeral,tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_prod(T_c,tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral))),c_Random_Orange(hAPP(c_Code__Numeral_Oof__nat,hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2)))),hAPP(hAPP(c_COMBB(T_c,tc_fun(tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_prod(T_c,tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral))),tc_Code__Numeral_Ocode__numeral),c_Product__Type_OPair(T_c,tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral))),hAPP(hAPP(c_COMBB(tc_Nat_Onat,T_c,tc_Code__Numeral_Ocode__numeral),hAPP(c_List_Onth(T_c),V_xs_2)),c_Code__Numeral_Onat__of)))) # label(fact_select__def) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2339 (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]. 5.89/5.99 2340 (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]. 5.89/5.99 2341 (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]. 5.89/5.99 2342 (all V_i_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_k_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_i_2),V_k_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k_2),V_i_2))))) # label(fact_pos__imp__zdiv__pos__iff) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2343 (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_2),V_com_2),V_fun2_2) = hAPP(hAPP(hAPP(c_Hoare__Mirabelle_Otriple_Otriple(T_c),V_fun1_H_2),V_com_H_2),V_fun2_H_2) <-> V_com_2 = V_com_H_2 & V_fun2_H_2 = V_fun2_2 & V_fun1_2 = V_fun1_H_2)) # label(fact_triple_Oinject) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2344 (all V_fs_2 all T_c c_List_Omeasures(T_c,V_fs_2) = hAPP(hAPP(c_Relation_Oinv__image(tc_List_Olist(tc_Nat_Onat),T_c),c_List_Olex(tc_Nat_Onat,c_Wellfounded_Oless__than)),hAPP(hAPP(c_COMBC(T_c,tc_List_Olist(tc_fun(T_c,tc_Nat_Onat)),tc_List_Olist(tc_Nat_Onat)),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_c,tc_Nat_Onat),tc_Nat_Onat),tc_fun(tc_List_Olist(tc_fun(T_c,tc_Nat_Onat)),tc_List_Olist(tc_Nat_Onat)),T_c),c_List_Omap(tc_fun(T_c,tc_Nat_Onat),tc_Nat_Onat)),hAPP(c_COMBC(tc_fun(T_c,tc_Nat_Onat),T_c,tc_Nat_Onat),c_COMBI(tc_fun(T_c,tc_Nat_Onat))))),V_fs_2))) # label(fact_measures__def) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2345 (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]. 5.89/5.99 2346 (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_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)) = 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))) # label(fact_zip__map2) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2347 (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]. 5.89/5.99 2348 (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]. 5.89/5.99 2349 (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]. 5.89/5.99 2350 (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]. 5.89/5.99 2351 (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]. 5.89/5.99 2352 (all V_n_2 all V_k_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m_2),V_k_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n_2),V_k_2))) <-> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_k_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2))))) # label(fact_mult__le__cancel2) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2353 (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]. 5.89/5.99 2354 (all V_r_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)),c_List_Olist_ONil(T_c)),V_y_2)),c_List_Olexord(T_c,V_r_2))) <-> (exists B_a exists B_x V_y_2 = hAPP(hAPP(c_List_Olist_OCons(T_c),B_a),B_x)))) # label(fact_lexord__Nil__left) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2355 (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]. 5.89/5.99 2356 (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]. 5.89/5.99 2357 (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]. 5.89/5.99 2358 (all V_F_2 all V_E_2 all V_B_2 all V_A_2 all T_d all T_c all T_f all T_e hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_prod(T_e,T_f),tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),hAPP(hAPP(c_Product__Type_OSigma(T_e,T_f),V_A_2),hAPP(c_COMBK(tc_fun(T_f,tc_HOL_Obool),T_e),V_B_2))),hAPP(c_Product__Type_Oprod_Oprod__case(T_e,T_f,tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),hAPP(hAPP(c_COMBC(T_e,tc_fun(T_f,tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool))),tc_fun(T_f,tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool)),tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),tc_fun(tc_fun(T_f,tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool))),tc_fun(T_f,tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool))),T_e),c_COMBB(tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool)),tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),T_f)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool)),tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),T_e),c_Product__Type_OSigma(T_c,T_d)),V_E_2))),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool)),T_f),c_COMBK(tc_fun(T_d,tc_HOL_Obool),T_c)),V_F_2)))) = hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_e,tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_E_2)),hAPP(c_COMBK(tc_fun(T_d,tc_HOL_Obool),T_c),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_f,tc_fun(T_d,tc_HOL_Obool)),V_B_2),V_F_2)))) # label(fact_UN__Times__distrib) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2359 (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))) -> c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_a),V_b)))) # label(fact_mod__pos__neg__trivial) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2360 (all V_R_2 all T_d all T_c V_R_2 = hAPP(hAPP(c_Relation_Orel__comp(T_c,T_d,T_d),V_R_2),c_Relation_OId(T_d))) # label(fact_R__O__Id) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2361 (all V_ys_2 all V_xs_2 all V_f_2 all T_c all T_d (class_Orderings_Olinorder(T_d) -> (c_Fun_Oinj__on(T_c,T_d,V_f_2,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_List_Oset(T_c),V_xs_2)),hAPP(c_List_Oset(T_c),V_ys_2))) -> (c_List_Olinorder__class_Osorted(T_d,hAPP(hAPP(c_List_Omap(T_c,T_d),V_f_2),V_xs_2)) -> (hBOOL(hAPP(c_List_Odistinct(T_d),hAPP(hAPP(c_List_Omap(T_c,T_d),V_f_2),V_xs_2))) -> (c_List_Olinorder__class_Osorted(T_d,hAPP(hAPP(c_List_Omap(T_c,T_d),V_f_2),V_ys_2)) -> (hBOOL(hAPP(c_List_Odistinct(T_d),hAPP(hAPP(c_List_Omap(T_c,T_d),V_f_2),V_ys_2))) -> (hAPP(c_List_Oset(T_c),V_xs_2) = hAPP(c_List_Oset(T_c),V_ys_2) -> V_ys_2 = V_xs_2)))))))) # label(fact_map__sorted__distinct__set__unique) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2362 (all V_A_2 all T_c ((-hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> c_Nitpick_Ocard_H(T_c,V_A_2) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) & (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)))) # label(fact_Nitpick_Ocard_H__def) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2363 (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]. 5.89/5.99 2364 (all V_Qa_2 all V_Pa_2 (-hBOOL(hAPP(hAPP(c_fconj,V_Pa_2),V_Qa_2)) | hBOOL(V_Qa_2))) # label(help_c__fconj__3) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2365 (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]. 5.89/5.99 2366 (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]. 5.89/5.99 2367 (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]. 5.89/5.99 2368 (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_OgreaterThanLessThan(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)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_c,V_m_2),V_u_2)))))) # label(fact_ivl__disj__un_I15_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2369 (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]. 5.89/5.99 2370 (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,V_g_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_Oeq__fold) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2371 (all V_l_2 all V_k_2 (V_l_2 = V_k_2 <-> c_Int_OBit1(V_l_2) = c_Int_OBit1(V_k_2))) # label(fact_rel__simps_I51_J) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2372 (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]. 5.89/5.99 2373 (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]. 5.89/5.99 2374 (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]. 5.89/5.99 2375 (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_b_2),V_a_2)),V_e_2)),V_d_2) = V_c_2))) # label(fact_eq__add__iff2) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2376 (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]. 5.89/5.99 2377 (all V_n_2 all V_k_2 all V_m_2 (V_n_2 = V_m_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]. 5.89/5.99 2378 (all V_R_2 all T_c c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)) = hAPP(c_Relation_ORange(T_c,T_c),c_Transitive__Closure_Ortrancl(T_c,V_R_2))) # label(fact_Range__rtrancl) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2379 (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_z1),B_z2) = hAPP(hAPP(V_f_2,B_y1),B_y2))))) -> 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]. 5.89/5.99 2380 (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_B_2)) -> hBOOL(hAPP(V_Pa_2,B_x)))) & (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),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]. 5.89/5.99 2381 (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 -> V_y_2 = hAPP(c_Fun_Ofun__upd(T_c,T_d,V_f_2,V_x_2,V_y_2),V_z_2)) & (V_x_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) = hAPP(V_f_2,V_z_2)))) # label(fact_fun__upd__apply) # label(axiom) # label(non_clause). [assumption]. 5.89/5.99 2382 (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]. 5.89/5.99 2383 (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]. 5.89/6.00 2384 (all V_A_2 all V_g_2 all V_f_2 all T_d all T_c all T_e 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))) = 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)) # label(fact_image__split__eq__Sigma) # label(axiom) # label(non_clause). [assumption]. 5.89/6.00 2385 (all V_q all T_a c_Nitpick_Oinverse__frac(T_a,V_q) = hAPP(hAPP(c_Nitpick_Ofrac(T_a),hAPP(c_Nitpick_Odenom(T_a),V_q)),hAPP(c_Nitpick_Onum(T_a),V_q))) # label(fact_Nitpick_Oinverse__frac__def) # label(axiom) # label(non_clause). [assumption]. 5.89/6.00 2386 (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]. 5.89/6.00 2387 (all V_Pa_2 all V_xs_2 all T_c ((exists B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(c_List_Oset(T_c),V_xs_2))) & hBOOL(hAPP(V_Pa_2,B_x)))) <-> c_List_Olist__ex(T_c,V_Pa_2,V_xs_2))) # label(fact_Bex__set__list__ex) # label(axiom) # label(non_clause). [assumption]. 5.89/6.00 2388 (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]. 5.89/6.00 2389 (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]. 5.89/6.00 2390 (all V_b all V_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_a),V_b)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin)))) # label(fact_div__pos__neg__trivial) # label(axiom) # label(non_clause). [assumption]. 5.89/6.00 2391 (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]. 5.89/6.00 2392 (all V_R_2 all V_n_2 all V_z_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_z_2)),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_c,T_c),tc_HOL_Obool),V_n_2),V_R_2))) -> ((c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n_2 -> V_x_2 != V_z_2) -> -(all B_y all B_m (V_n_2 = hAPP(c_Nat_OSuc,B_m) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),B_y)),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]. 5.89/6.00 2393 (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]. 5.89/6.00 2394 (all V_f_2 all T_d all T_c (class_Groups_Ocomm__monoid__add(T_c) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_c),V_f_2),c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool))) = c_Groups_Ozero__class_Ozero(T_c))) # label(fact_setsum__empty) # label(axiom) # label(non_clause). [assumption]. 5.89/6.00 2395 (all V_y_2 all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> (hAPP(hAPP(c_Orderings_Oord__class_Omax(T_c),V_x_2),V_y_2) = V_y_2 <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_x_2),V_y_2))))) # label(fact_min__max_Ole__iff__sup) # label(axiom) # label(non_clause). [assumption]. 5.89/6.00 2396 (all V_z_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),hAPP(c_Int_Onat,V_z_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),V_z_2)))) # label(fact_one__less__nat__eq) # label(axiom) # label(non_clause). [assumption]. 5.89/6.00 2397 (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]. 5.89/6.00 2398 (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]. 5.89/6.00 2399 (all V_y all V_x all T_a (class_Lattices_Osemilattice__sup(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y)) -> V_y = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y)))) # label(fact_sup__absorb2) # label(axiom) # label(non_clause). [assumption]. 5.89/6.00 2400 (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]. 5.89/6.00 2401 (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]. 5.89/6.00 2402 (all V_B_2 all V_A_2 all T_c hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBS(T_c,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_c),c_fconj),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_A_2))),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_B_2))) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) # label(fact_Int__def) # label(axiom) # label(non_clause). [assumption]. 5.89/6.00 2403 (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]. 5.89/6.00 2404 (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]. 5.89/6.00 2405 (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]. 5.89/6.00 2406 (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_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),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)) -> 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)))))) # label(fact_min__max_OInf__fin_Oremove) # label(axiom) # label(non_clause). [assumption]. 5.89/6.00 2407 (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]. 5.89/6.00 2408 (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_ys_2) = hAPP(c_List_Oset(T_c),V_xs_2) -> V_xs_2 = V_ys_2))))))) # label(fact_sorted__distinct__set__unique) # label(axiom) # label(non_clause). [assumption]. 5.89/6.00 2409 (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]. 5.89/6.00 2410 (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]. 5.89/6.00 2411 (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]. 5.89/6.00 2412 (all V_xs_2 all V_p_2 all T_c c_List_Oconcat(T_c,hAPP(hAPP(c_List_Omap(tc_List_Olist(T_c),tc_List_Olist(T_c)),c_List_Ofilter(T_c,V_p_2)),V_xs_2)) = hAPP(c_List_Ofilter(T_c,V_p_2),c_List_Oconcat(T_c,V_xs_2))) # label(fact_filter__concat) # label(axiom) # label(non_clause). [assumption]. 5.89/6.00 2413 (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]. 5.89/6.00 2414 (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]. 5.89/6.00 2415 (all V_b_2 all V_a_2 all V_c_2 all T_c (class_Rings_Olinordered__ring__strict(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_c_2),c_Groups_Ozero__class_Ozero(T_c))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_b_2),V_a_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_c_2),V_a_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_c_2),V_b_2))))))) # label(fact_mult__less__cancel__left__neg) # label(axiom) # label(non_clause). [assumption]. 5.89/6.00 2416 (all V_f_2 all V_B_2 all V_A_2 all T_c all T_d (class_Groups_Oab__group__add(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_f_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_d),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_d),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_f_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_f_2),V_B_2))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_f_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))))))) # label(fact_setsum__Un) # label(axiom) # label(non_clause). [assumption]. 5.89/6.00 2417 (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]. 5.89/6.00 2418 (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]. 5.89/6.00 2419 (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]. 5.89/6.00 2420 (all V_x all V_xs all T_a (hBOOL(hAPP(c_List_Odistinct(T_a),V_xs)) -> c_List_Oremove1(T_a,V_x,V_xs) = hAPP(c_List_OremoveAll(T_a,V_x),V_xs))) # label(fact_distinct__remove1__removeAll) # label(axiom) # label(non_clause). [assumption]. 5.89/6.00 2421 (all V_Ga_2 all V_F_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_F_2)) | hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_Ga_2)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_F_2),V_Ga_2))))) # label(fact_finite__Int) # label(axiom) # label(non_clause). [assumption]. 5.89/6.00 2422 (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]. 5.89/6.00 2423 (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]. 5.89/6.00 2424 (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]. 5.89/6.00 2425 (all V_ys_2 all V_xs_2 all V_Pa_2 all T_c (c_List_Olist__all(T_c,V_Pa_2,V_xs_2) & c_List_Olist__all(T_c,V_Pa_2,V_ys_2) <-> c_List_Olist__all(T_c,V_Pa_2,hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_ys_2)))) # label(fact_list__all__append) # label(axiom) # label(non_clause). [assumption]. 5.89/6.00 2426 (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]. 5.89/6.00 2427 (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_OSup__fin(T_c,V_A_2) = c_Complete__Lattice_OSup__class_OSup(T_c,V_A_2))))) # label(fact_Sup__fin__Sup) # label(axiom) # label(non_clause). [assumption]. 5.89/6.00 2428 (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]. 5.89/6.00 2429 (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]. 5.89/6.00 2430 (all V_a_2 all V_b_2 all V_c_2 all T_c (class_Fields_Olinordered__field(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_c_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_c_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_b_2),V_c_2)),V_a_2)))))) # label(fact_pos__divide__le__eq) # label(axiom) # label(non_clause). [assumption]. 5.89/6.00 2431 (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]. 5.89/6.00 2432 (all V_A_2 all V_g_2 all V_f_2 all T_d all T_c (class_Groups_Oab__group__add(T_c) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_c),hAPP(hAPP(c_COMBS(T_d,T_c,T_c),hAPP(hAPP(c_COMBB(T_c,tc_fun(T_c,T_c),T_d),c_Groups_Ominus__class_Ominus(T_c)),V_f_2)),V_g_2)),V_A_2) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_c),V_f_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_c),V_g_2),V_A_2)))) # label(fact_setsum__subtractf) # label(axiom) # label(non_clause). [assumption]. 5.89/6.00 2433 (all V_z_2 all V_g_2 all V_f_2 all T_d all T_c 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))) = V_z_2) # label(fact_fold__image__empty) # label(axiom) # label(non_clause). [assumption]. 5.89/6.00 2434 (all V_x_2 all V_f_2 all T_c all T_d all V_k_2 ((c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = V_k_2 -> hAPP(c_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)) & (c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != V_k_2 -> hAPP(c_Random_Oiterate(T_d,T_c,V_k_2,V_f_2),V_x_2) = hAPP(hAPP(c_Product__Type_Oscomp(T_c,T_d,T_c,tc_prod(T_d,T_c)),hAPP(V_f_2,V_x_2)),c_Random_Oiterate(T_d,T_c,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Code__Numeral_Ocode__numeral),V_k_2),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),V_f_2))))) # label(fact_iterate_Osimps) # label(axiom) # label(non_clause). [assumption]. 5.89/6.00 2435 (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_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__iff) # label(axiom) # label(non_clause). [assumption]. 5.89/6.00 2436 (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]. 5.89/6.00 2437 (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]. 5.89/6.00 2438 (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]. 5.89/6.00 2439 (all V_m all V_l c_List_Oreturn__list(hAPP(hAPP(c_List_Oappend(tc_Int_Oint),c_List_Oembed__list(V_l)),c_List_Oembed__list(V_m))) = hAPP(hAPP(c_List_Oappend(tc_Nat_Onat),V_l),V_m)) # label(fact_transfer__nat__int__list__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.89/6.00 2440 (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]. 5.89/6.00 2441 (all V_B_2 all T_d all V_A_2 all T_c ((-(hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),V_B_2)) & hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2))) -> 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(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__conv__if) # label(axiom) # label(non_clause). [assumption]. 5.89/6.00 2442 (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]. 5.89/6.00 2443 (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]. 5.89/6.00 2444 (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]. 5.89/6.00 2445 (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]. 5.89/6.00 2446 (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]. 5.89/6.00 2447 (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]. 5.89/6.00 2448 (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]. 5.89/6.00 2449 (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]. 5.89/6.00 2450 (all V_x_2 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_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)) -> -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]. 5.89/6.00 2451 (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]. 5.89/6.00 2452 (all V_y_2 all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> (V_y_2 = V_x_2 <-> hAPP(c_SetInterval_Oord__class_OlessThan(T_c),V_x_2) = hAPP(c_SetInterval_Oord__class_OlessThan(T_c),V_y_2)))) # label(fact_lessThan__eq__iff) # label(axiom) # label(non_clause). [assumption]. 5.89/6.00 2453 (all V_x_2 all V_f_2 all T_c all T_d (class_Orderings_Olinorder(T_d) -> hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),c_List_Olist_ONil(T_c)) = hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_c,T_d,V_f_2),V_x_2),c_List_Olist_ONil(T_c)))) # label(fact_insort__key_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.89/6.00 2454 (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_a = V_x)) # label(fact_fst__eqD) # label(axiom) # label(non_clause). [assumption]. 5.89/6.00 2455 (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]. 5.89/6.00 2456 (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_c),V_A_2) = hAPP(c_Finite__Set_Ocard(T_d),V_B_2)))))))) # label(fact_card__bij__eq) # label(axiom) # label(non_clause). [assumption]. 5.89/6.00 2457 (all V_j_2 all V_i_2 all V_Pa_2 (c_List_Olist__ex(tc_Int_Oint,V_Pa_2,c_List_Oupto(V_i_2,V_j_2)) <-> -c_List_Oall__interval__int(hAPP(c_Fun_Ocomp(tc_HOL_Obool,tc_HOL_Obool,tc_Int_Oint,c_fNot),V_Pa_2),V_i_2,V_j_2))) # label(fact_list__ex__iff__not__all__inverval__int) # label(axiom) # label(non_clause). [assumption]. 5.89/6.00 2458 (all V_B_2 all V_A_2 all V_a_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_A_2)),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_B_2)) = hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))) # label(fact_insert__inter__insert) # label(axiom) # label(non_clause). [assumption]. 5.89/6.00 2459 (all V_n all V_m (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) = c_Groups_Oone__class_Oone(tc_Int_Oint))) # label(fact_abs__zmult__eq__1) # label(axiom) # label(non_clause). [assumption]. 5.89/6.00 2460 (all V_n_2 all V_d_2 all V_a_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)))) = 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)))) # label(fact_arith__series__int) # label(axiom) # label(non_clause). [assumption]. 5.89/6.00 2461 (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]. 5.89/6.00 2462 (all V_m all V_n all V_k c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_n)),V_m),V_n) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n)) # label(fact_mod__mult__self3) # label(axiom) # label(non_clause). [assumption]. 5.89/6.00 2463 (all V_C_2 all V_B_2 all V_A_2 all T_e all T_c all T_d hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_e,tc_fun(T_d,tc_HOL_Obool)),V_A_2),V_B_2)),V_C_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_e,tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBC(T_e,tc_fun(T_d,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(tc_fun(T_d,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_c,tc_HOL_Obool)),T_e),c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool))),V_B_2)),V_C_2))) # label(fact_UN__UN__flatten) # label(axiom) # label(non_clause). [assumption]. 5.89/6.00 2464 (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]. 5.89/6.00 2465 (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]. 5.89/6.00 2466 (all V_xs_2 all V_f_2 all T_d all V_Pa_2 all T_c hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),hAPP(c_List_Ofilter(T_c,V_Pa_2),hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_xs_2))) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_d)),hAPP(c_List_Ofilter(T_d,hAPP(c_Fun_Ocomp(T_c,tc_HOL_Obool,T_d,V_Pa_2),V_f_2)),V_xs_2))) # label(fact_length__filter__map) # label(axiom) # label(non_clause). [assumption]. 5.89/6.00 2467 (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]. 5.89/6.00 2468 (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]. 5.89/6.00 2469 (all V_N_2 (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),V_N_2)) <-> (exists B_m all B_x (hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),V_N_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B_x),B_m)))))) # label(fact_finite__nat__set__iff__bounded__le) # label(axiom) # label(non_clause). [assumption]. 5.89/6.00 2470 (all V_a_2 all V_S_2 all T_c (class_Orderings_Olinorder(T_c) & class_Complete__Lattice_Ocomplete__lattice(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]. 5.89/6.00 2471 (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]. 5.99/6.00 2472 (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]. 5.99/6.00 2473 (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]. 5.99/6.00 2474 (all T_a (class_Rings_Oring__1(T_a) -> c_Groups_Oone__class_Oone(T_a) = hAPP(c_Int_Oring__1__class_Oof__int(T_a),c_Groups_Oone__class_Oone(tc_Int_Oint)))) # label(fact_of__int__1) # label(axiom) # label(non_clause). [assumption]. 5.99/6.00 2475 (all V_xs all V_x all T_a hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),c_List_Olist_ONil(T_a)) = hAPP(hAPP(c_List_Otake(T_a),c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs))) # label(fact_take__1__Cons) # label(axiom) # label(non_clause). [assumption]. 5.99/6.00 2476 (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,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_OCons(T_c),V_x_2),V_xs_2),hAPP(hAPP(c_List_Olist_OCons(T_d),V_y_2),V_ys_2)))) # label(fact_list__all2__Cons) # label(axiom) # label(non_clause). [assumption]. 5.99/6.00 2477 (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]. 5.99/6.00 2478 (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]. 5.99/6.00 2479 (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]. 5.99/6.00 2480 (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]. 5.99/6.00 2481 (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]. 5.99/6.00 2482 (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_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 | (exists B_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_i),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_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),B_i),V_y_2) = hAPP(hAPP(c_List_Otake(T_c),B_i),V_x_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_x_2),B_i)),hAPP(hAPP(c_List_Onth(T_c),V_y_2),B_i))),V_r_2)))))) # label(fact_lexord__take__index__conv) # label(axiom) # label(non_clause). [assumption]. 5.99/6.00 2483 (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]. 5.99/6.00 2484 (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 -> hAPP(hAPP(c_List_Olist__update(T_a,c_List_Obutlast(T_a,V_xs)),V_k),V_x) = c_List_Obutlast(T_a,hAPP(hAPP(c_List_Olist__update(T_a,V_xs),V_k),V_x))) & (V_k = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)),c_Groups_Oone__class_Oone(tc_Nat_Onat)) -> c_List_Obutlast(T_a,hAPP(hAPP(c_List_Olist__update(T_a,V_xs),V_k),V_x)) = c_List_Obutlast(T_a,V_xs)))) # label(fact_butlast__list__update) # label(axiom) # label(non_clause). [assumption]. 5.99/6.00 2485 (all V_x_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> c_Big__Operators_Olinorder__class_OMin(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),V_x_2),c_Big__Operators_Olinorder__class_OMin(T_c,V_A_2)))))) # label(fact_Min__insert) # label(axiom) # label(non_clause). [assumption]. 5.99/6.00 2486 (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) & c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) = V_a)) # label(fact_DIVISION__BY__ZERO) # label(axiom) # label(non_clause). [assumption]. 5.99/6.00 2487 (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]. 5.99/6.00 2488 (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_Otranspose(T_c,c_List_Olist_ONil(tc_List_Olist(T_c))) = c_List_Olist_ONil(tc_List_Olist(T_c)))) # label(fact_transpose_Opsimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.00 2489 (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]. 5.99/6.00 2490 (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]. 5.99/6.00 2491 (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]. 5.99/6.00 2492 (all V_j_2 all V_i_2 c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_i_2),hAPP(c_Nat_OSuc,V_j_2)) = hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Nat_Onat,c_Nat_OSuc),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_i_2,V_j_2))) # label(fact_image__Suc__atLeastAtMost) # label(axiom) # label(non_clause). [assumption]. 5.99/6.00 2493 (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),V_a_2),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_a_2)))))) # label(fact_le__minus__self__iff) # label(axiom) # label(non_clause). [assumption]. 5.99/6.00 2494 (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]. 5.99/6.00 2495 (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]. 5.99/6.00 2496 (all V_zs all V_y all V_x all T_a c_List_Oremove1(T_a,V_x,c_List_Oremove1(T_a,V_y,V_zs)) = c_List_Oremove1(T_a,V_y,c_List_Oremove1(T_a,V_x,V_zs))) # label(fact_remove1__commute) # label(axiom) # label(non_clause). [assumption]. 5.99/6.00 2497 (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]. 5.99/6.00 2498 (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]. 5.99/6.00 2499 (all V_y all V_x all T_a (class_Lattices_Olattice(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_y),V_x))) # label(fact_inf__sup__aci_I5_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.00 2500 (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]. 5.99/6.00 2501 (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_Olist_ONil(T_c) != V_ys_2 & c_List_Obutlast(T_c,V_ys_2) = V_xs_2 & c_List_Olast(T_c,V_ys_2) = V_x_2)) # label(fact_snoc__eq__iff__butlast) # label(axiom) # label(non_clause). [assumption]. 5.99/6.00 2502 (all V_l_2 all T_c (class_Orderings_Olinorder(T_c) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_c),V_l_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))),hAPP(c_SetInterval_Oord__class_OgreaterThan(T_c),V_l_2)) = hAPP(c_SetInterval_Oord__class_OatLeast(T_c),V_l_2))) # label(fact_ivl__disj__un_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.00 2503 (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]. 5.99/6.00 2504 (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) -> hAPP(hAPP(c_List_Onth(T_a),V_xs),V_n) = V_y)) # label(fact_nth__via__drop) # label(axiom) # label(non_clause). [assumption]. 5.99/6.00 2505 (all V_y_2 all V_x_2 all V_Pa_2 ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_y_2),V_x_2)) -> hBOOL(hAPP(V_Pa_2,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_x_2)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_y_2))))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_x_2),V_y_2)) -> hBOOL(hAPP(V_Pa_2,c_Groups_Ozero__class_Ozero(tc_Int_Oint)))) <-> hBOOL(hAPP(V_Pa_2,hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_x_2),V_y_2)))))) # label(fact_zdiff__int__split) # label(axiom) # label(non_clause). [assumption]. 5.99/6.00 2506 (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_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]. 5.99/6.00 2507 (all V_xs_2 all V_a_2 all V_f_2 all T_d all T_c c_List_Ofoldr(T_d,T_c,hAPP(c_COMBC(T_c,T_d,T_c),V_f_2),hAPP(c_List_Orev(T_d),V_xs_2),V_a_2) = hAPP(c_List_Ofoldl(T_c,T_d,V_f_2,V_a_2),V_xs_2)) # label(fact_foldl__foldr) # label(axiom) # label(non_clause). [assumption]. 5.99/6.00 2508 (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]. 5.99/6.00 2509 (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]. 5.99/6.00 2510 (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]. 5.99/6.00 2511 (all V_n_2 hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_n_2)) = hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2)) # label(fact_less__eq__Suc__le__raw) # label(axiom) # label(non_clause). [assumption]. 5.99/6.00 2512 (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]. 5.99/6.00 2513 (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]. 5.99/6.00 2514 (all V_n_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2)) | V_m_2 = hAPP(c_Nat_OSuc,V_n_2) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),hAPP(c_Nat_OSuc,V_n_2))))) # label(fact_le__Suc__eq) # label(axiom) # label(non_clause). [assumption]. 5.99/6.00 2515 (all V_xs_2 all V_x_2 all V_f_2 all T_c all T_d hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),hAPP(hAPP(c_List_Olist_OCons(T_d),V_x_2),V_xs_2)) = 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))) # label(fact_map_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.00 2516 (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__less__linear) # label(axiom) # label(non_clause). [assumption]. 5.99/6.00 2517 (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]. 5.99/6.00 2518 (all V_c all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),V_a) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.00 2519 (all V_B_2 all V_A_2 all T_c hAPP(c_Set_OPow(T_c),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool)),hAPP(c_Set_OPow(T_c),V_A_2)),hAPP(c_Set_OPow(T_c),V_B_2))) # label(fact_Pow__Int__eq) # label(axiom) # label(non_clause). [assumption]. 5.99/6.00 2520 (all V_n_2 all V_m_2 all V_k_2 (hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_k_2),V_n_2) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_k_2),V_m_2) <-> V_n_2 = V_m_2)) # label(fact_nat__add__left__cancel) # label(axiom) # label(non_clause). [assumption]. 5.99/6.00 2521 (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]. 5.99/6.00 2522 (all V_x_2 all V_f_2 all T_c all T_d all V_k_2 ((c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = V_k_2 -> hAPP(hAPP(c_Lazy__Sequence_Oanamorph(T_d,T_c,V_f_2),V_k_2),V_x_2) = hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),T_d),c_List_Olist_ONil(T_c)),V_x_2)) & (V_k_2 != c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) -> hAPP(hAPP(c_Lazy__Sequence_Oanamorph(T_d,T_c,V_f_2),V_k_2),V_x_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))))) # label(fact_anamorph_Osimps) # label(axiom) # label(non_clause). [assumption]. 5.99/6.00 2523 (all V_A_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),V_A_2)) <-> hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_b_2),V_A_2))))) # label(fact_insert__iff) # label(axiom) # label(non_clause). [assumption]. 5.99/6.00 2524 (all V_B_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_B_2 -> c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),V_A_2)),c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),V_B_2)))))))) # label(fact_min__max_OInf__fin_Ounion__idem) # label(axiom) # label(non_clause). [assumption]. 5.99/6.00 2525 (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) -> 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))) & (hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v) != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> 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]. 5.99/6.00 2526 (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]. 5.99/6.00 2527 (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]. 5.99/6.00 2528 (all V_xs_2 all T_c all V_x_2 all V_Pa_2 (-hBOOL(hAPP(V_Pa_2,V_x_2)) -> hAPP(c_List_OremoveAll(T_c,V_x_2),hAPP(c_List_Ofilter(T_c,V_Pa_2),V_xs_2)) = hAPP(c_List_Ofilter(T_c,V_Pa_2),V_xs_2))) # label(fact_removeAll__filter__not) # label(axiom) # label(non_clause). [assumption]. 5.99/6.00 2529 (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]. 5.99/6.00 2530 (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]. 5.99/6.00 2531 (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) # label(axiom) # label(non_clause). [assumption]. 5.99/6.00 2532 (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]. 5.99/6.00 2533 (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(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)) = 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)) # label(fact_split__comp) # label(axiom) # label(non_clause). [assumption]. 5.99/6.00 2534 (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]. 5.99/6.00 2535 (all V_b_2 all V_a_2 all T_c (class_Rings_Oidom(T_c) -> (V_a_2 = hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_b_2) | V_a_2 = 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]. 5.99/6.00 2536 (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]. 5.99/6.00 2537 (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]. 5.99/6.00 2538 (all V_v_H_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_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_v_2),V_v_H_2)) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_v_2),V_v_H_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OPls),V_v_H_2))))) # label(fact_less__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 5.99/6.00 2539 (all V_h_2 all V_B_2 all V_f_2 all T_d all T_c all T_e (class_Groups_Ocomm__monoid__add(T_e) -> (c_Fun_Oinj__on(T_c,T_d,V_f_2,V_B_2) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_e),hAPP(c_Fun_Ocomp(T_d,T_e,T_c,V_h_2),V_f_2)),V_B_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_e),V_h_2),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_B_2))))) # label(fact_setsum__reindex) # label(axiom) # label(non_clause). [assumption]. 5.99/6.00 2540 (all V_n all V_m ((V_m != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),c_Groups_Oone__class_Oone(tc_Nat_Onat))),V_n))) & (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_m -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_mult__eq__if) # label(axiom) # label(non_clause). [assumption]. 5.99/6.00 2541 (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_H,V_b) = c_Divides_Odiv__class_Omod(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) = c_Divides_Odiv__class_Omod(T_a,hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a),V_b)))) # label(fact_mod__minus__cong) # label(axiom) # label(non_clause). [assumption]. 5.99/6.00 2542 (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]. 5.99/6.00 2543 (all V_R_2 all T_c (c_Wellfounded_Owf(T_c,hAPP(hAPP(c_Relation_Orel__comp(T_c,T_c,T_c),V_R_2),V_R_2)) <-> c_Wellfounded_Owf(T_c,V_R_2))) # label(fact_wf__comp__self) # label(axiom) # label(non_clause). [assumption]. 5.99/6.00 2544 (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]. 5.99/6.00 2545 (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]. 5.99/6.00 2546 (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]. 5.99/6.00 2547 (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]. 5.99/6.00 2548 (all V_j all V_k all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),V_k)),V_j)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),V_k) = hAPP(hAPP(c_List_Onth(tc_Nat_Onat),c_List_Oupt(V_i,V_j)),V_k))) # label(fact_nth__upt) # label(axiom) # label(non_clause). [assumption]. 5.99/6.00 2549 (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]. 5.99/6.00 2550 (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]. 5.99/6.00 2551 (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)) = 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))))) & (hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w_2))) -> c_Groups_Ozero__class_Ozero(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]. 5.99/6.00 2552 (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))) -> c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = V_A_2))) # label(fact_wfE__pf) # label(axiom) # label(non_clause). [assumption]. 5.99/6.00 2553 (all V_b_2 all V_a_2 all V_f1_2 all T_c all T_e all T_d c_Nitpick_Opair__box_Opair__box__case(T_d,T_e,T_c,V_f1_2,c_Nitpick_Opair__box_OPairBox(T_d,T_e,V_a_2,V_b_2)) = hAPP(hAPP(V_f1_2,V_a_2),V_b_2)) # label(fact_pair__box_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.00 2554 (all V_i_2 all V_j_2 ((-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_j_2),V_i_2)) -> c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,V_i_2,V_j_2) = hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),V_i_2),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_i_2),c_Groups_Oone__class_Oone(tc_Int_Oint)),V_j_2))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_j_2),V_i_2)) -> c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) = c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,V_i_2,V_j_2)))) # label(fact_simp__from__to) # label(axiom) # label(non_clause). [assumption]. 5.99/6.00 2555 (all V_b_2 all V_a_2 all V_Pa_2 (-((exists B_d (hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_b_2),B_d) = V_a_2 & -hBOOL(hAPP(V_Pa_2,B_d)))) | -hBOOL(hAPP(V_Pa_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_a_2),V_b_2))) <-> 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]. 5.99/6.00 2556 (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]. 5.99/6.00 2557 (all V_xs all T_a (class_Orderings_Olinorder(T_a) -> (V_xs != c_List_Olist_ONil(T_a) -> (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]. 5.99/6.00 2558 (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]. 5.99/6.00 2559 (all T_d all T_c c_Product__Type_Omap__pair(T_c,T_c,T_d,T_d,c_Fun_Oid(T_c),c_Fun_Oid(T_d)) = c_Fun_Oid(tc_prod(T_c,T_d))) # label(fact_map__pair_Oid) # label(axiom) # label(non_clause). [assumption]. 5.99/6.00 2560 (all V_z_2 all V_w_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onat,V_w_2)),hAPP(c_Int_Onat,V_z_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_w_2),V_z_2)))) # label(fact_zless__nat__conj) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2561 (all V_a2_2 all V_a1_2 all V_R_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_a1_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(V_R_2,B_x),B_xa)) & hBOOL(hAPP(hAPP(c_member(T_c),B_xa),V_a2_2)))))) <-> c_Wellfounded_Omax__extp(T_c,V_R_2,V_a1_2,V_a2_2))) # label(fact_max__extp_Oequations) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2562 (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_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]. 5.99/6.01 2563 (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]. 5.99/6.01 2564 (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]. 5.99/6.01 2565 (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]. 5.99/6.01 2566 (all V_Pa_2 all T_c (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = hAPP(c_Set_OCollect(T_c),V_Pa_2) <-> (all B_x -hBOOL(hAPP(V_Pa_2,B_x))))) # label(fact_Collect__empty__eq) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2567 (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]. 5.99/6.01 2568 (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]. 5.99/6.01 2569 (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]. 5.99/6.01 2570 (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]. 5.99/6.01 2571 (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]. 5.99/6.01 2572 (all V_A_2 all V_f_2 all T_c 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)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,tc_Nat_Onat),V_f_2),V_A_2)) # label(fact_transfer__nat__int__sum__prod2_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2573 (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]. 5.99/6.01 2574 (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]. 5.99/6.01 2575 (all V_y_2 all V_x_2 all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,V_A_2) -> (V_y_2 != V_x_2 -> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_y_2),V_A_2)) -> hAPP(V_f_2,V_y_2) != hAPP(V_f_2,V_x_2)))))) # label(fact_inj__on__contraD) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2576 (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]. 5.99/6.01 2577 (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]. 5.99/6.01 2578 (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]. 5.99/6.01 2579 (all V_seed_2 all V_size_2 all V_nrandom_2 all V_x_2 all T_c c_New__DSequence_Opos__single(T_c,V_x_2) = hAPP(hAPP(hAPP(hAPP(c_New__Random__Sequence_Opos__single(T_c),V_x_2),V_nrandom_2),V_size_2),V_seed_2)) # label(fact_pos__single__def) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2580 (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]. 5.99/6.01 2581 (all V_f_2 all T_c all T_d c_Lazy__Sequence_Omap(T_d,T_c,V_f_2,c_Lazy__Sequence_Olazy__sequence_OEmpty(T_d)) = c_Lazy__Sequence_Olazy__sequence_OEmpty(T_c)) # label(fact__07) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2582 (all V_k hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Int_OPls),V_k) = V_k) # label(fact_add__Pls) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2583 (all V_y_2 all V_x_2 all T_c (class_Int_Onumber__ring(T_c) & class_Int_Oring__char__0(T_c) -> (hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_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]. 5.99/6.01 2584 (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]. 5.99/6.01 2585 (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]. 5.99/6.01 2586 (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]. 5.99/6.01 2587 (all V_c all V_b all V_a all T_a (class_Rings_Odivision__ring(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),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_Otimes__class_Otimes(T_a),V_a),V_b)),V_c))) # label(fact_times__divide__eq__right) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2588 (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]. 5.99/6.01 2589 (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]. 5.99/6.01 2590 (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]. 5.99/6.01 2591 (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]. 5.99/6.01 2592 (all V_r_2 all T_c (c_Nitpick_Owf_H(T_c,V_r_2) <-> (c_Nitpick_Ounknown(tc_HOL_Obool) | hBOOL(hAPP(c_Finite__Set_Ofinite(tc_prod(T_c,T_c)),V_r_2))) & c_Wellfounded_Oacyclic(T_c,V_r_2))) # label(fact_Nitpick_Owf_H__def) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2593 (all V_ys_2 all V_g_2 all T_e all V_xs_2 all V_f_2 all T_c all T_d (hAPP(hAPP(c_List_Omap(T_e,T_c),V_g_2),V_ys_2) = hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_xs_2) -> hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_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]. 5.99/6.01 2594 (all V_w_2 all V_v_2 all T_c (class_Int_Onumber__ring(T_c) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_v_2)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(V_w_2))) = hAPP(hAPP(c_HOL_OLet(T_c,T_c),hAPP(hAPP(c_Power_Opower__class_Opower(T_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_v_2)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_w_2))),hAPP(hAPP(c_COMBS(T_c,T_c,T_c),c_Groups_Otimes__class_Otimes(T_c)),c_COMBI(T_c))))) # label(fact_power__number__of__even__number__of) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2595 (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]. 5.99/6.01 2596 (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)))) -> c_List_OtakeWhile(T_c,V_Pa_2,V_xs_2) = hAPP(c_List_Ofilter(T_c,V_Pa_2),V_xs_2))) # label(fact_takeWhile__eq__filter) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2597 (all V_b_H all V_b all V_a_H all V_c all V_a all T_a (class_Divides_Osemiring__div(T_a) -> (c_Divides_Odiv__class_Omod(T_a,V_a,V_c) = c_Divides_Odiv__class_Omod(T_a,V_a_H,V_c) -> (c_Divides_Odiv__class_Omod(T_a,V_b_H,V_c) = c_Divides_Odiv__class_Omod(T_a,V_b,V_c) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a_H),V_b_H),V_c))))) # label(fact_mod__add__cong) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2598 (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__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)) = hAPP(hAPP(V_f2_2,V_a_2),V_lazy__sequence_2)) # label(fact_lazy__sequence_Osimps_I5_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2599 (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]. 5.99/6.01 2600 (all V_ys_2 all V_xs_2 all V_A_2 all T_c (hBOOL(hAPP(c_List_Olistsp(T_c,V_A_2),V_ys_2)) & hBOOL(hAPP(c_List_Olistsp(T_c,V_A_2),V_xs_2)) <-> hBOOL(hAPP(c_List_Olistsp(T_c,V_A_2),hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_ys_2))))) # label(fact_append__in__listsp__conv) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2601 (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]. 5.99/6.01 2602 (all V_n all V_m hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),V_m)) # label(fact_nat__add__commute) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2603 (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]. 5.99/6.01 2604 (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]. 5.99/6.01 2605 (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]. 5.99/6.01 2606 (all V_r_2 all T_d all T_c (hAPP(c_Relation_ODomain(T_c,T_d),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_c,T_d),tc_HOL_Obool)) = V_r_2)) # label(fact_Domain__empty__iff) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2607 (all V_b_2 all V_a_2 (V_a_2 = V_b_2 -> hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool))) = 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]. 5.99/6.01 2608 (all V_A_2 all T_c hAPP(c_Set_OCollect(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBC(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool))),V_A_2)) = hAPP(c_Set_OPow(T_c),V_A_2)) # label(fact_Pow__def) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2609 (all V_y_2 all V_f_2 all T_d all V_A_2 all V_x_2 all T_c ((-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> hAPP(c_Set_Oimage(T_c,T_d,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)) & (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> hAPP(c_Set_Oimage(T_c,T_d,c_Fun_Ofun__upd(T_c,T_d,V_f_2,V_x_2,V_y_2)),V_A_2) = hAPP(hAPP(c_Set_Oinsert(T_d),V_y_2),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))))))) # label(fact_fun__upd__image) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2610 (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]. 5.99/6.01 2611 (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]. 5.99/6.01 2612 (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]. 5.99/6.01 2613 (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]. 5.99/6.01 2614 (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]. 5.99/6.01 2615 (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]. 5.99/6.01 2616 (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]. 5.99/6.01 2617 (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]. 5.99/6.01 2618 (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]. 5.99/6.01 2619 (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]. 5.99/6.01 2620 (all V_b all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,c_Divides_Odiv__class_Omod(T_a,V_a,V_b),V_b) = c_Divides_Odiv__class_Omod(T_a,V_a,V_b))) # label(fact_mod__mod__trivial) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2621 (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]. 5.99/6.01 2622 (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]. 5.99/6.01 2623 (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]. 5.99/6.01 2624 (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]. 5.99/6.01 2625 (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]. 5.99/6.01 2626 (all V_B_2 all V_A_2 all T_d all T_c all V_C_2 ((V_C_2 = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool)),T_c),c_Groups_Ominus__class_Ominus(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_Groups_Ominus__class_Ominus(tc_fun(T_d,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),V_A_2)),V_B_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool)),T_c),c_Groups_Ominus__class_Ominus(tc_fun(T_d,tc_HOL_Obool))),V_A_2)),V_B_2))))) # label(fact_INT__simps_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2627 (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]. 5.99/6.01 2628 (all V_f_2 all T_c all T_d c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = hAPP(c_Set_Oimage(T_d,T_c,V_f_2),c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool)))) # label(fact_image__empty) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2629 (all V_n all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) -> (hAPP(c_Nat_OSuc,V_m) != V_n -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),V_n))))) # label(fact_Suc__lessI) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2630 (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]. 5.99/6.01 2631 (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_A_2) = hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_B_2) <-> V_B_2 = V_A_2))) # label(fact_inj__on__Un__image__eq__iff) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2632 (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]. 5.99/6.01 2633 (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]. 5.99/6.01 2634 (all V_c all V_a all V_b all T_a (class_Groups_Ocancel__semigroup__add(T_a) -> (hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_c),V_a) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),V_a) -> V_b = V_c))) # label(fact_add__right__imp__eq) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2635 (all V_y_2 all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> (V_y_2 = V_x_2 <-> hAPP(c_SetInterval_Oord__class_OgreaterThan(T_c),V_x_2) = hAPP(c_SetInterval_Oord__class_OgreaterThan(T_c),V_y_2)))) # label(fact_greaterThan__eq__iff) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2636 (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)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_d,tc_HOL_Obool)),V_A_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)) = 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]. 5.99/6.01 2637 (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]. 5.99/6.01 2638 (all T_a all V_xs (c_List_Olist_ONil(T_a) != V_xs -> V_xs = hAPP(hAPP(c_List_Oappend(T_a),c_List_Obutlast(T_a,V_xs)),hAPP(hAPP(c_List_Olist_OCons(T_a),c_List_Olast(T_a,V_xs)),c_List_Olist_ONil(T_a))))) # label(fact_append__butlast__last__id) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2639 (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]. 5.99/6.01 2640 (all V_n all V_m ((V_m = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n) = V_n) & (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != V_m -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n) = hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),c_Groups_Oone__class_Oone(tc_Nat_Onat))),V_n))))) # label(fact_add__eq__if) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2641 (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]. 5.99/6.01 2642 (all V_xs_2 all T_c (c_List_Olist_ONil(T_c) = hAPP(c_List_Orev(T_c),V_xs_2) <-> c_List_Olist_ONil(T_c) = V_xs_2)) # label(fact_Nil__is__rev__conv) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2643 (all V_b all V_a hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a)),V_b) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_b))) # label(fact_zdiv__zminus2) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2644 (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]. 5.99/6.01 2645 (all V_c_2 all V_b_2 all V_a_2 all T_c (class_Divides_Osemiring__div(T_c) -> (c_Groups_Ozero__class_Ozero(T_c) != V_a_2 -> (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_c),V_a_2),V_b_2)) -> (V_b_2 = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_c_2),V_a_2) <-> V_c_2 = hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_c),V_b_2),V_a_2)))))) # label(fact_dvd__div__eq__mult) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2646 (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]. 5.99/6.01 2647 (all V_R_2 all V_a2_2 all V_a1_2 all T_c (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(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(c_Finite__Set_Ofinite(T_c),V_a2_2)) & hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_a1_2)) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),V_a1_2),V_a2_2)),c_Wellfounded_Omax__ext(T_c,V_R_2))))) # label(fact_max__ext_Osimps) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2648 (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]. 5.99/6.01 2649 (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]. 5.99/6.01 2650 (all V_B_2 all V_C_2 all V_a_2 all T_c (-hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_C_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_B_2)),V_C_2) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_C_2))) # label(fact_Int__insert__left__if0) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2651 (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]. 5.99/6.01 2652 (all V_B_2 all V_A_2 all T_c (V_A_2 = V_B_2 | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) <-> 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]. 5.99/6.01 2653 (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]. 5.99/6.01 2654 (all V_I_2 all V_xs_2 all T_c hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),c_List_Osublist(T_c,V_xs_2,V_I_2)) = hAPP(c_Finite__Set_Ocard(tc_Nat_Onat),hAPP(c_Set_OCollect(tc_Nat_Onat),hAPP(hAPP(c_COMBS(tc_Nat_Onat,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Nat_Onat),c_fconj),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_HOL_Obool),c_Orderings_Oord__class_Oless(tc_Nat_Onat)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2)))),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool),c_member(tc_Nat_Onat)),V_I_2))))) # label(fact_length__sublist) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2655 (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]. 5.99/6.01 2656 (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]. 5.99/6.01 2657 (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]. 5.99/6.01 2658 (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]. 5.99/6.01 2659 (all V_r_2 all V_y_2 all V_ys_2 all V_x_2 all V_xs_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_y_2)),V_r_2)) & V_xs_2 = V_ys_2 | V_y_2 = V_x_2 & hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_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_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]. 5.99/6.01 2660 (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]. 5.99/6.01 2661 (all V_xs all T_a all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))) -> hAPP(hAPP(c_List_Otake(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),c_Groups_Oone__class_Oone(tc_Nat_Onat))),V_xs) = c_List_Obutlast(T_a,hAPP(hAPP(c_List_Otake(T_a),V_n),V_xs)))) # label(fact_butlast__take) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2662 (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]. 5.99/6.01 2663 (all V_Pa_2 all T_c ((all B_x -hBOOL(hAPP(V_Pa_2,B_x))) <-> c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = hAPP(c_Set_OCollect(T_c),V_Pa_2))) # label(fact_empty__Collect__eq) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2664 (all V_xs_2 all T_c hAPP(c_List_Oset(T_c),c_List_Oremdups(T_c,V_xs_2)) = hAPP(c_List_Oset(T_c),V_xs_2)) # label(fact_set__remdups) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2665 (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]. 5.99/6.01 2666 (all V_r_2 all V_x_2 all V_y_2 all T_c (-hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_y_2)),c_Transitive__Closure_Ortrancl(T_c,V_r_2))) & c_Wellfounded_Owf(T_c,V_r_2) <-> 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)))) # label(fact_wf__insert) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2667 (all V_m all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n),V_m)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),V_n) = hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)))) # label(fact_Suc__diff__le) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2668 (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]. 5.99/6.01 2669 (all V_A_2 all T_c hAPP(c_Set_Ovimage(T_c,T_c,c_Fun_Oid(T_c)),V_A_2) = V_A_2) # label(fact_vimage__id) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2670 (all V_ry all V_rx all V_ly all V_lx all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_rx),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),V_ly)),V_ry)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),V_ly)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_rx),V_ry)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2671 (all V_lazy__sequence_2 all V_a_2 all V_f2_2 all V_f1_2 all T_d all T_c hAPP(hAPP(hAPP(V_f2_2,V_a_2),V_lazy__sequence_2),c_Lazy__Sequence_Olazy__sequence_Olazy__sequence__rec(T_c,T_d,V_f1_2,V_f2_2,V_lazy__sequence_2)) = c_Lazy__Sequence_Olazy__sequence_Olazy__sequence__rec(T_c,T_d,V_f1_2,V_f2_2,c_Lazy__Sequence_Olazy__sequence_OInsert(T_d,V_a_2,V_lazy__sequence_2))) # label(fact_lazy__sequence_Orecs_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2672 (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]. 5.99/6.01 2673 (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]. 5.99/6.01 2674 (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]. 5.99/6.01 2675 (all V_b_2 all V_B_2 all V_a_2 all T_c ((-hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_B_2)) -> V_a_2 = V_b_2) -> hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_b_2),V_B_2))))) # label(fact_insertCI) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2676 (all V_Qa_2 all V_Pa_2 (hBOOL(V_Pa_2) | -hBOOL(hAPP(hAPP(c_fconj,V_Pa_2),V_Qa_2)))) # label(help_c__fconj__2) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2677 (all V_x_2 all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> hAPP(c_Finite__Set_Ocard(T_c),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_c),V_A_2)),c_Groups_Oone__class_Oone(tc_Nat_Onat))))) # label(fact_card__Diff__singleton) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2678 (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]. 5.99/6.01 2679 (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)) -> (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_Lattices_Osemilattice__inf__class_Oinf(T_c)),V_A_2)))))))) # label(fact_below__fold1__iff) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2680 (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]. 5.99/6.01 2681 (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]. 5.99/6.01 2682 (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]. 5.99/6.01 2683 (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]. 5.99/6.01 2684 (all V_n c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n)) # label(fact_diff__0__eq__0) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2685 (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]. 5.99/6.01 2686 (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]. 5.99/6.01 2687 (all V_r_2 all T_c hAPP(hAPP(c_Relation_Orel__comp(T_c,T_c,T_c),V_r_2),c_Transitive__Closure_Ortrancl(T_c,V_r_2)) = hAPP(hAPP(c_Relation_Orel__comp(T_c,T_c,T_c),c_Transitive__Closure_Ortrancl(T_c,V_r_2)),V_r_2)) # label(fact_r__comp__rtrancl__eq) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2688 (all V_b all V_a V_a = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b))),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b))) # label(fact_zmod__zdiv__equality) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2689 (all V_x_2 all V_g_2 all V_f_2 all T_c all T_f all T_e all T_d hAPP(hAPP(V_g_2,hAPP(c_Product__Type_Ofst(T_e,T_f),hAPP(V_f_2,V_x_2))),hAPP(c_Product__Type_Osnd(T_e,T_f),hAPP(V_f_2,V_x_2))) = hAPP(hAPP(hAPP(c_Product__Type_Oscomp(T_d,T_e,T_f,T_c),V_f_2),V_g_2),V_x_2)) # label(fact_scomp__unfold) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2690 (all V_k_2 (V_k_2 = c_Int_OMin <-> c_Int_OBit1(V_k_2) = c_Int_OMin)) # label(fact_rel__simps_I47_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2691 (all V_i_2 all V_M_2 (hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_M_2)) -> hAPP(c_Finite__Set_Ocard(tc_Nat_Onat),hAPP(c_Set_OCollect(tc_Nat_Onat),hAPP(hAPP(c_COMBS(tc_Nat_Onat,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Nat_Onat),c_fconj),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool),c_member(tc_Nat_Onat)),V_M_2))),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_HOL_Obool),c_Orderings_Oord__class_Oless(tc_Nat_Onat)),hAPP(c_Nat_OSuc,V_i_2))))) = hAPP(c_Nat_OSuc,hAPP(c_Finite__Set_Ocard(tc_Nat_Onat),hAPP(c_Set_OCollect(tc_Nat_Onat),hAPP(hAPP(c_COMBS(tc_Nat_Onat,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Nat_Onat),c_fconj),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool),tc_Nat_Onat),c_member(tc_Nat_Onat)),c_Nat_OSuc)),V_M_2))),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_HOL_Obool),c_Orderings_Oord__class_Oless(tc_Nat_Onat)),V_i_2))))))) # label(fact_card__less__Suc) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2692 (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]. 5.99/6.01 2693 (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]. 5.99/6.01 2694 (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_Olinorder__class_OMax(T_c,V_A_2)) = c_Big__Operators_Olinorder__class_OMax(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)))))) # label(fact_Max__insert) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2695 (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]. 5.99/6.01 2696 (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]. 5.99/6.01 2697 (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]. 5.99/6.01 2698 (all V_k (hAPP(c_Code__Numeral_Oof__nat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) != V_k -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Code__Numeral_Onat__of,V_k)),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))),hAPP(c_Code__Numeral_Onat__of,V_k))))) # label(fact_code__numeral__decr) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2699 (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]. 5.99/6.01 2700 (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]. 5.99/6.01 2701 (all V_a_2 all T_c (class_Rings_Olinordered__idom(T_c) -> (c_Groups_Oone__class_Oone(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__1__pos) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2702 (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__eq(T_c),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_b_2),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2))),V_a_2)) <-> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2))) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2)),c_Groups_Ozero__class_Ozero(T_c))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_a_2))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2)),c_Groups_Ozero__class_Ozero(T_c))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2))),V_b_2)))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2)))))))) # label(fact_divide__le__eq__number__of1) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2703 (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]. 5.99/6.01 2704 (all V_z_2 all T_c (class_Rings_Osemiring__1(T_c) -> (c_Int_Oiszero(T_c,V_z_2) <-> c_Groups_Ozero__class_Ozero(T_c) = V_z_2))) # label(fact_iszero__def) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2705 (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]. 5.99/6.01 2706 (all V_u_2 all V_l_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_l_2),V_u_2)) -> hAPP(c_SetInterval_Oord__class_OatMost(T_c),V_u_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(T_c),V_l_2)),c_SetInterval_Oord__class_OatLeastAtMost(T_c,V_l_2,V_u_2))))) # label(fact_ivl__disj__un_I10_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2707 (all V_x_2 all T_c (class_Groups_Ozero(T_c) -> (c_Groups_Ozero__class_Ozero(T_c) = V_x_2 <-> V_x_2 = c_Groups_Ozero__class_Ozero(T_c)))) # label(fact_zero__reorient) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2708 (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]. 5.99/6.01 2709 (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]. 5.99/6.01 2710 (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]. 5.99/6.01 2711 (all V_c all V_b all V_a c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_a),V_b),V_c) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_a),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_b,V_c)),V_c)) # label(fact_zmod__zmult1__eq) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2712 (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]. 5.99/6.01 2713 (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]. 5.99/6.01 2714 (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]. 5.99/6.01 2715 (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]. 5.99/6.01 2716 (all T_a (class_Rings_Olinordered__idom(T_a) -> hAPP(c_Groups_Oabs__class_Oabs(T_a),c_Groups_Oone__class_Oone(T_a)) = c_Groups_Oone__class_Oone(T_a))) # label(fact_abs__one) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2717 (all V_xs all V_x all T_a hAPP(hAPP(c_List_Oappend(T_a),V_x),c_List_Oconcat(T_a,V_xs)) = c_List_Oconcat(T_a,hAPP(hAPP(c_List_Olist_OCons(tc_List_Olist(T_a)),V_x),V_xs))) # label(fact_concat_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2718 (all V_g_2 all V_B_2 all V_A_2 all T_c all T_d (class_Groups_Ocomm__monoid__mult(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_d),hAPP(hAPP(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]. 5.99/6.01 2719 (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]. 5.99/6.01 2720 (all V_b 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_Divides_Odiv__class_Omod(T_a,V_a,V_b)),V_b))) # label(fact_mod__div__trivial) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2721 (all V_b_2 all V_a_2 all V_c_2 all T_c (class_Groups_Oordered__ab__semigroup__add__imp__le(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_a_2),V_b_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),V_c_2),V_a_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),V_c_2),V_b_2)))))) # label(fact_add__less__cancel__left) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2722 (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_s_2) = hAPP(c_Product__Type_Osnd(T_c,T_d),V_t_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]. 5.99/6.01 2723 (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]. 5.99/6.01 2724 (all V_xs_2 all V_x_2 all T_c (hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),c_List_Olist_ONil(T_c)) = hAPP(c_List_Orev(T_c),V_xs_2) <-> V_xs_2 = hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),c_List_Olist_ONil(T_c)))) # label(fact_singleton__rev__conv) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2725 (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__empty2) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2726 (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(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__idem_Oinsert__idem) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2727 (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]. 5.99/6.01 2728 (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]. 5.99/6.01 2729 (all V_l all V_k hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),c_Int_OBit0(V_k)),c_Int_OBit0(V_l)) = c_Int_OBit0(hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_k),V_l))) # label(fact_diff__bin__simps_I7_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2730 (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]. 5.99/6.01 2731 (all V_c all V_b all V_a all T_a (class_Divides_Oring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_b),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),c_Divides_Odiv__class_Omod(T_a,V_a,V_c)),c_Divides_Odiv__class_Omod(T_a,V_b,V_c)),V_c))) # label(fact_mod__diff__eq) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2732 (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)) -> V_z = 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)) -> 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))))) # label(fact_int__nat__eq) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2733 (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]. 5.99/6.01 2734 (all V_A_2 all V_y_2 all V_x_2 all T_c (V_x_2 = V_y_2 & hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_y_2)),c_Relation_OId__on(T_c,V_A_2))))) # label(fact_Id__on__iff) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2735 (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,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))) = hAPP(hAPP(V_f_2,hAPP(V_F_2,V_A_2)),hAPP(V_F_2,V_B_2)))))) # label(fact_folding__image__simple_Ounion__inter) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2736 (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_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_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))) # label(fact_INT__simps_I10_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2737 (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]. 5.99/6.01 2738 (all V_k hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Int_OBit0(V_k)) = c_Int_OBit0(hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_k))) # label(fact_minus__Bit0) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2739 (all V_xs_2 all V_Pa_2 all T_c all V_j_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_j_2),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),c_List_OdropWhile(T_c,V_Pa_2,V_xs_2)))) -> hAPP(hAPP(c_List_Onth(T_c),c_List_OdropWhile(T_c,V_Pa_2,V_xs_2)),V_j_2) = hAPP(hAPP(c_List_Onth(T_c),V_xs_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_j_2),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),c_List_OtakeWhile(T_c,V_Pa_2,V_xs_2)))))) # label(fact_dropWhile__nth) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2740 (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]. 5.99/6.01 2741 (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]. 5.99/6.01 2742 (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]. 5.99/6.01 2743 (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]. 5.99/6.01 2744 (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_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) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2745 (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]. 5.99/6.01 2746 (all V_c all V_b all V_v all T_b (class_Int_Onumber(T_b) & class_Rings_Oring(T_b) -> hAPP(hAPP(c_Groups_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)) = 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)))) # label(fact_right__diff__distrib__number__of) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2747 (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]. 5.99/6.01 2748 (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]. 5.99/6.01 2749 (all V_x_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_A_2 -> hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),V_x_2),c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),V_A_2)) = c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2))))))) # label(fact_min__max_OInf__fin_Oinsert) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2750 (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]. 5.99/6.01 2751 (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]. 5.99/6.01 2752 (all V_x_2 all V_g_2 all V_f_2 all T_f all T_e all T_d all T_c hAPP(c_Product__Type_Ofst(T_c,T_d),hAPP(c_Product__Type_Omap__pair(T_e,T_c,T_f,T_d,V_f_2,V_g_2),V_x_2)) = hAPP(V_f_2,hAPP(c_Product__Type_Ofst(T_e,T_f),V_x_2))) # label(fact_fst__map__pair) # label(axiom) # label(non_clause). [assumption]. 5.99/6.01 2753 (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]. 5.99/6.01 2754 (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]. 5.99/6.01 2755 (all V_B_2 all V_A_2 all T_c (class_Lattices_Odistrib__lattice(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_A_2 -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_B_2 -> c_Big__Operators_Olattice__class_OSup__fin(T_c,hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool,T_c),c_HOL_OEx(T_c)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_c,tc_HOL_Obool),T_c),hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool,T_c),c_HOL_OEx(T_c))),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_c,tc_fun(T_c,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool))),tc_fun(tc_fun(T_c,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_c,tc_fun(T_c,tc_HOL_Obool))),T_c),c_COMBS(T_c,tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(T_c,tc_fun(tc_HOL_Obool,tc_HOL_Obool))),tc_fun(T_c,tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool))),T_c),hAPP(c_COMBB(tc_fun(T_c,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),T_c),c_COMBS(T_c,tc_HOL_Obool,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_c,tc_fun(T_c,tc_fun(tc_HOL_Obool,tc_HOL_Obool))),T_c),hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),T_c),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_c),c_fconj))),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_fun(T_c,T_c)),tc_fun(T_c,tc_fun(T_c,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_c,T_c),tc_fun(T_c,tc_HOL_Obool)),tc_fun(tc_fun(T_c,tc_fun(T_c,T_c)),tc_fun(T_c,tc_fun(T_c,tc_HOL_Obool))),T_c),c_COMBB(tc_fun(T_c,T_c),tc_fun(T_c,tc_HOL_Obool),T_c)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_c,T_c),tc_fun(T_c,tc_HOL_Obool)),T_c),c_COMBB(T_c,tc_HOL_Obool,T_c)),c_fequal))),c_Lattices_Osemilattice__inf__class_Oinf(T_c)))))),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),T_c),c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_c)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_c),c_fconj),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_A_2)))),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_B_2))))))) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_c),c_Big__Operators_Olattice__class_OSup__fin(T_c,V_A_2)),c_Big__Operators_Olattice__class_OSup__fin(T_c,V_B_2)))))))) # label(fact_inf__Sup2__distrib) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2756 (all T_c c_Relation_Oantisym(T_c,c_Relation_OId(T_c))) # label(fact_antisym__Id) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2757 (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]. 5.99/6.02 2758 (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]. 5.99/6.02 2759 (all V_n_2 all V_k_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m_2),V_k_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n_2),V_k_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),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)))) # label(fact_mult__less__cancel2) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2760 (all V_y_2 all T_c (class_Int_Onumber__ring(T_c) & class_Rings_Olinordered__idom(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)),V_y_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Oone__class_Oone(T_c)),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_y_2)))))) # label(fact_less__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2761 (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]. 5.99/6.02 2762 (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)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_M_2 -> (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]. 5.99/6.02 2763 (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]. 5.99/6.02 2764 (all V_r_2 all V_x_2 all V_y_2 all T_c (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))) <-> c_Wellfounded_Oacyclic(T_c,hAPP(hAPP(c_Set_Oinsert(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_y_2),V_x_2)),V_r_2)))) # label(fact_acyclic__insert) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2765 (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]. 5.99/6.02 2766 (all V_c all V_a all V_b all T_a (class_Divides_Osemiring__div(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_a),V_b))),c_Divides_Odiv__class_Omod(T_a,V_a,V_b))),V_c))) # label(fact_div__mod__equality2) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2767 (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]. 5.99/6.02 2768 (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(V_f_2,V_x_2),hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_d,V_f_2),V_z_2),V_A_2))))) # label(fact_fun__left__comm__idem_Ofold__insert__idem) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2769 (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]. 5.99/6.02 2770 (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]. 5.99/6.02 2771 (all V_A_2 all V_B_2 all V_f_2 all T_c all T_d (c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)) = hAPP(c_Set_Oimage(T_d,T_c,V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_d,tc_HOL_Obool)),hAPP(c_Set_Ovimage(T_d,T_c,V_f_2),V_B_2)),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_B_2),hAPP(c_Set_Oimage(T_d,T_c,V_f_2),V_A_2)))))) # label(fact_vimage__subsetD) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2772 (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]. 5.99/6.02 2773 (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]. 5.99/6.02 2774 (all V_x_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x_2)) <-> -hBOOL(hAPP(c_Nat__Numeral_Oneg,V_x_2)))) # label(fact_not__neg__eq__ge__0) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2775 (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]. 5.99/6.02 2776 (all V_B_2 all V_a_2 all T_d all T_c all V_C_2 ((V_C_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool),T_c),hAPP(c_Set_Oinsert(T_d),V_a_2)),V_B_2)) = hAPP(hAPP(c_Set_Oinsert(T_d),V_a_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),V_B_2))) & (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)) = c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool))))) # label(fact_UN__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2777 (all V_B_2 all V_A_2 (c_Nat__Transfer_Onat__set(V_A_2) -> (c_Nat__Transfer_Onat__set(V_B_2) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Int_Oint,tc_HOL_Obool)),V_A_2),V_B_2) = hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),V_A_2)),hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),V_B_2)))))) # label(fact_Nat__Transfer_Otransfer__int__nat__set__functions_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2778 (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]. 5.99/6.02 2779 (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]. 5.99/6.02 2780 (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]. 5.99/6.02 2781 (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_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) = 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))) # label(fact_setsum__commute) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2782 (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]. 5.99/6.02 2783 (all V_q all V_p all T_b all T_a (hAPP(c_Product__Type_Ofst(T_a,T_b),V_p) = hAPP(c_Product__Type_Ofst(T_a,T_b),V_q) -> (hAPP(c_Product__Type_Osnd(T_a,T_b),V_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]. 5.99/6.02 2784 (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]. 5.99/6.02 2785 (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_Nat_Osemiring__1__class_Oof__nat(T_a),hAPP(c_Int_Onat,V_z)) = hAPP(c_Int_Oring__1__class_Oof__int(T_a),V_z)))) # label(fact_of__nat__nat) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2786 (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]. 5.99/6.02 2787 (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(V_F_2,V_A_2) = hAPP(hAPP(V_f_2,hAPP(V_F_2,V_B_2)),hAPP(V_F_2,V_A_2)))))) # label(fact_folding__image__simple__idem_Osubset__idem) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2788 (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]. 5.99/6.02 2789 (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]. 5.99/6.02 2790 (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]. 5.99/6.02 2791 (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]. 5.99/6.02 2792 (all V_S_2 all V_R_2 all T_c (V_R_2 = V_S_2 <-> hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_R_2) = hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_S_2))) # label(fact_pred__equals__eq) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2793 (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]. 5.99/6.02 2794 (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]. 5.99/6.02 2795 (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),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_B_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)))) # label(fact_Int__insert__right) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2796 (all V_k_2 all V_j_2 all V_i_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i_2),V_j_2)) -> hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_i_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_j_2),V_k_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_i_2),V_j_2)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_j_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_j_2),V_k_2))))) # label(fact_atLeastLessThan__add__Un) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2797 (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]. 5.99/6.02 2798 (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]. 5.99/6.02 2799 (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_y_2) != hAPP(V_f_2,V_x_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)) = 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))))) # label(fact_insort__key__left__comm) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2800 (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]. 5.99/6.02 2801 (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]. 5.99/6.02 2802 (all V_xs_2 all V_f_2 all T_d all V_n_2 all T_c 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)) = 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))) # label(fact_take__map) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2803 (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]. 5.99/6.02 2804 (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,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,hAPP(c_Nat_OSuc,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]. 5.99/6.02 2805 (all V_x_2 all V_n_2 all T_c all V_f_2 hAPP(V_f_2,hAPP(hAPP(hAPP(c_Nat_Ocompow(T_c,T_c),V_n_2),V_f_2),V_x_2)) = hAPP(hAPP(hAPP(c_Nat_Ocompow(T_c,T_c),V_n_2),V_f_2),hAPP(V_f_2,V_x_2))) # label(fact_funpow__swap1) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2806 (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),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_i),V_j)),V_u)),V_m)),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__eq1) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2807 (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_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))))) # label(fact_id__take__nth__drop) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2808 (all V_n_2 all V_m_2 all V_k_2 (V_m_2 = V_n_2 <-> hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_k_2)),V_n_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_k_2)),V_m_2))) # label(fact_Suc__mult__cancel1) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2809 (all V_xs all T_a all V_y all V_x ((V_y != V_x -> hAPP(hAPP(c_List_Olist_OCons(T_a),V_y),c_List_Oremove1(T_a,V_x,V_xs)) = c_List_Oremove1(T_a,V_x,hAPP(hAPP(c_List_Olist_OCons(T_a),V_y),V_xs))) & (V_y = V_x -> V_xs = c_List_Oremove1(T_a,V_x,hAPP(hAPP(c_List_Olist_OCons(T_a),V_y),V_xs))))) # label(fact_remove1_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2810 (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]. 5.99/6.02 2811 (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]. 5.99/6.02 2812 (all V_Pa_2 all V_a_2 all V_n_2 all T_c ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(c_List_Oset(T_c),c_List_Oreplicate(T_c,V_n_2,V_a_2)))) -> hBOOL(hAPP(V_Pa_2,B_x)))) <-> hBOOL(hAPP(V_Pa_2,V_a_2)) | V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_Ball__set__replicate) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2813 (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_c_2),V_d_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_a_2),V_b_2)))))) # label(fact_diff__eq__diff__less) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2814 (all V_l all V_k hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),c_Int_OBit1(V_k)),c_Int_OBit0(V_l)) = c_Int_OBit1(hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_k),V_l))) # label(fact_diff__bin__simps_I9_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2815 (all T_a c_List_Olist_ONil(T_a) = hAPP(c_List_Orev(T_a),c_List_Olist_ONil(T_a))) # label(fact_rev_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2816 (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]. 5.99/6.02 2817 (all V_z_2 all V_y_2 all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_x_2),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),V_y_2),V_z_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_x_2),V_z_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_x_2),V_y_2))))) # label(fact_min__max_Ole__inf__iff) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2818 (all V_w all V_z hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_z),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_w)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_z),V_w)) # label(fact_diff__int__def) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2819 (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]. 5.99/6.02 2820 (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]. 5.99/6.02 2821 (all V_xa_2 all V_x_2 all V_r_2 all T_c (c_List_Olistrelp(T_c,hAPP(hAPP(c_COMBC(T_c,tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),T_c),c_COMBC(T_c,tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_prod(T_c,T_c)),tc_fun(T_c,tc_fun(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_HOL_Obool)),T_c),hAPP(c_COMBB(tc_prod(T_c,T_c),tc_fun(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_HOL_Obool),T_c),c_member(tc_prod(T_c,T_c)))),c_Product__Type_OPair(T_c,T_c)))),V_r_2),V_x_2,V_xa_2) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_x_2),V_xa_2)),c_List_Olistrel(T_c,V_r_2))))) # label(fact_listrelp__listrel__eq) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2822 (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]. 5.99/6.02 2823 (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]. 5.99/6.02 2824 (all V_z all V_y all V_x hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_y),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_x),V_z)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_x),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_y),V_z))) # label(fact_nat__add__left__commute) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2825 (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]. 5.99/6.02 2826 (all T_c c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)) = 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)))) # label(fact_surj__id) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2827 (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]. 5.99/6.02 2828 (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]. 5.99/6.02 2829 (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 (hAPP(c_Set_Oimage(T_d,T_c,V_f_2),V_A_2) = V_A_H_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]. 5.99/6.02 2830 (all V_s_2 all V_g_2 all V_h_2 all V_f_2 all T_d all T_e all V_xs_2 all T_c ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(c_List_Oset(T_c),V_xs_2))) -> hAPP(c_Fun_Ocomp(T_d,T_e,T_d,V_h_2),hAPP(V_g_2,B_x)) = hAPP(c_Fun_Ocomp(T_e,T_e,T_d,hAPP(V_f_2,B_x)),V_h_2))) -> hAPP(V_h_2,hAPP(c_List_Ofoldl(T_d,T_c,hAPP(c_COMBC(T_c,T_d,T_d),V_g_2),V_s_2),V_xs_2)) = hAPP(c_List_Ofoldl(T_e,T_c,hAPP(c_COMBC(T_c,T_e,T_e),V_f_2),hAPP(V_h_2,V_s_2)),V_xs_2))) # label(fact_foldl__apply) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2831 (all V_n all V_xs all T_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)),c_Groups_Oone__class_Oone(tc_Nat_Onat))) -> V_xs = hAPP(c_List_Orotate(T_a,V_n),V_xs))) # label(fact_rotate__length01) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2832 (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]. 5.99/6.02 2833 (all V_A_2 all T_c c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)))) # label(fact_Un__UNIV__right) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2834 (all V_x all V_xs all T_a all V_j all V_i (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))) # label(fact_nth__list__update__neq) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2835 (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]. 5.99/6.02 2836 (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]. 5.99/6.02 2837 (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]. 5.99/6.02 2838 (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_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)) = 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))))) # label(fact_fold__inf__insert) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2839 (all V_a_2 all V_b_2 all V_c_2 all T_c (class_Fields_Olinordered__field(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_c_2),c_Groups_Ozero__class_Ozero(T_c))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_b_2),V_c_2)),V_a_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_c_2)),V_b_2)))))) # label(fact_neg__divide__le__eq) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2840 (all V_B_2 all V_a_2 all V_A_2 all T_c hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_B_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))),V_B_2)) # label(fact_Diff__insert2) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2841 (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]. 5.99/6.02 2842 (all V_m all V_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OPls)),V_n)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),V_n))) # label(fact_Suc__diff__eq__diff__pred) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2843 (all V_n all V_m all V_u all V_j all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i),V_j)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_i),V_u)),V_m)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_j),V_u)),V_n)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j),V_i)),V_u)),V_n)))) # label(fact_nat__diff__add__eq2) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2844 (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]. 5.99/6.02 2845 (all V_g_2 all V_B_2 all V_A_2 all T_c all T_d (class_Groups_Ocomm__monoid__add(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))) -> c_Groups_Ozero__class_Ozero(T_d) = hAPP(V_g_2,B_x))) -> hAPP(hAPP(c_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]. 5.99/6.02 2846 (all V_z (hBOOL(hAPP(c_Nat__Numeral_Oneg,V_z)) -> hAPP(c_Int_Onat,V_z) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_neg__nat) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2847 (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]. 5.99/6.02 2848 (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]. 5.99/6.02 2849 (all V_A_2 all V_x_2 all T_c (-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) = V_A_2)) # label(fact_Diff__insert__absorb) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2850 (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_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)) = 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)))))) # label(fact_setsum__head) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2851 (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]. 5.99/6.02 2852 (all V_k all V_b all V_a hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_a),V_k) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b)),V_b)),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b))),V_k)) # label(fact_zdiv__zmod__equality2) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2853 (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]. 5.99/6.02 2854 (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]. 5.99/6.02 2855 (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]. 5.99/6.02 2856 (all V_k_2 all V_l_2 all T_c (class_Orderings_Oorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_l_2),V_k_2)) -> c_SetInterval_Oord__class_OgreaterThanLessThan(T_c,V_k_2,V_l_2) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))) # label(fact_greaterThanLessThan__empty) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2857 (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_c,T_d,V_Pa_2,hAPP(hAPP(c_List_Omap(T_e,T_c),V_f_2),V_as_2),V_bs_2) <-> 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))) # label(fact_list__all2__map1) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2858 (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]. 5.99/6.02 2859 (all V_vs_2 all V_us_2 all V_Pa_2 all V_ys_2 all T_d all V_xs_2 all T_c (hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_d)),V_ys_2) -> (c_List_Olist__all2(T_c,T_d,V_Pa_2,V_us_2,V_vs_2) & c_List_Olist__all2(T_c,T_d,V_Pa_2,V_xs_2,V_ys_2) <-> c_List_Olist__all2(T_c,T_d,V_Pa_2,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]. 5.99/6.02 2860 (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]. 5.99/6.02 2861 (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]. 5.99/6.02 2862 (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]. 5.99/6.02 2863 (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]. 5.99/6.02 2864 (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]. 5.99/6.02 2865 (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]. 5.99/6.02 2866 (all V_B_2 all V_A_2 all V_x_2 all T_c (-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_B_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))))) # label(fact_subset__insert) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2867 (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]. 5.99/6.02 2868 (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]. 5.99/6.02 2869 (all V_list_2 all V_g_2 all T_e all V_f_2 all T_c all T_d hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),hAPP(hAPP(c_List_Omap(T_e,T_d),V_g_2),V_list_2)) = hAPP(hAPP(c_List_Omap(T_e,T_c),hAPP(c_Fun_Ocomp(T_d,T_c,T_e,V_f_2),V_g_2)),V_list_2)) # label(fact_List_Omap_Ocompositionality) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2870 (all V_B_2 all V_a_2 all V_C_2 all T_c all T_d 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)) = 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))) # label(fact_INT__simps_I5_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2871 (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]. 5.99/6.02 2872 (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]. 5.99/6.02 2873 (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]. 5.99/6.02 2874 (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]. 5.99/6.02 2875 (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_bij__def) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2876 (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]. 5.99/6.02 2877 (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]. 5.99/6.02 2878 (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]. 5.99/6.02 2879 (all V_r_2 all V_S_2 all T_e all T_c all T_d hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_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)) = 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))) # label(fact_converse__UNION) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2880 (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) -> ((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)) & (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_divide__eq__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2881 (all V_r_2 all T_c c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c)),tc_HOL_Obool)) = hAPP(c_List_Olexn(T_c,V_r_2),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_lexn_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2882 (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]. 5.99/6.02 2883 (all V_r_2 all T_c (c_Relation_Otrans(T_c,V_r_2) -> c_Transitive__Closure_Otrancl(T_c,V_r_2) = V_r_2)) # label(fact_trancl__id) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2884 (all V_d_2 hAPP(c_Set_OCollect(tc_prod(tc_Int_Oint,tc_Int_Oint)),hAPP(c_Product__Type_Oprod_Oprod__case(tc_Int_Oint,tc_Int_Oint,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_fun(tc_Int_Oint,tc_HOL_Obool),tc_fun(tc_Int_Oint,tc_HOL_Obool),tc_Int_Oint),hAPP(c_COMBS(tc_Int_Oint,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Int_Oint),c_fconj),hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_d_2)))),c_Orderings_Oord__class_Oless(tc_Int_Oint)))) = c_Int_Oint__ge__less__than2(V_d_2)) # label(fact_int__ge__less__than2__def) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2885 (all V_k all V_l ((V_l = c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> c_Divides_Opdivmod(V_k,V_l) = hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),V_k))) & (c_Groups_Ozero__class_Ozero(tc_Int_Oint) != V_l -> c_Divides_Opdivmod(V_k,V_l) = c_Divides_OposDivAlg(hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),V_k),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),V_l))))) # label(fact_pdivmod__posDivAlg) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2886 (all V_f_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)))) -> (c_Fun_Oinj__on(T_c,T_c,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> hAPP(c_Set_Oimage(T_c,T_c,V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) = c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))))) # label(fact_finite__UNIV__inj__surj) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2887 (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_singleton__iff) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2888 (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]. 5.99/6.02 2889 (all V_Qa_2 all V_Pa_2 (-hBOOL(V_Pa_2) | hBOOL(V_Qa_2) | -hBOOL(hAPP(hAPP(c_fimplies,V_Pa_2),V_Qa_2)))) # label(help_c__fimplies__3) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2890 (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]. 5.99/6.02 2891 (all V_a all T_a (class_Rings_Odivision__ring(T_a) -> c_Groups_Ozero__class_Ozero(T_a) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a))) # label(fact_divide__zero__left) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2892 (all V_x_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),V_x_2),c_Big__Operators_Olinorder__class_OMin(T_c,V_A_2)) = c_Big__Operators_Olinorder__class_OMin(T_c,V_A_2))))) # label(fact_Min_Oin__idem) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2893 (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),c_List_Obutlast(T_d,V_xs_2)) = c_List_Obutlast(T_c,hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_xs_2))) # label(fact_map__butlast) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2894 (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]. 5.99/6.02 2895 (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]. 5.99/6.02 2896 (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]. 5.99/6.02 2897 (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]. 5.99/6.02 2898 (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]. 5.99/6.02 2899 (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]. 5.99/6.02 2900 (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]. 5.99/6.02 2901 (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]. 5.99/6.02 2902 (all V_xs_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2))) -> c_List_Oinsert(T_c,V_x_2,V_xs_2) = V_xs_2)) # label(fact_in__set__insert) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2903 (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(c_SetInterval_Oord__class_OatLeastLessThan(T_c,V_n_2),V_m_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))))) # label(fact_ivl__diff) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2904 (all V_z all V_z_H (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z_H)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_z_H),V_z)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Int_Onat,V_z)),hAPP(c_Int_Onat,V_z_H)) = hAPP(c_Int_Onat,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_z),V_z_H))))) # label(fact_nat__diff__distrib) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2905 (all T_d all V_A_2 all V_c_2 all T_c ((-hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),V_A_2)) -> hAPP(c_Set_Ovimage(T_d,T_c,hAPP(c_COMBK(T_c,T_d),V_c_2)),V_A_2) = c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool))) & (hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),V_A_2)) -> c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool)) = hAPP(c_Set_Ovimage(T_d,T_c,hAPP(c_COMBK(T_c,T_d),V_c_2)),V_A_2)))) # label(fact_vimage__const) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2906 (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]. 5.99/6.02 2907 (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_y) = hAPP(V_f_2,B_z))) -> c_Equiv__Relations_Ocongruent(T_c,T_d,V_r_2,V_f_2))) # label(fact_congruentI) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2908 (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]. 5.99/6.02 2909 (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]. 5.99/6.02 2910 (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]. 5.99/6.02 2911 (all V_x all T_a (class_Fields_Ofield(T_a) & class_Int_Onumber__ring(T_a) -> hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OBit1(c_Int_OPls))) = V_x)) # label(fact_divide__Numeral1) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2912 (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]. 5.99/6.02 2913 (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__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]. 5.99/6.02 2914 (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]. 5.99/6.02 2915 (all V_xs_2 all V_x_2 all V_a_2 all V_f_2 all T_d all T_c hAPP(c_List_Ofoldl(T_c,T_d,V_f_2,V_a_2),hAPP(hAPP(c_List_Olist_OCons(T_d),V_x_2),V_xs_2)) = hAPP(c_List_Ofoldl(T_c,T_d,V_f_2,hAPP(hAPP(V_f_2,V_a_2),V_x_2)),V_xs_2)) # label(fact_foldl__Cons) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2916 (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]. 5.99/6.02 2917 (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]. 5.99/6.02 2918 (all V_m_2 all V_w_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_w_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onat,V_w_2)),V_m_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_w_2),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m_2)))))) # label(fact_nat__less__iff) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2919 (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]. 5.99/6.02 2920 (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]. 5.99/6.02 2921 (all V_z_2 all V_y_2 all V_x_2 all T_c (class_Lattices_Osemilattice__sup(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_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_Lattices_Osemilattice__sup__class_Osup(T_c),V_x_2),V_y_2)),V_z_2))))) # label(fact_le__sup__iff) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2922 (all V_k c_Int_Osucc(c_Int_OBit1(V_k)) = c_Int_OBit0(c_Int_Osucc(V_k))) # label(fact_succ__Bit1) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2923 (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]. 5.99/6.02 2924 (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]. 5.99/6.02 2925 (all V_Pa_2 ((all B_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B_x)) -> hBOOL(hAPP(V_Pa_2,B_x)))) <-> (all B_x hBOOL(hAPP(V_Pa_2,hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),B_x)))))) # label(fact_transfer__int__nat__quantifiers_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2926 (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]. 5.99/6.02 2927 (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]. 5.99/6.02 2928 (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]. 5.99/6.02 2929 (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_xs_092_060_094isub_0621_2)),V_ys_092_060_094isub_0621_2) = V_xs_092_060_094isub_0621_2 & V_xs_092_060_094isub_0622_2 = hAPP(hAPP(c_List_Oappend(T_c),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)) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_092_060_094isub_0621_2)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_ys_092_060_094isub_0621_2))) -> V_ys_092_060_094isub_0621_2 = hAPP(hAPP(c_List_Otake(T_c),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_ys_092_060_094isub_0621_2)),V_xs_092_060_094isub_0621_2) & V_ys_092_060_094isub_0622_2 = hAPP(hAPP(c_List_Oappend(T_c),hAPP(hAPP(c_List_Odrop(T_c),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_ys_092_060_094isub_0621_2)),V_xs_092_060_094isub_0621_2)),V_xs_092_060_094isub_0622_2)) <-> hAPP(hAPP(c_List_Oappend(T_c),V_xs_092_060_094isub_0621_2),V_xs_092_060_094isub_0622_2) = hAPP(hAPP(c_List_Oappend(T_c),V_ys_092_060_094isub_0621_2),V_ys_092_060_094isub_0622_2))) # label(fact_append__eq__append__conv__if) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2930 (all T_a (class_Lattices_Oboolean__algebra(T_a) -> c_Orderings_Otop__class_Otop(T_a) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),c_Orderings_Obot__class_Obot(T_a)))) # label(fact_compl__bot__eq) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2931 (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]. 5.99/6.02 2932 (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]. 5.99/6.02 2933 (all V_xs_2 all V_Pa_2 all T_c hAPP(hAPP(c_List_Oappend(T_c),c_List_OtakeWhile(T_c,V_Pa_2,V_xs_2)),c_List_OdropWhile(T_c,V_Pa_2,V_xs_2)) = V_xs_2) # label(fact_takeWhile__dropWhile__id) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2934 (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]. 5.99/6.02 2935 (all V_x_2 all V_g_2 all V_f_2 all T_d all T_c (class_Lattices_Olattice(T_c) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_c),hAPP(V_f_2,V_x_2)),hAPP(V_g_2,V_x_2)) = hAPP(hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_d,T_c)),V_f_2),V_g_2),V_x_2))) # label(fact_inf__fun__def) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2936 (all V_y_2 all V_x_2 all V_A_2 all V_Abs_2 all V_Rep_2 all T_d all T_c (c_Typedef_Otype__definition(T_c,T_d,V_Rep_2,V_Abs_2,V_A_2) -> (V_y_2 = V_x_2 <-> hAPP(V_Rep_2,V_y_2) = hAPP(V_Rep_2,V_x_2)))) # label(fact_type__definition_ORep__inject) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2937 (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]. 5.99/6.02 2938 (all V_x all T_a (class_Lattices_Obounded__lattice__bot(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),c_Orderings_Obot__class_Obot(T_a)) = c_Orderings_Obot__class_Obot(T_a))) # label(fact_inf__bot__right) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2939 (all V_x all T_a all V_n (V_n != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> 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) = hAPP(c_List_Otl(T_a),c_List_Oreplicate(T_a,V_n,V_x)))) # label(fact_tl__replicate) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2940 (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]. 5.99/6.02 2941 (all 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))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z_2)))) # label(fact_zero__less__nat__eq) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2942 (all V_R_2 all T_d all T_e all T_c c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)) = hAPP(hAPP(c_Relation_Orel__comp(T_c,T_e,T_d),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_c,T_e),tc_HOL_Obool))),V_R_2)) # label(fact_rel__comp__empty1) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2943 (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]. 5.99/6.02 2944 (all V_xs_2 all V_n_2 all T_c hAPP(c_List_Oset(tc_List_Olist(T_c)),c_Enum_On__lists(T_c,V_n_2,V_xs_2)) = hAPP(c_Set_OCollect(tc_List_Olist(T_c)),hAPP(hAPP(c_COMBS(tc_List_Olist(T_c),tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_List_Olist(T_c)),c_fconj),hAPP(hAPP(c_COMBC(tc_List_Olist(T_c),tc_Nat_Onat,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_List_Olist(T_c)),c_fequal),c_Nat_Osize__class_Osize(tc_List_Olist(T_c)))),V_n_2))),hAPP(hAPP(c_COMBC(tc_List_Olist(T_c),tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),tc_List_Olist(T_c)),c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool))),c_List_Oset(T_c))),hAPP(c_List_Oset(T_c),V_xs_2))))) # label(fact_set__n__lists) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2945 (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]. 5.99/6.02 2946 (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))) -> hAPP(V_g_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_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__inter__neutral) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2947 (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]. 5.99/6.02 2948 (all V_n_2 hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),V_n_2) = c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_n_2)) # label(fact_atLeast0AtMost) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2949 (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]. 5.99/6.02 2950 (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]. 5.99/6.02 2951 (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]. 5.99/6.02 2952 (all T_a (class_Groups_Omonoid__add(T_a) -> c_Groups_Ozero__class_Ozero(T_a) = hAPP(c_List_Omonoid__add__class_Olistsum(T_a),c_List_Olist_ONil(T_a)))) # label(fact_listsum__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2953 (all V_M_2 all V_n_2 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_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))) = 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)) # label(fact_UN__le__eq__Un0) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2954 (all V_B_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_B_2 -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_A_2)) -> c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),V_A_2) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),V_B_2)),c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),V_A_2))))))) # label(fact_min__max_OInf__fin_Osubset__idem) # label(axiom) # label(non_clause). [assumption]. 5.99/6.02 2955 (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),V_z_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) <-> 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)))))) # label(fact_of__int__le__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 2956 (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]. 5.99/6.03 2957 (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]. 5.99/6.03 2958 (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]. 5.99/6.03 2959 (all V_xs_2 all T_c (V_xs_2 = c_List_Olist_ONil(T_c) <-> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2))) # label(fact_length__0__conv) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 2960 (all V_A_2 all T_c (class_Lattices_Olattice(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> c_Big__Operators_Olattice__class_OSup__fin(T_c,V_A_2) = hAPP(c_Finite__Set_Ofold1(T_c,c_Lattices_Osemilattice__sup__class_Osup(T_c)),V_A_2)))) # label(fact_Sup__fin_OF__eq) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 2961 (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),c_List_Oconcat(T_d,V_xs_2)) = 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))) # label(fact_map__concat) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 2962 (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]. 5.99/6.03 2963 (all V_z (-hBOOL(hAPP(c_Nat__Numeral_Oneg,V_z)) -> hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Int_Onat,V_z)) = V_z)) # label(fact_not__neg__nat) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 2964 (all V_x_2 all V_A_2 all V_F_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofolding(T_c,T_d,V_f_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(c_Fun_Ocomp(T_d,T_d,T_d,hAPP(V_F_2,V_A_2)),hAPP(V_f_2,V_x_2)) = hAPP(c_Fun_Ocomp(T_d,T_d,T_d,hAPP(V_f_2,V_x_2)),hAPP(V_F_2,V_A_2))))) # label(fact_folding_Ocommute__comp_H) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 2965 (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]. 5.99/6.03 2966 (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]. 5.99/6.03 2967 (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]. 5.99/6.03 2968 (all V_f_2 all T_d all V_A_2 all T_c (-hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2))) -> (exists B_x (-hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBS(T_c,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_c),c_fconj),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_A_2))),hAPP(hAPP(c_COMBC(T_c,T_d,tc_HOL_Obool),hAPP(hAPP(c_COMBB(T_d,tc_fun(T_d,tc_HOL_Obool),T_c),c_fequal),V_f_2)),hAPP(V_f_2,B_x)))))) & hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2))))))) # label(fact_pigeonhole__infinite) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 2969 (all V_n_2 (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__eq__0__conv) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 2970 (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]. 5.99/6.03 2971 (all V_S_2 all V_R_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_R_2),V_S_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool)),V_S_2),c_Transitive__Closure_Ortrancl(T_c,V_R_2))) -> c_Transitive__Closure_Ortrancl(T_c,V_S_2) = c_Transitive__Closure_Ortrancl(T_c,V_R_2)))) # label(fact_rtrancl__subset) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 2972 (all V_com_H_2 all V_fun_H_2 c_Com_Ocom_OSKIP != c_Com_Ocom_OWhile(V_fun_H_2,V_com_H_2)) # label(fact_com_Osimps_I16_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 2973 (all V_k_2 all V_C_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(T_c,tc_HOL_Obool)),V_C_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,tc_HOL_Obool),V_C_2))) -> ((all B_c (hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),B_c),V_C_2)) -> hAPP(c_Finite__Set_Ocard(T_c),B_c) = V_k_2)) -> ((all B_c1 (hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),B_c1),V_C_2)) -> (all B_c2 (hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),B_c2),V_C_2)) -> (B_c2 != B_c1 -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),B_c1),B_c2) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))))) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),hAPP(c_Finite__Set_Ocard(tc_fun(T_c,tc_HOL_Obool)),V_C_2)) = hAPP(c_Finite__Set_Ocard(T_c),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,tc_HOL_Obool),V_C_2))))))) # label(fact_card__partition) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 2974 (all V_b all V_a ((c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b) != c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a),V_b) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_b),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b))) & (c_Groups_Ozero__class_Ozero(tc_Int_Oint) = c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b) -> c_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)))) # label(fact_zmod__zminus1__eq__if) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 2975 (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]. 5.99/6.03 2976 (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]. 5.99/6.03 2977 (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]. 5.99/6.03 2978 (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_uminus__apply) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 2979 (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]. 5.99/6.03 2980 (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]. 5.99/6.03 2981 (all V_p_2 all V_g_2 all T_f all V_f_2 all T_c all T_d all T_e hAPP(c_Product__Type_Oapfst(T_f,T_c,T_d,V_g_2),hAPP(c_Product__Type_Oapsnd(T_e,T_d,T_f,V_f_2),V_p_2)) = hAPP(c_Product__Type_Oapsnd(T_e,T_d,T_c,V_f_2),hAPP(c_Product__Type_Oapfst(T_f,T_c,T_e,V_g_2),V_p_2))) # label(fact_apsnd__apfst__commute) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 2982 (all V_xs_2 all V_x_2 all T_c (-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2))) -> c_List_Oinsert(T_c,V_x_2,V_xs_2) = hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2))) # label(fact_not__in__set__insert) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 2983 (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]. 5.99/6.03 2984 (all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k_2),c_Int_OMin)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OBit0(V_k_2)),c_Int_OMin)))) # label(fact_rel__simps_I28_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 2985 (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]. 5.99/6.03 2986 (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]. 5.99/6.03 2987 (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]. 5.99/6.03 2988 (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]. 5.99/6.03 2989 (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]. 5.99/6.03 2990 (all V_a_2 all T_c (class_Rings_Olinordered__idom(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_even__less__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 2991 (all V_y_2 all T_c (class_Int_Onumber__ring(T_c) -> (c_Int_Oiszero(T_c,hAPP(c_Int_Onumber__class_Onumber__of(T_c),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_y_2)))) <-> hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_y_2) = c_Groups_Oone__class_Oone(T_c)))) # label(fact_eq__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 2992 (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]. 5.99/6.03 2993 (all V_B_2 all V_A_2 all V_C_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,V_C_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_C_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_C_2)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_d,tc_HOL_Obool)),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2)),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_B_2)) = hAPP(c_Set_Oimage(T_c,T_d,V_f_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)))))) # label(fact_inj__on__image__set__diff) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 2994 (all V_A_2 (c_Nat__Transfer_Onat__set(V_A_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),V_A_2))) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),V_A_2))))) # label(fact_transfer__int__nat__set__relations_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 2995 (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]. 5.99/6.03 2996 (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]. 5.99/6.03 2997 (all V_n_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),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_c),V_n_2)),c_Groups_Oone__class_Oone(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))))) # label(fact_gauss__sum) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 2998 (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]. 5.99/6.03 2999 (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]. 5.99/6.03 3000 (all V_xs_2 all T_c hAPP(c_List_Oset(T_c),hAPP(c_List_Orev(T_c),V_xs_2)) = hAPP(c_List_Oset(T_c),V_xs_2)) # label(fact_set__rev) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3001 (all V_xss_2 all V_xs_2 all T_c hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),c_List_Oconcat(T_c,V_xss_2)) = 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)) # label(fact_foldl__conv__concat) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3002 (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]. 5.99/6.03 3003 (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]. 5.99/6.03 3004 (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),V_v)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_c),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w))) = 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))) # label(fact_add__number__of__diff2) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3005 (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]. 5.99/6.03 3006 (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]. 5.99/6.03 3007 (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]. 5.99/6.03 3008 (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]. 5.99/6.03 3009 (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]. 5.99/6.03 3010 (all V_ys_2 all V_Pa_2 all T_d all T_c (V_ys_2 = c_List_Olist_ONil(T_d) <-> c_List_Olist__all2(T_c,T_d,V_Pa_2,c_List_Olist_ONil(T_c),V_ys_2))) # label(fact_list__all2__Nil) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3011 (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]. 5.99/6.03 3012 (all V_seed_2 all V_size_2 all V_nrandom_2 all V_f_2 all V_R_2 all T_c all T_d c_New__Random__Sequence_Oneg__decr__bind(T_d,T_c,V_R_2,V_f_2,V_nrandom_2,V_size_2,V_seed_2) = c_New__DSequence_Oneg__decr__bind(T_d,T_c,hAPP(hAPP(hAPP(V_R_2,V_nrandom_2),V_size_2),V_seed_2),hAPP(hAPP(c_COMBC(T_d,tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Lazy__Sequence_Olazy__sequence(tc_Option_Ooption(T_c)))),hAPP(hAPP(c_COMBC(T_d,tc_Code__Numeral_Ocode__numeral,tc_fun(tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Lazy__Sequence_Olazy__sequence(tc_Option_Ooption(T_c))))),hAPP(hAPP(c_COMBC(T_d,tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Lazy__Sequence_Olazy__sequence(tc_Option_Ooption(T_c)))))),V_f_2),V_nrandom_2)),V_size_2)),V_seed_2))) # label(fact_New__Random__Sequence_Oneg__decr__bind__def) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3013 (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]. 5.99/6.03 3014 (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]. 5.99/6.03 3015 (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)),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__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))) # label(fact_Diff__Int) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3016 (all T_c c_Relation_Osym(T_c,c_Relation_OId(T_c))) # label(fact_sym__Id) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3017 (all V_z hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_z),V_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_z)) # label(fact_nat__mult__2) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3018 (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]. 5.99/6.03 3019 (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))) -> 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) = 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)))) # label(fact_filter__equals__takeWhile__sorted__rev) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3020 (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]. 5.99/6.03 3021 (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]. 5.99/6.03 3022 (all V_w all V_v hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_v),V_w)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w))) # label(fact_plus__numeral__code_I9_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3023 (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]. 5.99/6.03 3024 (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]. 5.99/6.03 3025 (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]. 5.99/6.03 3026 (all V_r all V_q all V_n all V_m (hBOOL(hAPP(c_Divides_Odivmod__nat__rel(V_m,V_n),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_q),V_r))) -> c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n) = V_r)) # label(fact_mod__eq) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3027 (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]. 5.99/6.03 3028 (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]. 5.99/6.03 3029 (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]. 5.99/6.03 3030 (all V_n all V_m all T_a (class_Rings_Osemiring__1(T_a) -> hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),V_m)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),V_n)))) # label(fact_of__nat__mult) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3031 (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]. 5.99/6.03 3032 (all V_n_2 all V_xs_2 all V_x_2 all T_c c_Nat_Onat_Onat__case(T_c,V_x_2,hAPP(c_List_Onth(T_c),V_xs_2),V_n_2) = hAPP(hAPP(c_List_Onth(T_c),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2)),V_n_2)) # label(fact_nth_Osimps) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3033 (all V_a all T_a (class_Int_Onumber__ring(T_a) & class_Fields_Ofield(T_a) -> hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OBit1(c_Int_OPls))) = V_a)) # label(fact_divide__numeral__1) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3034 (all V_b_2 all V_a_2 all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> c_Complete__Lattice_OInf__class_OInf(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_b_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_c),V_a_2),V_b_2))) # label(fact_Inf__binary) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3035 (all V_n_2 all V_f_2 all T_c all V_A_2 (hAPP(c_Set_Oimage(tc_Nat_Onat,T_c,V_f_2),hAPP(c_Set_OCollect(tc_Nat_Onat),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_HOL_Obool),c_Orderings_Oord__class_Oless(tc_Nat_Onat)),V_n_2))) = V_A_2 -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)))) # label(fact_nat__seg__image__imp__finite) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3036 (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]. 5.99/6.03 3037 (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]. 5.99/6.03 3038 (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]. 5.99/6.03 3039 (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]. 5.99/6.03 3040 (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]. 5.99/6.03 3041 (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]. 5.99/6.03 3042 (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]. 5.99/6.03 3043 (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]. 5.99/6.03 3044 (all V_A_2 all V_a_2 all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> c_Complete__Lattice_OSup__class_OSup(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_A_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_c),V_a_2),c_Complete__Lattice_OSup__class_OSup(T_c,V_A_2)))) # label(fact_Sup__insert) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3045 (all V_r_2 all T_c all T_d hAPP(c_Relation_Oconverse(T_d,T_c),V_r_2) = hAPP(c_Set_OCollect(tc_prod(T_c,T_d)),hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,tc_HOL_Obool),hAPP(hAPP(c_COMBC(T_c,tc_fun(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)))) # label(fact_converse__def) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3046 (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]. 5.99/6.03 3047 (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]. 5.99/6.03 3048 (all V_r_2 all T_c (hAPP(c_Relation_Oconverse(T_c,T_c),V_r_2) = V_r_2 <-> c_Relation_Osym(T_c,V_r_2))) # label(fact_sym__conv__converse__eq) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3049 (all V_d_2 all V_m_2 (c_Groups_Ozero__class_Ozero(tc_Int_Oint) = c_Divides_Odiv__class_Omod(tc_Int_Oint,V_m_2,V_d_2) <-> (exists B_q hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_d_2),B_q) = V_m_2))) # label(fact_zmod__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3050 (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_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_Set_Oinsert(T_d),V_a_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),V_B_2)) = hAPP(hAPP(c_Set_Oinsert(T_d),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool)))))) # label(fact_UN__extend__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3051 (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))) -> (V_a != c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> c_Groups_Ozero__class_Ozero(tc_Int_Oint) = V_r))) # label(fact_self__remainder) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3052 (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]. 5.99/6.03 3053 (all V_xs_2 all V_Pa_2 all T_c (c_List_Olist__ex(T_c,V_Pa_2,V_xs_2) <-> (exists B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(c_List_Oset(T_c),V_xs_2))) & hBOOL(hAPP(V_Pa_2,B_x)))))) # label(fact_list__ex__iff__raw) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3054 (all V_x_2 all T_c 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))) = c_List_OremoveAll(T_c,V_x_2)) # label(fact_removeAll__filter__not__eq) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3055 (all V_y all V_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_y)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Int_Onat,V_x)),hAPP(c_Int_Onat,V_y)) = hAPP(c_Int_Onat,c_Nat__Transfer_Otsub(V_x,V_y))))) # label(fact_Nat__Transfer_Otransfer__nat__int__functions_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3056 (all V_B_2 all V_C_2 all T_d all V_a_2 all T_c 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)) = 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))) # label(fact_INT__extend__simps_I5_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3057 (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]. 5.99/6.03 3058 (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]. 5.99/6.03 3059 (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]. 5.99/6.03 3060 (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]. 5.99/6.03 3061 (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]. 5.99/6.03 3062 (all V_j_2 all V_i_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_j_2),V_i_2)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_j_2 <-> c_List_Olist_ONil(tc_Nat_Onat) = c_List_Oupt(V_i_2,V_j_2))) # label(fact_upt__eq__Nil__conv) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3063 (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]. 5.99/6.03 3064 (all V_z all V_y all V_x 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)) = 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))) # label(fact_zpower__zadd__distrib) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3065 (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]. 5.99/6.03 3066 (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__empty) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3067 (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]. 5.99/6.03 3068 (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]. 5.99/6.03 3069 (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]. 5.99/6.03 3070 (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]. 5.99/6.03 3071 (all V_ys_2 all V_xs_2 all T_c (class_Orderings_Olinorder(T_c) -> (c_List_Olinorder__class_Osorted(T_c,hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_ys_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))) -> (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_ys_2)))) # label(fact_sorted__append) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3072 (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]. 5.99/6.03 3073 (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]. 5.99/6.03 3074 (all V_x all T_a (class_Rings_Ocomm__ring__1(T_a) -> hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_x) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),c_Groups_Oone__class_Oone(T_a))),V_x))) # label(fact_comm__ring__1__class_Onormalizing__ring__rules_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3075 (all V_list_H all V_a_H all T_a hAPP(hAPP(c_List_Olist_OCons(T_a),V_a_H),V_list_H) != c_List_Olist_ONil(T_a)) # label(fact_list_Osimps_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3076 (all V_a all T_a (class_Groups_Omonoid__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__left) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3077 (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]. 5.99/6.03 3078 (all V_xs all T_a (hBOOL(hAPP(c_List_Odistinct(T_a),V_xs)) -> V_xs = c_List_Oremdups(T_a,V_xs))) # label(fact_distinct__remdups__id) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3079 (all V_n all V_m 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_m)),hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),V_n)) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n)))) # label(fact_power__add) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3080 (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]. 5.99/6.03 3081 (all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(c_Finite__Set_Ofold1(T_c,c_Orderings_Oord__class_Omin(T_c)),V_A_2) = c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),V_A_2)))) # label(fact_min__max_OInf__fin_OF__eq) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3082 (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]. 5.99/6.03 3083 (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]. 5.99/6.03 3084 (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]. 5.99/6.03 3085 (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]. 5.99/6.03 3086 (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]. 5.99/6.03 3087 (all V_c all V_a all V_b all T_a (class_Divides_Osemiring__div(T_a) -> (V_b != c_Groups_Ozero__class_Ozero(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(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_Oplus__class_Oplus(T_a),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c))),V_b)))) # label(fact_div__mult__self2) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3088 (all T_a (class_Int_Onumber__ring(T_a) -> hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OPls) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_number__of__Pls) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3089 (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]. 5.99/6.03 3090 (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]. 5.99/6.03 3091 (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]. 5.99/6.03 3092 (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]. 5.99/6.03 3093 (all T_c c_Predicate_Oconversep(T_c,T_c,c_fequal) = c_fequal) # label(fact_conversep__eq) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3094 (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),V_a),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_b),V_c)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omax(T_a),V_a),V_b)),V_c))) # label(fact_min__max_Osup_Oassoc) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3095 (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 -> V_x_2 = hAPP(c_Hilbert__Choice_Oinv__into(T_c,T_d,V_A_2,V_f_2),V_y_2))))) # label(fact_inv__into__f__eq) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3096 (all V_xs_2 all V_Pa_2 all T_c (c_List_Olist__ex(T_c,V_Pa_2,V_xs_2) <-> c_List_Olist__ex(T_c,V_Pa_2,hAPP(c_List_Orev(T_c),V_xs_2)))) # label(fact_list__ex__rev) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3097 (all V_r_2 all T_c hAPP(c_Relation_OField(T_c),V_r_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Relation_ODomain(T_c,T_c),V_r_2)),hAPP(c_Relation_ORange(T_c,T_c),V_r_2))) # label(fact_Field__def) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3098 (all V_v all V_b all V_a all T_b (class_Int_Onumber(T_b) & class_Rings_Oring(T_b) -> hAPP(hAPP(c_Groups_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]. 5.99/6.03 3099 (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]. 5.99/6.03 3100 (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) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),V_x_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_add__nonneg__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3101 (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),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))) = hAPP(c_Nat_Osize__class_Osize(tc_Com_Ocom),c_Com_Ocom_OCond(V_fun_2,V_com1_2,V_com2_2))) # label(fact_com_Osize_I13_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3102 (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]. 5.99/6.03 3103 (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]. 5.99/6.03 3104 (all V_z all V_y all V_x all T_a (class_Groups_Oordered__ab__group__add(T_a) -> hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_x),V_z)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_y),V_z)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),V_y)),V_z))) # label(fact_min__diff__distrib__left) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3105 (all V_u_2 all V_l_2 hAPP(c_Int_Onat,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_u_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_l_2),c_Groups_Oone__class_Oone(tc_Int_Oint)))) = hAPP(c_Finite__Set_Ocard(tc_Int_Oint),c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Int_Oint,V_l_2,V_u_2))) # label(fact_card__greaterThanLessThan__int) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3106 (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]. 5.99/6.03 3107 (all V_b_2 all V_a_2 (hBOOL(hAPP(c_Wellfounded_Oaccp(tc_prod(tc_Int_Oint,tc_Int_Oint),c_Divides_OposDivAlg__rel),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_a_2),V_b_2))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_a_2),V_b_2)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_b_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> c_Divides_OposDivAlg(V_a_2,V_b_2) = hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a_2)) & (-(hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__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(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)))) # label(fact_posDivAlg_Opsimps) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3108 (all V_b all V_a all T_a (class_Groups_Oab__group__add(T_a) -> hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_b)))) # label(fact_minus__add__distrib) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3109 (all V_com_H_2 all V_fun_H_2 c_Com_Ocom_OSKIP != c_Com_Ocom_OWhile(V_fun_H_2,V_com_H_2)) # label(fact_com_Osimps_I17_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3110 (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]. 5.99/6.03 3111 (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]. 5.99/6.03 3112 (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]. 5.99/6.03 3113 (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]. 5.99/6.03 3114 (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)) -> 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)) = c_List_Otranspose(T_c,V_xs_2)))) # label(fact_transpose__rectangle) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3115 (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]. 5.99/6.03 3116 (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]. 5.99/6.03 3117 (all V_b_2 all V_a_2 all T_c (class_Groups_Ogroup__add(T_c) -> (hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_c),V_a_2),V_b_2) = c_Groups_Ozero__class_Ozero(T_c) <-> V_b_2 = V_a_2))) # label(fact_right__minus__eq) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3118 (all V_b all V_c all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Divides_Odiv__class_Omod(T_a,V_a,V_c)),V_b),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b),V_c))) # label(fact_zmod__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3119 (all V_b_2 all V_a_2 all V_Pa_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)))) <-> hBOOL(hAPP(V_Pa_2,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_a_2),V_b_2))))) # label(fact_nat__diff__split) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3120 (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))) -> (V_a != c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> V_q = c_Groups_Oone__class_Oone(tc_Int_Oint)))) # label(fact_self__quotient) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3121 (all V_xs_2 all V_Pa_2 all T_c hAPP(c_List_Orev(T_c),hAPP(c_List_Ofilter(T_c,V_Pa_2),V_xs_2)) = hAPP(c_List_Ofilter(T_c,V_Pa_2),hAPP(c_List_Orev(T_c),V_xs_2))) # label(fact_rev__filter) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3122 (all V_k2_2 all V_k1_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OBit1(V_k1_2)),c_Int_OBit0(V_k2_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k1_2),V_k2_2)))) # label(fact_less__eq__int__code_I15_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3123 (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]. 5.99/6.03 3124 (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]. 5.99/6.03 3125 (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),V_c_2),c_Groups_Ozero__class_Ozero(T_c))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_a_2),c_Groups_Ozero__class_Ozero(T_c)))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_c_2),c_Groups_Ozero__class_Ozero(T_c))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_c_2))))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_c_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_c_2)),V_b_2))) <-> 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]. 5.99/6.03 3126 (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]. 5.99/6.03 3127 (all T_c all V_r_2 all T_d ((all B_i c_Wellfounded_OwfP(T_d,hAPP(V_r_2,B_i))) -> ((all B_i all B_j (hAPP(V_r_2,B_j) != hAPP(V_r_2,B_i) -> 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)),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_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]. 5.99/6.03 3128 (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]. 5.99/6.03 3129 (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]. 5.99/6.03 3130 (all V_v_2 (hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_v_2),c_Int_OPls)))) # label(fact_eq__0__number__of) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3131 (all V_xs all T_a all V_j all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),V_j)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))) -> hAPP(hAPP(c_List_Oappend(T_a),hAPP(hAPP(c_List_Otake(T_a),V_i),V_xs)),hAPP(hAPP(c_List_Otake(T_a),V_j),hAPP(hAPP(c_List_Odrop(T_a),V_i),V_xs))) = hAPP(hAPP(c_List_Otake(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_i),V_j)),V_xs))) # label(fact_take__add) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3132 (all V_xs_2 all V_Pa_2 all T_c (hAPP(c_List_Ofilter(T_c,V_Pa_2),V_xs_2) = c_List_Olist_ONil(T_c) <-> (all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(c_List_Oset(T_c),V_xs_2))) -> -hBOOL(hAPP(V_Pa_2,B_x)))))) # label(fact_filter__empty__conv) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3133 (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]. 5.99/6.03 3134 (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]. 5.99/6.03 3135 (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]. 5.99/6.03 3136 (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_y_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_x_2))) # label(fact_rtrancl__listrel1__eq__len) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3137 (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]. 5.99/6.03 3138 (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]. 5.99/6.03 3139 (all V_xs_2 all V_Pa_2 all T_c ((exists B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(c_List_Oset(T_c),V_xs_2))) & hBOOL(hAPP(V_Pa_2,B_x)) & (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)))) <-> c_List_Olist__ex1(T_c,V_Pa_2,V_xs_2))) # label(fact_list__ex1__iff) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3140 (all V_zs_2 all V_ys_2 all V_xs_2 all T_c (V_ys_2 = hAPP(hAPP(c_List_Odrop(T_c),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2)),V_zs_2) & V_xs_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_zs_2 = hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_ys_2))) # label(fact_append__eq__conv__conj) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3141 (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]. 5.99/6.03 3142 (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]. 5.99/6.03 3143 (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]. 5.99/6.03 3144 (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]. 5.99/6.03 3145 (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]. 5.99/6.03 3146 (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,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))) # label(fact_transpose_Opsimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3147 (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_Otrancl(T_c,V_r_2)),c_Relation_OId(T_c))) # label(fact_reflcl__trancl) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3148 (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(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(T_c),V_b_2),V_d_2)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_c_2),V_a_2))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_b_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(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)))))) # label(fact_atLeastatMost__psubset__iff) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3149 (all V_b_2 all V_a_2 all T_c (class_Orderings_Oorder(T_c) -> (V_a_2 = V_b_2 -> hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) = c_SetInterval_Oord__class_OatLeastAtMost(T_c,V_a_2,V_b_2)))) # label(fact_atLeastAtMost__singleton_H) # label(axiom) # label(non_clause). [assumption]. 5.99/6.03 3150 (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]. 5.99/6.03 3151 (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]. 5.99/6.04 3152 (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]. 5.99/6.04 3153 (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]. 5.99/6.04 3154 (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]. 5.99/6.04 3155 (all V_x_2 all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(c_List_Olistsp(T_c,V_A_2),V_x_2)) & hBOOL(hAPP(c_List_Olistsp(T_c,V_B_2),V_x_2)) <-> 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)))) # label(fact_listsp__conj__eq) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3156 (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]. 5.99/6.04 3157 (all V_m hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,V_m))),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(c_Nat_OSuc,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) # label(fact_div2__Suc__Suc) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3158 (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]. 5.99/6.04 3159 (all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m_2,hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) <-> c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m_2,hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = c_Groups_Oone__class_Oone(tc_Nat_Onat))) # label(fact_mod2__gr__0) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3160 (all V_m hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m)) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m)) # label(fact_abs__int__eq) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3161 (all V_k_2 all V_l_2 (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_l_2),V_k_2)) -> c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = c_SetInterval_Oord_OgreaterThanLessThan(tc_Nat_Onat,hAPP(hAPP(c_COMBS(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),tc_Nat_Onat),c_COMBS(tc_Nat_Onat,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_Nat_Onat),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Nat_Onat),c_fconj)),c_Rings_Odvd__class_Odvd(tc_Nat_Onat)))),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_Nat_Onat),hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Nat_Onat),c_fNot)),hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_HOL_Obool),c_Rings_Odvd__class_Odvd(tc_Nat_Onat)))),V_k_2,V_l_2))) # label(fact_dvd_OgreaterThanLessThan__empty) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3162 (all V_A_2 all T_c c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,tc_HOL_Obool),hAPP(c_Set_OPow(T_c),V_A_2)) = V_A_2) # label(fact_Union__Pow__eq) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3163 (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]. 5.99/6.04 3164 (all V_n_2 hAPP(c_List_Oset(tc_Nat_Onat),c_List_Oupt(c_Groups_Ozero__class_Ozero(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_n_2))) = hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),V_n_2)) # label(fact_atMost__upto) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3165 (all V_r_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_prod(T_c,T_c)),V_r_2)) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(tc_prod(T_c,T_c)),c_Transitive__Closure_Otrancl(T_c,V_r_2))))) # label(fact_finite__trancl) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3166 (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_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_I39_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3167 (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]. 5.99/6.04 3168 (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]. 5.99/6.04 3169 (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]. 5.99/6.04 3170 (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(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_y)),V_x) = hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_y),V_x))))) # label(fact_abs__mult__pos) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3171 (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]. 5.99/6.04 3172 (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]. 5.99/6.04 3173 (all V_y_2 all V_f_2 all T_c all T_d (c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)) = hAPP(c_Set_Oimage(T_d,T_c,V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool))) -> 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]. 5.99/6.04 3174 (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_Groups_Otimes__class_Otimes(T_d),hAPP(V_g_2,V_x_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_g_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_g_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2))))) # label(fact_setprod_Oinsert__remove) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3175 (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))) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_X_2),V_Y_2) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) | V_X_2 = V_Y_2)))) # label(fact_quotient__disj) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3176 (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,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_Oin__idem) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3177 (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]. 5.99/6.04 3178 (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)),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]. 5.99/6.04 3179 (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]. 5.99/6.04 3180 (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__add__class_Osetsum(tc_Int_Oint,T_c),hAPP(hAPP(c_COMBB(tc_Nat_Onat,T_c,tc_Int_Oint),V_f_2),c_Int_Onat)),hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),V_A_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_c),V_f_2),V_A_2))) # label(fact_transfer__nat__int__sum__prod_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3181 (all V_xs all V_n all T_a hAPP(c_List_Otl(T_a),hAPP(hAPP(c_List_Odrop(T_a),V_n),V_xs)) = hAPP(hAPP(c_List_Odrop(T_a),V_n),hAPP(c_List_Otl(T_a),V_xs))) # label(fact_tl__drop) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3182 (all V_y_2 all V_x_2 all V_z_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_z_2),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),V_x_2),V_y_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_z_2),V_y_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_z_2),V_x_2))))) # label(fact_min__less__iff__conj) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3183 (all V_xs all V_n all T_a hAPP(c_List_Orotate(T_a,hAPP(c_Nat_OSuc,V_n)),V_xs) = hAPP(c_List_Orotate1(T_a),hAPP(c_List_Orotate(T_a,V_n),V_xs))) # label(fact_rotate__Suc) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3184 (all T_c c_Wellfounded_Ofinite__psubset(T_c) = hAPP(c_Set_OCollect(tc_prod(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool))),hAPP(c_Product__Type_Oprod_Oprod__case(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),hAPP(hAPP(c_COMBC(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool)),tc_fun(T_c,tc_HOL_Obool)),c_COMBS(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(T_c,tc_HOL_Obool)),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),c_fconj)),c_Orderings_Oord__class_Oless(tc_fun(T_c,tc_HOL_Obool))))),c_Finite__Set_Ofinite(T_c))))) # label(fact_finite__psubset__def) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3185 (all V_a all T_a (class_Groups_Ocomm__monoid__add(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a) = V_a)) # label(fact_add__0) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3186 (all V_n all V_m ((-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_m)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_n))) -> c_List_Oupt(hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_m),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_n)) = c_List_Olist_ONil(tc_Nat_Onat)) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_m)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_n))) -> hAPP(hAPP(c_List_Olist_OCons(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_m)),c_List_Oupt(hAPP(c_Nat_OSuc,hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_m)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_n))) = c_List_Oupt(hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_m),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_n))))) # label(fact_upt__rec__number__of) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3187 (all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_A_2 -> hBOOL(hAPP(hAPP(c_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]. 5.99/6.04 3188 (all V_x_2 all V_z_2 all V_A_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofun__left__comm__idem(T_c,T_d,V_f_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_d,V_f_2),hAPP(hAPP(V_f_2,V_x_2),V_z_2)),V_A_2) = hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_d,V_f_2),V_z_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2))))) # label(fact_fun__left__comm__idem_Ofold__insert__idem2) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3189 (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]. 5.99/6.04 3190 (all V_g_2 all V_f_2 all T_c all T_d (class_Orderings_Oord(T_d) -> ((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)))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,T_d)),V_f_2),V_g_2))))) # label(fact_le__fun__def) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3191 (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]. 5.99/6.04 3192 (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]. 5.99/6.04 3193 (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]. 5.99/6.04 3194 (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]. 5.99/6.04 3195 (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 -> 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_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))))) # label(fact_UN__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3196 (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]. 5.99/6.04 3197 (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,B_i) = hAPP(V_f_2,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B_i),B_k)))))) # label(fact_wf__weak__decr__stable) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3198 (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]. 5.99/6.04 3199 (all V_ys_2 all V_xs_2 all V_Pa_2 all T_c (c_List_Olist__ex(T_c,V_Pa_2,V_xs_2) | c_List_Olist__ex(T_c,V_Pa_2,V_ys_2) <-> c_List_Olist__ex(T_c,V_Pa_2,hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_ys_2)))) # label(fact_list__ex__append) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3200 (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),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_e,T_d),tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool))) # label(fact_rel__comp__empty2) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3201 (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]. 5.99/6.04 3202 (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]. 5.99/6.04 3203 (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]. 5.99/6.04 3204 (all V_xs all V_x all T_a V_x = hAPP(hAPP(c_List_Onth(T_a),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_nth__Cons__0) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3205 (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]. 5.99/6.04 3206 (all V_k_2 all T_c (class_Orderings_Olinorder(T_c) -> hAPP(c_SetInterval_Oord__class_OgreaterThan(T_c),V_k_2) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatMost(T_c),V_k_2)))) # label(fact_Compl__atMost) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3207 (all V_b_2 all V_c_2 all V_a_2 all T_c (class_Rings_Olinordered__ring__strict(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_c_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_b_2),V_c_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_b_2),V_a_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_c_2),c_Groups_Ozero__class_Ozero(T_c))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),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))))) # label(fact_mult__less__cancel__right__disj) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3208 (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]. 5.99/6.04 3209 (all V_xs_2 all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> hAPP(c_List_Ofoldl(T_c,T_c,c_Orderings_Oord__class_Omin(T_c),V_x_2),V_xs_2) = c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),hAPP(c_List_Oset(T_c),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2))))) # label(fact_min__max_OInf__fin__set__fold) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3210 (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]. 5.99/6.04 3211 (all V_c all V_b all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),c_Divides_Odiv__class_Omod(T_a,V_a,V_c)),V_b),V_c))) # label(fact_mod__add__left__eq) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3212 (all V_a_2 all T_c (class_Groups_Oordered__ab__group__add(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_a_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_a_2)),c_Groups_Ozero__class_Ozero(T_c)))))) # label(fact_neg__less__0__iff__less) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3213 (all V_xs all V_n all T_a hAPP(hAPP(c_List_Odrop(T_a),V_n),c_List_Obutlast(T_a,V_xs)) = c_List_Obutlast(T_a,hAPP(hAPP(c_List_Odrop(T_a),V_n),V_xs))) # label(fact_butlast__drop) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3214 (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]. 5.99/6.04 3215 (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]. 5.99/6.04 3216 (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]. 5.99/6.04 3217 (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]. 5.99/6.04 3218 (all V_b_2 all V_B_2 all V_A_2 all T_d all T_c ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)) -> hBOOL(hAPP(hAPP(V_B_2,B_x),V_b_2)))) <-> hBOOL(hAPP(hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_HOL_Obool)),V_A_2),V_B_2),V_b_2)))) # label(fact_INF1__iff) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3219 (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]. 5.99/6.04 3220 (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]. 5.99/6.04 3221 (all V_y all V_x all T_a (class_Orderings_Oorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),V_x),V_y)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y)) | V_x = V_y))) # label(fact_order__le__imp__less__or__eq) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3222 (all V_xs_2 all V_x_2 all T_c ((exists B_ys exists B_zs (-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),B_zs))) & V_xs_2 = hAPP(hAPP(c_List_Oappend(T_c),B_ys),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),B_zs)))) <-> hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2))))) # label(fact_in__set__conv__decomp__last) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3223 (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]. 5.99/6.04 3224 (all V_n all V_m hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),V_n) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),hAPP(c_Nat_OSuc,V_n))) # label(fact_add__Suc__shift) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3225 (all V_f_2 all V_A_2 all T_c all T_d (class_Groups_Ocomm__monoid__add(T_d) -> (-hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_f_2),V_A_2) = c_Groups_Ozero__class_Ozero(T_d)))) # label(fact_setsum__infinite) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3226 (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]. 5.99/6.04 3227 (all V_C_2 all V_B_2 all V_A_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_C_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),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__assoc) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3228 (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]. 5.99/6.04 3229 (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]. 5.99/6.04 3230 (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]. 5.99/6.04 3231 (all V_y_2 all V_x_2 (hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_y_2) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_x_2) <-> V_y_2 = V_x_2)) # label(fact_transfer__int__nat__relations_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3232 (all V_u_2 all V_l_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_l_2),V_u_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(T_c),V_l_2)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_c,V_l_2),V_u_2)) = hAPP(c_SetInterval_Oord__class_OlessThan(T_c),V_u_2)))) # label(fact_ivl__disj__un_I8_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3233 (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]. 5.99/6.04 3234 (all V_A_2 all V_Abs_2 all V_Rep_2 all T_d all T_c (c_Typedef_Otype__definition(T_c,T_d,V_Rep_2,V_Abs_2,V_A_2) -> V_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))))) # label(fact_type__definition_ORep__range) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3235 (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]. 5.99/6.04 3236 (all T_c c_List_Olists(T_c,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) = c_Orderings_Otop__class_Otop(tc_fun(tc_List_Olist(T_c),tc_HOL_Obool))) # label(fact_lists__UNIV) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3237 (all V_c all V_b all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Divides_Odiv__class_Omod(T_a,V_a,V_c)),V_b),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b),V_c))) # label(fact_mod__mult__left__eq) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3238 (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]. 5.99/6.04 3239 (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]. 5.99/6.04 3240 (all V_xs all V_x all T_a all V_i ((-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),V_i),hAPP(c_Product__Type_Ofst(tc_Code__Numeral_Ocode__numeral,T_a),V_x))) -> hAPP(c_Random_Opick(T_a,hAPP(hAPP(c_List_Olist_OCons(tc_prod(tc_Code__Numeral_Ocode__numeral,T_a)),V_x),V_xs)),V_i) = hAPP(c_Random_Opick(T_a,V_xs),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Code__Numeral_Ocode__numeral),V_i),hAPP(c_Product__Type_Ofst(tc_Code__Numeral_Ocode__numeral,T_a),V_x)))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),V_i),hAPP(c_Product__Type_Ofst(tc_Code__Numeral_Ocode__numeral,T_a),V_x))) -> hAPP(c_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]. 5.99/6.04 3241 (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]. 5.99/6.04 3242 (all V_a all V_xs all T_a all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))) -> hAPP(hAPP(c_List_Oappend(T_a),hAPP(hAPP(c_List_Otake(T_a),V_i),V_xs)),hAPP(hAPP(c_List_Olist_OCons(T_a),V_a),hAPP(hAPP(c_List_Odrop(T_a),hAPP(c_Nat_OSuc,V_i)),V_xs))) = hAPP(hAPP(c_List_Olist__update(T_a,V_xs),V_i),V_a))) # label(fact_upd__conv__take__nth__drop) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3243 (all V_b_2 all V_c_2 all V_a_2 all T_c (class_Rings_Oidom(T_c) -> (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_c_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_b_2),V_c_2))) <-> hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_c),V_a_2),V_b_2)) | c_Groups_Ozero__class_Ozero(T_c) = V_c_2))) # label(fact_dvd__mult__cancel__right) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3244 (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]. 5.99/6.04 3245 (all V_B_2 all V_A_2 all V_r_2 all T_d all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_d,T_c,V_r_2),V_A_2)),V_B_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_d,tc_HOL_Obool)),V_A_2),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_d,tc_HOL_Obool)),hAPP(c_Relation_OImage(T_c,T_d,hAPP(c_Relation_Oconverse(T_d,T_c),V_r_2)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),V_B_2))))))) # label(fact_Image__subset__eq) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3246 (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(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 <-> (c_Groups_Ozero__class_Ozero(T_c) != hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2) -> V_b_2 = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2))) & (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))))) # label(fact_eq__divide__eq__number__of1) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3247 (all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_A_2)) -> V_B_2 = V_A_2))) # label(fact_equalityI) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3248 (all V_b all V_a all V_c all T_a (class_Fields_Ofield__inverse__zero(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) != V_c -> hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_a)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),V_b)) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_b)))) # label(fact_mult__divide__mult__cancel__left) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3249 (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]. 5.99/6.04 3250 (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]. 5.99/6.04 3251 (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]. 5.99/6.04 3252 (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]. 5.99/6.04 3253 (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]. 5.99/6.04 3254 (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]. 5.99/6.04 3255 (all V_f_2 all T_c all T_d c_List_Omaps(T_d,T_c,V_f_2,c_List_Olist_ONil(T_d)) = c_List_Olist_ONil(T_c)) # label(fact_maps__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3256 (all V_y all V_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_y)) -> hAPP(c_Int_Onat,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_x),V_y)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(c_Int_Onat,V_x)),hAPP(c_Int_Onat,V_y))))) # label(fact_Divides_Otransfer__nat__int__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3257 (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]. 5.99/6.04 3258 (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__1) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3259 (all V_b_2 all V_a_2 all T_c (class_Groups_Oordered__ab__group__add__abs(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(c_Groups_Oabs__class_Oabs(T_c),V_a_2)),V_b_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),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))))) # label(fact_abs__le__iff) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3260 (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]. 5.99/6.04 3261 (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)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_a_2),V_b_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)),V_a_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_b_2))))) # label(fact_zero__le__divide__iff) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3262 (all V_n hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),V_n) = hAPP(c_Nat_OSuc,V_n)) # label(fact_Suc__eq__plus1__left) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3263 (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]. 5.99/6.04 3264 (all V_B_2 all V_A_2 all V_c_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))) <-> hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),V_B_2)) | hBOOL(hAPP(hAPP(c_member(T_c),V_c_2),V_A_2)))) # label(fact_Un__iff) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3265 (all T_d all T_c c_Product__Type_Oapfst(T_c,T_c,T_d,c_Fun_Oid(T_c)) = c_Fun_Oid(tc_prod(T_c,T_d))) # label(fact_apfst__id) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3266 (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]. 5.99/6.04 3267 (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]. 5.99/6.04 3268 (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]. 5.99/6.04 3269 (all V_b all V_a all T_a (class_Groups_Ogroup__add(T_a) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_b) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_b)))) # label(fact_diff__def) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3270 (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]. 5.99/6.04 3271 (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]. 5.99/6.04 3272 (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]. 5.99/6.04 3273 (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]. 5.99/6.04 3274 (all V_xa_2 all V_x_2 all V_R_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_x_2),V_xa_2)),c_Wellfounded_Omax__ext(T_c,V_R_2))) <-> 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))) # label(fact_max__extp__max__ext__eq) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3275 (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]. 5.99/6.04 3276 (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]. 5.99/6.04 3277 (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),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(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_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2))))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2)),c_Groups_Ozero__class_Ozero(T_c))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_a_2),c_Groups_Ozero__class_Ozero(T_c)))))))) # label(fact_less__divide__eq__number__of1) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3278 (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]. 5.99/6.04 3279 (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]. 5.99/6.04 3280 (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_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)) = 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_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]. 5.99/6.04 3281 (all V_f_2 all V_T_2 all V_S_2 all T_c all T_d (class_Groups_Ocomm__monoid__mult(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_S_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_T_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_S_2),V_T_2))) -> c_Groups_Oone__class_Oone(T_d) = hAPP(V_f_2,B_x))) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_d),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_f_2),V_S_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_f_2),V_T_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),V_f_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_S_2),V_T_2))))))) # label(fact_setprod__Un__one) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3282 (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]. 5.99/6.04 3283 (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]. 5.99/6.04 3284 (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]. 5.99/6.04 3285 (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]. 5.99/6.04 3286 (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))) <-> (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)))) & 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__iff__nth) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3287 (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]. 5.99/6.04 3288 (all V_c_2 all V_b_2 all V_a_2 all T_c (class_Fields_Ofield__inverse__zero(T_c) -> ((c_Groups_Ozero__class_Ozero(T_c) = V_c_2 -> V_a_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),V_a_2),V_c_2)) <-> hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_b_2),V_c_2) = V_a_2))) # label(fact_eq__divide__eq) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3289 (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]. 5.99/6.04 3290 (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_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) = 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)))) # label(fact_setsum__right__distrib) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3291 (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]. 5.99/6.04 3292 (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]. 5.99/6.04 3293 (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]. 5.99/6.04 3294 (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]. 5.99/6.04 3295 (all V_xs all V_x all T_a (class_Groups_Omonoid__add(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_x),hAPP(c_List_Omonoid__add__class_Olistsum(T_a),V_xs)) = hAPP(c_List_Omonoid__add__class_Olistsum(T_a),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)))) # label(fact_listsum__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3296 (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_minus__apply) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3297 (all V_k V_k = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_k),c_Int_OPls)) # label(fact_add__Pls__right) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3298 (all V_n 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)),hAPP(c_Nat_OSuc,V_n)) = hAPP(hAPP(c_List_Onth(T_a),V_xs),V_n)) # label(fact_nth__Cons__Suc) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3299 (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]. 5.99/6.04 3300 (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]. 5.99/6.04 3301 (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]. 5.99/6.04 3302 (all T_a (class_Rings_Ozero__neq__one(T_a) -> c_Groups_Ozero__class_Ozero(T_a) != c_Groups_Oone__class_Oone(T_a))) # label(fact_one__neq__zero) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3303 (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]. 5.99/6.04 3304 (all V_x all T_a all V_n all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),V_n)) -> hAPP(hAPP(c_List_Onth(T_a),c_List_Oreplicate(T_a,V_n,V_x)),V_i) = V_x)) # label(fact_nth__replicate) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3305 (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]. 5.99/6.04 3306 (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]. 5.99/6.04 3307 (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_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__le) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3308 (all V_v_2 all V_f_2 all V_a_2 all T_c hAPP(hAPP(c_HOL_OLet(tc_Int_Oint,T_c),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_Opred(V_v_2))),hAPP(hAPP(c_COMBS(tc_Int_Oint,T_c,T_c),hAPP(hAPP(c_COMBC(tc_Int_Oint,T_c,tc_fun(T_c,T_c)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(T_c,tc_fun(T_c,T_c)),tc_Int_Oint),c_If(T_c)),c_Nat__Numeral_Oneg)),V_a_2)),hAPP(hAPP(c_COMBB(tc_Nat_Onat,T_c,tc_Int_Oint),V_f_2),c_Int_Onat))) = c_Nat_Onat_Onat__case(T_c,V_a_2,V_f_2,hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2))) # label(fact_nat__case__number__of) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3309 (all V_n_2 all V_x_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),V_x_2)),V_n_2))) <-> V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != V_x_2)) # label(fact_zero__less__zpower__abs__iff) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3310 (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]. 5.99/6.04 3311 (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]. 5.99/6.04 3312 (all V_y_2 all V_x_2 all V_f_2 all V_A_2 all T_d all T_c (hAPP(c_Hilbert__Choice_Oinv__into(T_c,T_d,V_A_2,V_f_2),V_y_2) = hAPP(c_Hilbert__Choice_Oinv__into(T_c,T_d,V_A_2,V_f_2),V_x_2) -> (hBOOL(hAPP(hAPP(c_member(T_d),V_x_2),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2))) -> (hBOOL(hAPP(hAPP(c_member(T_d),V_y_2),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_A_2))) -> V_x_2 = V_y_2)))) # label(fact_inv__into__injective) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3313 (all V_x all 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)) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_y),V_x)) -> c_Nat__Transfer_Otsub(V_x,V_y) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_x),V_y)))) # label(fact_tsub__def) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3314 (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]. 5.99/6.04 3315 (all V_u_2 all V_l_2 hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_l_2),hAPP(c_Nat_OSuc,V_u_2)) = c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_l_2,V_u_2)) # label(fact_atLeastLessThanSuc__atLeastAtMost) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3316 (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]. 5.99/6.04 3317 (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]. 5.99/6.04 3318 (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]. 5.99/6.04 3319 (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]. 5.99/6.04 3320 (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]. 5.99/6.04 3321 (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]. 5.99/6.04 3322 (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]. 5.99/6.04 3323 (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]. 5.99/6.04 3324 (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]. 5.99/6.04 3325 (all V_a all V_b all T_a (class_Rings_Odivision__ring(T_a) -> (V_b != c_Groups_Ozero__class_Ozero(T_a) -> hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_b)) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_b)))) # label(fact_nonzero__minus__divide__divide) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3326 (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)) -> 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]. 5.99/6.04 3327 (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]. 5.99/6.04 3328 (all T_c hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) # label(fact_Compl__UNIV__eq) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3329 (all V_b all V_a all T_a (class_Rings_Ono__zero__divisors(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) != V_a -> (c_Groups_Ozero__class_Ozero(T_a) != V_b -> c_Groups_Ozero__class_Ozero(T_a) != hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b))))) # label(fact_no__zero__divisors) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3330 (all V_B_2 all V_A_2 all T_c (class_Lattices_Olattice(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (V_B_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_A_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_c),c_Big__Operators_Olattice__class_OSup__fin(T_c,V_B_2)),c_Big__Operators_Olattice__class_OSup__fin(T_c,V_A_2)) = c_Big__Operators_Olattice__class_OSup__fin(T_c,V_A_2)))))) # label(fact_Sup__fin_Osubset__idem) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3331 (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]. 5.99/6.04 3332 (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]. 5.99/6.04 3333 (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]. 5.99/6.04 3334 (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]. 5.99/6.04 3335 (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 (V_a_2 = hAPP(hAPP(c_List_Olist_OCons(T_c),B_x),B_xs) & c_List_Olinorder__class_Osorted(T_c,B_xs) & (all B_xa (hBOOL(hAPP(hAPP(c_member(T_c),B_xa),hAPP(c_List_Oset(T_c),B_xs))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),B_x),B_xa)))))) | c_List_Olist_ONil(T_c) = V_a_2))) # label(fact_sorted_Osimps) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3336 (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]. 5.99/6.04 3337 (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),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,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(c_Finite__Set_Ofinite(T_d),hAPP(V_B_2,B_x)))))))) # label(fact_finite__UN) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3338 (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]. 5.99/6.04 3339 (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(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__idem_Ounion__idem) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3340 (all V_com2_H_2 all V_com1_H_2 all V_com2_2 all V_com1_2 (c_Com_Ocom_OSemi(V_com1_2,V_com2_2) = c_Com_Ocom_OSemi(V_com1_H_2,V_com2_H_2) <-> V_com1_H_2 = V_com1_2 & V_com2_H_2 = V_com2_2)) # label(fact_com_Osimps_I3_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3341 (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_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)))) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b)))) # label(fact_div__neg__neg) # label(axiom) # label(non_clause). [assumption]. 5.99/6.04 3342 (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]. 5.99/6.04 3343 (all V_i all V_j all V_k (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_k),V_j)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_j),V_k)),V_i) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_j),V_i)),V_k))) # label(fact_add__diff__assoc2) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3344 (all V_xs_2 all V_f_2 all T_d all T_c hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_xs_2)) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_d)),V_xs_2)) # label(fact_length__map) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3345 (all V_b all V_a all T_a (class_Rings_Olinordered__idom(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Osgn__class_Osgn(T_a,V_a)),c_Groups_Osgn__class_Osgn(T_a,V_b)) = c_Groups_Osgn__class_Osgn(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)))) # label(fact_sgn__times) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3346 (all V_f_2 all V_r_2 all T_d all V_y_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_y_2)),hAPP(hAPP(c_Relation_Oinv__image(T_d,T_c),V_r_2),V_f_2))) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(T_d,T_d)),hAPP(hAPP(c_Product__Type_OPair(T_d,T_d),hAPP(V_f_2,V_x_2)),hAPP(V_f_2,V_y_2))),V_r_2)))) # label(fact_in__inv__image) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3347 (all V_v_2 all V_w_2 all T_c (class_Int_Onumber__ring(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w_2))) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_v_2)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(V_w_2))) = hAPP(hAPP(c_HOL_OLet(T_c,T_c),hAPP(hAPP(c_Power_Opower__class_Opower(T_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_v_2)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_w_2))),hAPP(hAPP(c_COMBS(T_c,T_c,T_c),hAPP(hAPP(c_COMBB(T_c,tc_fun(T_c,T_c),T_c),c_Groups_Otimes__class_Otimes(T_c)),hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_v_2)))),c_COMBI(T_c)))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w_2))) -> c_Groups_Oone__class_Oone(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_OBit1(V_w_2)))))) # label(fact_power__number__of__odd__number__of) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3348 (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(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_OatLeastAtMost(T_c,V_l_2,V_u_2)))) # label(fact_ivl__disj__un_I6_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3349 (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]. 5.99/6.05 3350 (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]. 5.99/6.05 3351 (all V_i_2 all V_inc_2 all T_c (class_Rings_Osemiring__1(T_c) -> V_i_2 = c_Nat_Osemiring__1__class_Oof__nat__aux(T_c,V_inc_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_i_2))) # label(fact_of__nat__aux_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3352 (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]. 5.99/6.05 3353 (all V_lazy__sequence_H all V_a_H all T_a c_Lazy__Sequence_Olazy__sequence_OInsert(T_a,V_a_H,V_lazy__sequence_H) != c_Lazy__Sequence_Olazy__sequence_OEmpty(T_a)) # label(fact_lazy__sequence_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3354 (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]. 5.99/6.05 3355 (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_Omin(T_c),V_x_2),V_y_2)),V_z_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_y_2),V_z_2)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_x_2),V_z_2))))) # label(fact_min__less__iff__disj) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3356 (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]. 5.99/6.05 3357 (all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OMin),c_Int_OBit0(V_k_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OMin),V_k_2)))) # label(fact_rel__simps_I25_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3358 (all V_n_2 all V_m_2 hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),V_m_2),hAPP(c_Nat_OSuc,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(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)) # label(fact_max__Suc2) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3359 (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 (c_Com_Ocom_OLocal(V_loc_H_2,V_fun_H_2,V_com_H_2) = c_Com_Ocom_OLocal(V_loc_2,V_fun_2,V_com_2) <-> V_com_2 = V_com_H_2 & V_fun_H_2 = V_fun_2 & V_loc_H_2 = V_loc_2)) # label(fact_com_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3360 (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]. 5.99/6.05 3361 (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]. 5.99/6.05 3362 (all V_B_2 all V_A_2 all T_d all V_b_2 all T_c ((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))))) <-> 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))))) # label(fact_INT__iff) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3363 (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]. 5.99/6.05 3364 (all V_u_2 all V_l_2 all V_less_2 all V_i_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_i_2),c_SetInterval_Oord_OgreaterThanLessThan(T_c,V_less_2,V_l_2,V_u_2))) <-> hBOOL(hAPP(hAPP(V_less_2,V_i_2),V_u_2)) & hBOOL(hAPP(hAPP(V_less_2,V_l_2),V_i_2)))) # label(fact_ord_OgreaterThanLessThan__iff) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3365 (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]. 5.99/6.05 3366 (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]. 5.99/6.05 3367 (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]. 5.99/6.05 3368 (all V_x all T_a all V_n (V_n != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> V_x = c_List_Olast(T_a,c_List_Oreplicate(T_a,V_n,V_x)))) # label(fact_last__replicate) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3369 (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]. 5.99/6.05 3370 (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]. 5.99/6.05 3371 (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_g_2,B_x) = hAPP(V_h_2,B_x))) -> hAPP(hAPP(V_F_2,V_h_2),V_A_2) = hAPP(hAPP(V_F_2,V_g_2),V_A_2))))) # label(fact_folding__image_Ocong) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3372 (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]. 5.99/6.05 3373 (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]. 5.99/6.05 3374 (all V_b all V_a all T_a (class_Lattices_Osemilattice__inf(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_a),V_b) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_b),V_a))) # label(fact_inf_Ocommute) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3375 (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]. 5.99/6.05 3376 (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]. 5.99/6.05 3377 (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]. 5.99/6.05 3378 (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,hAPP(c_Fun_Ocomp(T_d,T_c,T_d,V_g_2),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)))))) # label(fact_folding__image_Oreindex) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3379 (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]. 5.99/6.05 3380 (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_OBit1(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)))))) # label(fact_number__of1) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3381 (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]. 5.99/6.05 3382 (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),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))) & (-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))) -> (c_Groups_Ozero__class_Ozero(tc_Int_Oint) != V_b_1 -> c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a_1,V_b_1) = V_y)))) # label(fact_divmod__int__rel__mod__eq) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3383 (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]. 5.99/6.05 3384 (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]. 5.99/6.05 3385 (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]. 5.99/6.05 3386 (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]. 5.99/6.05 3387 (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]. 5.99/6.05 3388 (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]. 5.99/6.05 3389 (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 | V_b = c_Groups_Ozero__class_Ozero(T_a)))) # label(fact_divisors__zero) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3390 (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]. 5.99/6.05 3391 (all V_s_2 all V_r_2 all V_I_2 all T_f all T_d all T_e all T_c hAPP(hAPP(c_Relation_Orel__comp(T_c,T_e,T_d),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_f,tc_fun(tc_prod(T_c,T_e),tc_HOL_Obool)),V_I_2),V_r_2)),V_s_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_f,tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),V_I_2),hAPP(hAPP(c_COMBC(T_f,tc_fun(tc_prod(T_e,T_d),tc_HOL_Obool),tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_prod(T_c,T_e),tc_HOL_Obool),tc_fun(tc_fun(tc_prod(T_e,T_d),tc_HOL_Obool),tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),T_f),c_Relation_Orel__comp(T_c,T_e,T_d)),V_r_2)),V_s_2))) # label(fact_rel__comp__UNION__distrib2) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3392 (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_b) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_c) -> V_b = V_c))) # label(fact_add__imp__eq) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3393 (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]. 5.99/6.05 3394 (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]. 5.99/6.05 3395 (all V_x_2 all V_A_2 all V_F_2 all V_g_2 all V_z_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofolding__image__simple__idem(T_c,T_d,V_f_2,V_z_2,V_g_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),V_A_2)) -> hAPP(V_F_2,hAPP(hAPP(c_Set_Oinsert(T_d),V_x_2),V_A_2)) = hAPP(hAPP(V_f_2,hAPP(V_g_2,V_x_2)),hAPP(V_F_2,V_A_2))))) # label(fact_folding__image__simple__idem_Oinsert__idem) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3396 (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]. 5.99/6.05 3397 (all V_r all V_q all V_b all V_a (hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a,V_b),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_q),V_r))) -> (c_Groups_Ozero__class_Ozero(tc_Int_Oint) != V_b -> 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]. 5.99/6.05 3398 (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,hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_l_2))) = c_List_OtakeWhile(T_c,V_Pa_2,V_xs_2))) # label(fact_takeWhile__tail) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3399 (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]. 5.99/6.05 3400 (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]. 5.99/6.05 3401 (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__eq(tc_Int_Oint),c_Int_OBit1(V_k_2)),c_Int_OPls)))) # label(fact_rel__simps_I29_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3402 (all V_i all V_j (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_j),V_i)) -> c_List_Oupto(V_i,V_j) = c_List_Olist_ONil(tc_Int_Oint))) # label(fact_upto__empty) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3403 (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]. 5.99/6.05 3404 (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__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(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__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]. 5.99/6.05 3405 (all V_z_2 all V_w_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_w_2),V_z_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_w_2),c_Groups_Oone__class_Oone(tc_Int_Oint))),V_z_2)))) # label(fact_add1__zle__eq) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3406 (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]. 5.99/6.05 3407 (all V_v_2 all V_u_2 all V_y_2 all V_x_2 (hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_x_2),V_v_2) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_u_2),V_y_2) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat))),hAPP(hAPP(c_Product__Type_OPair(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_x_2),V_y_2)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_u_2),V_v_2))),c_Int_Ointrel)))) # label(fact_intrel__iff) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3408 (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]. 5.99/6.05 3409 (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]. 5.99/6.05 3410 (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]. 5.99/6.05 3411 (all V_y_2 all V_x_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_y_2)) -> (V_x_2 = V_y_2 <-> hAPP(c_Int_Onat,V_x_2) = hAPP(c_Int_Onat,V_y_2))))) # label(fact_transfer__nat__int__relations_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3412 (all V_l_2 all V_k_2 all V_f_2 all T_c (class_Groups_Ocomm__monoid__add(T_c) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,T_c),V_f_2),hAPP(c_List_Oset(tc_Int_Oint),c_List_Oupto(V_k_2,V_l_2))) = hAPP(c_List_Omonoid__add__class_Olistsum(T_c),hAPP(hAPP(c_List_Omap(tc_Int_Oint,T_c),V_f_2),c_List_Oupto(V_k_2,V_l_2))))) # label(fact_interv__listsum__conv__setsum__set__int) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3413 (all V_l all V_k (c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_k),V_l) != c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> c_Divides_Odiv__class_Omod(tc_Int_Oint,V_k,V_l) != c_Groups_Ozero__class_Ozero(tc_Int_Oint))) # label(fact_zmod__zminus1__not__zero) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3414 (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]. 5.99/6.05 3415 (all V_xq_2 all V_s_2 all T_c c_Option_Ooption_Ooption__case(tc_Nat_Onat,tc_prod(T_c,tc_Lazy__Sequence_Olazy__sequence(T_c)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat),hAPP(c_Product__Type_Oprod_Oprod__case(T_c,tc_Lazy__Sequence_Olazy__sequence(T_c),tc_Nat_Onat),hAPP(hAPP(c_COMBC(T_c,tc_Nat_Onat,tc_fun(tc_Lazy__Sequence_Olazy__sequence(T_c),tc_Nat_Onat)),hAPP(hAPP(c_COMBB(tc_fun(tc_Lazy__Sequence_Olazy__sequence(T_c),tc_fun(tc_Nat_Onat,tc_Nat_Onat)),tc_fun(tc_Nat_Onat,tc_fun(tc_Lazy__Sequence_Olazy__sequence(T_c),tc_Nat_Onat)),T_c),c_COMBC(tc_Lazy__Sequence_Olazy__sequence(T_c),tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_COMBB(tc_fun(tc_Lazy__Sequence_Olazy__sequence(T_c),tc_Nat_Onat),tc_fun(tc_Lazy__Sequence_Olazy__sequence(T_c),tc_fun(tc_Nat_Onat,tc_Nat_Onat)),T_c),hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat),tc_Lazy__Sequence_Olazy__sequence(T_c)),c_Groups_Oplus__class_Oplus(tc_Nat_Onat))),hAPP(hAPP(c_COMBC(T_c,tc_fun(tc_Lazy__Sequence_Olazy__sequence(T_c),tc_Nat_Onat),tc_fun(tc_Lazy__Sequence_Olazy__sequence(T_c),tc_Nat_Onat)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_Nat_Onat),tc_fun(tc_fun(tc_Lazy__Sequence_Olazy__sequence(T_c),tc_Nat_Onat),tc_fun(tc_Lazy__Sequence_Olazy__sequence(T_c),tc_Nat_Onat)),T_c),c_COMBB(tc_Nat_Onat,tc_Nat_Onat,tc_Lazy__Sequence_Olazy__sequence(T_c))),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat),T_c),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),V_s_2))),c_Lazy__Sequence_Olazy__sequence_Olazy__sequence__size(T_c,V_s_2))))),c_Groups_Oone__class_Oone(tc_Nat_Onat))),hAPP(c_Lazy__Sequence_Oyield(T_c),V_xq_2)) = hAPP(c_Lazy__Sequence_Olazy__sequence_Olazy__sequence__size(T_c,V_s_2),V_xq_2)) # label(fact_lazy__sequence__size__code) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3416 (all V_D_2 all V_C_2 all V_B_2 all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Obij__betw(T_c,T_d,V_f_2,V_A_2,V_B_2) -> (c_Fun_Obij__betw(T_c,T_d,V_f_2,V_C_2,V_D_2) -> (hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_d,tc_HOL_Obool)),V_B_2),V_D_2) = c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool)) -> c_Fun_Obij__betw(T_c,T_d,V_f_2,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_C_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_d,tc_HOL_Obool)),V_B_2),V_D_2)))))) # label(fact_bij__betw__combine) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3417 (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) = 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))) -> (hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v_H))) -> c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H)) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)) & (-hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v_H))) -> hAPP(c_Int_Onat,c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v_H))) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H)))))) # label(fact_mod__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3418 (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]. 5.99/6.05 3419 (all V_x all V_z all V_y all T_a (class_Lattices_Odistrib__lattice(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_y),V_x)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_z),V_x)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_y),V_z)),V_x))) # label(fact_sup__inf__distrib2) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3420 (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]. 5.99/6.05 3421 (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_number__of__mult) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3422 (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]. 5.99/6.05 3423 (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(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) = 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)))))) # label(fact_setprod__Sigma) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3424 (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]. 5.99/6.05 3425 (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]. 5.99/6.05 3426 (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]. 5.99/6.05 3427 (all V_x_2 all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,V_A_2) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> hAPP(c_Hilbert__Choice_Oinv__into(T_c,T_d,V_A_2,V_f_2),hAPP(V_f_2,V_x_2)) = V_x_2))) # label(fact_inv__into__f__f) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3428 (all V_B_2 all V_A_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_A_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) # label(fact_Un__Diff__cancel) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3429 (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)))) & (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)))))) # label(fact_refl__on__def_H) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3430 (all V_x all V_ys all V_xs all T_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_ys))) -> V_xs != hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_ys))) # label(fact_impossible__Cons) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3431 (all V_xs_2 all V_Pa_2 all T_c hAPP(c_List_Oset(T_c),hAPP(c_List_Ofilter(T_c,V_Pa_2),V_xs_2)) = hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBS(T_c,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_c),c_fconj),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),hAPP(c_List_Oset(T_c),V_xs_2)))),V_Pa_2))) # label(fact_set__filter) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3432 (all V_w_2 all V_z_2 (V_z_2 != V_w_2 & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_z_2),V_w_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_z_2),V_w_2)))) # label(fact_zless__le) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3433 (all V_f_2 all V_xs_2 all V_a_2 all T_c all T_d (class_Orderings_Olinorder(T_d) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),hAPP(c_List_Oset(T_c),V_xs_2))) -> (c_List_Olinorder__class_Osorted(T_d,hAPP(hAPP(c_List_Omap(T_c,T_d),V_f_2),V_xs_2)) -> (V_a_2 = hAPP(c_List_Ohd(T_c),hAPP(c_List_Ofilter(T_c,hAPP(hAPP(c_COMBB(T_d,tc_HOL_Obool,T_c),hAPP(c_fequal,hAPP(V_f_2,V_a_2))),V_f_2)),V_xs_2)) -> V_xs_2 = hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_c,T_d,V_f_2),V_a_2),c_List_Oremove1(T_c,V_a_2,V_xs_2))))))) # label(fact_insort__key__remove1) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3434 (all V_xs_2 all V_Pa_2 all T_c (c_List_Olist__all(T_c,V_Pa_2,hAPP(c_List_Orev(T_c),V_xs_2)) <-> c_List_Olist__all(T_c,V_Pa_2,V_xs_2))) # label(fact_list__all__rev) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3435 (all V_d_2 all V_b_2 all V_c_2 all V_e_2 all V_a_2 all T_c (class_Rings_Oordered__ring(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_e_2)),V_c_2)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_b_2),V_e_2)),V_d_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_c_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_c),V_b_2),V_a_2)),V_e_2)),V_d_2)))))) # label(fact_less__add__iff2) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3436 (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]. 5.99/6.05 3437 (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]. 5.99/6.05 3438 (all V_m hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_mult__0__right) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3439 (all V_b all V_a hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_a)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_b)) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_a),V_b))) # label(fact_zdiv__int) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3440 (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_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__eq(T_c),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),V_c_2))))))) # label(fact_neg__le__divide__eq) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3441 (all V_v_2 all V_n_2 hAPP(hAPP(c_HOL_OLet(tc_Int_Oint,tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_Opred(V_v_2))),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat)),tc_Int_Oint),c_If(tc_Nat_Onat)),c_Nat__Numeral_Oneg)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Nat_Onat,tc_Int_Oint),c_Nat_OSuc),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Nat_Onat,tc_Int_Oint),hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),V_n_2)),c_Int_Onat)))) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_n_2)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2))) # label(fact_min__number__of__Suc) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3442 (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]. 5.99/6.05 3443 (all V_x_2 all V_A_2 all T_c (class_Lattices_Olattice(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_c,c_Lattices_Osemilattice__inf__class_Oinf(T_c)),V_x_2),V_A_2) = c_Big__Operators_Olattice__class_OInf__fin(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)))))) # label(fact_Inf__fin_Oeq__fold_H) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3444 (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_Omax(T_c)),V_b_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),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))))) # label(fact_min__max_Ofold__sup__insert) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3445 (all V_z all V_w all T_a (class_Rings_Oring__1(T_a) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(c_Int_Oring__1__class_Oof__int(T_a),V_w)),hAPP(c_Int_Oring__1__class_Oof__int(T_a),V_z)) = hAPP(c_Int_Oring__1__class_Oof__int(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_w),V_z)))) # label(fact_of__int__diff) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3446 (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,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)))))) = hAPP(V_F_2,V_A_2))))) # label(fact_folding__image__simple_Oremove) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3447 (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]. 5.99/6.05 3448 (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]. 5.99/6.05 3449 (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) = 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))))) # label(fact_Ints__double__eq__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3450 (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_2,hAPP(V_g_2,V_x_2)) = hAPP(V_f_H_2,hAPP(V_g_H_2,V_x_H_2)) -> hAPP(hAPP(c_Fun_Ocomp(T_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]. 5.99/6.05 3451 (all T_d all T_c all V_x_2 all V_f_2 all V_b_2 (hAPP(V_f_2,V_x_2) = V_b_2 -> hBOOL(hAPP(hAPP(c_member(T_c),V_b_2),hAPP(c_Set_Oimage(T_d,T_c,V_f_2),c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool))))))) # label(fact_range__eqI) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3452 (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]. 5.99/6.05 3453 (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))) <-> V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_x_2)))) # label(fact_nat__zero__less__power__iff) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3454 (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]. 5.99/6.05 3455 (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]. 5.99/6.05 3456 (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]. 5.99/6.05 3457 (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]. 5.99/6.05 3458 (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]. 5.99/6.05 3459 (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]. 5.99/6.05 3460 (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]. 5.99/6.05 3461 (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]. 5.99/6.05 3462 (all T_a all V_ys all V_xs (V_ys = V_xs -> hAPP(hAPP(c_List_Oappend(T_a),c_List_Olist_ONil(T_a)),V_ys) = V_xs)) # label(fact_eq__Nil__appendI) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3463 (all V_b_2 all V_A_2 all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> (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)) <-> (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))))))) # label(fact_Sup__le__iff) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3464 (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]. 5.99/6.05 3465 (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(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))) <-> 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(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_lexord__cons__cons) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3466 (all V_k_2 all V_f_2 all V_n_2 ((all B_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_i),V_n_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(V_f_2,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B_i),c_Groups_Oone__class_Oone(tc_Nat_Onat)))),hAPP(V_f_2,B_i)))),c_Groups_Oone__class_Oone(tc_Int_Oint))))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(V_f_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),V_k_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k_2),hAPP(V_f_2,V_n_2))) -> (exists B_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B_i),V_n_2)) & V_k_2 = hAPP(V_f_2,B_i))))))) # label(fact_nat0__intermed__int__val) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3467 (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]. 5.99/6.05 3468 (all V_w_2 all V_z_2 hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_z_2),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(V_w_2))) = hAPP(hAPP(c_HOL_OLet(tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),V_z_2),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_w_2))),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_Int_Oint,tc_Int_Oint),c_Groups_Otimes__class_Otimes(tc_Int_Oint)),c_COMBI(tc_Int_Oint)))) # label(fact_zpower__number__of__even) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3469 (all V_M_2 all V_B_2 all V_A_2 all T_c all T_d hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(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)) = 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)) # label(fact_UN__Un) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3470 (all V_n all V_m ((-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_m)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_n))) -> c_List_Oupto(hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_m),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_n)) = c_List_Olist_ONil(tc_Int_Oint)) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_m)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_n))) -> c_List_Oupto(hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_m),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_n)) = hAPP(hAPP(c_List_Olist_OCons(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_m)),c_List_Oupto(hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_m)),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_n)))))) # label(fact_upto__rec__number__of) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3471 (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]. 5.99/6.05 3472 (all V_x_2 all T_c (class_Groups_Oordered__ab__group__add(T_c) -> hAPP(c_Set_Oimage(T_c,T_c,c_Groups_Ouminus__class_Ouminus(T_c)),hAPP(c_SetInterval_Oord__class_OlessThan(T_c),V_x_2)) = hAPP(c_SetInterval_Oord__class_OgreaterThan(T_c),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_x_2)))) # label(fact_image__uminus__lessThan) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3473 (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]. 5.99/6.05 3474 (all V_k hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_k) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Int_OMin),V_k)) # label(fact_mult__Min) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3475 (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]. 5.99/6.05 3476 (all V_R_2 all T_c c_Relation_OId(T_c) = hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_c,T_c),tc_HOL_Obool),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_R_2)) # label(fact_relpow_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3477 (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]. 5.99/6.05 3478 (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]. 5.99/6.05 3479 (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]. 5.99/6.05 3480 (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]. 5.99/6.05 3481 (all V_n all V_m all V_k ((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))) & (V_k = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_m)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_n)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_nat__mult__div__cancel__disj) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3482 (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(c_Nat__Numeral_Oneg,V_Z_2)))) # label(fact_neg__def) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3483 (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]. 5.99/6.05 3484 (all V_x all T_a (class_Lattices_Obounded__lattice__bot(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),c_Orderings_Obot__class_Obot(T_a)),V_x) = V_x)) # label(fact_sup__bot__left) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3485 (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]. 5.99/6.05 3486 (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]. 5.99/6.05 3487 (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)) & (V_ys = c_List_Olist_ONil(T_a) -> c_List_Olast(T_a,V_xs) = c_List_Olast(T_a,hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys))))) # label(fact_last__append) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3488 (all T_d all T_c c_Orderings_Otop__class_Otop(tc_fun(tc_sum(T_c,T_d),tc_HOL_Obool)) = c_Sum__Type_OPlus(T_c,T_d,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)),c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool)))) # label(fact_UNIV__Plus__UNIV) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3489 (all V_w all V_v all T_a (class_Int_Onumber__ring(T_a) -> hAPP(c_Int_Onumber__class_Onumber__of(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_v),V_w)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w)))) # label(fact_number__of__add) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3490 (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]. 5.99/6.05 3491 (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]. 5.99/6.05 3492 (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]. 5.99/6.05 3493 (all V_r_2 all V_xs_2 all V_ys_2 all V_y_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),hAPP(hAPP(c_List_Olist_OCons(T_c),V_y_2),V_ys_2)),V_xs_2)),c_List_Olistrel(T_c,V_r_2))) -> -(all B_y all B_ys (V_xs_2 = hAPP(hAPP(c_List_Olist_OCons(T_c),B_y),B_ys) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_y_2),B_y)),V_r_2)) -> -hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_ys_2),B_ys)),c_List_Olistrel(T_c,V_r_2)))))))) # label(fact_listrel__Cons1) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3494 (all V_r_2 all T_c (c_Wellfounded_OwfP(T_c,hAPP(hAPP(c_COMBC(T_c,tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),T_c),c_COMBC(T_c,tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_prod(T_c,T_c)),tc_fun(T_c,tc_fun(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_HOL_Obool)),T_c),hAPP(c_COMBB(tc_prod(T_c,T_c),tc_fun(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_HOL_Obool),T_c),c_member(tc_prod(T_c,T_c)))),c_Product__Type_OPair(T_c,T_c)))),V_r_2)) <-> c_Wellfounded_Owf(T_c,V_r_2))) # label(fact_wfP__wf__eq) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3495 (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]. 5.99/6.05 3496 (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 -> (V_ys = hAPP(hAPP(c_List_Oappend(T_a),V_xs1),V_us) -> 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]. 5.99/6.05 3497 (all V_w (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)))) -> hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w))) = c_Divides_OnegDivAlg(c_Groups_Oone__class_Oone(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)))) -> 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))))) # label(fact_negDivAlg__eqn__1__number__of) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3498 (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]. 5.99/6.05 3499 (all T_a c_List_Otranspose(T_a,c_List_Olist_ONil(tc_List_Olist(T_a))) = c_List_Olist_ONil(tc_List_Olist(T_a))) # label(fact_transpose_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3500 (all V_u_2 all V_l_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_l_2),V_u_2)) -> hAPP(c_SetInterval_Oord__class_OlessThan(T_c),V_u_2) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatMost(T_c),V_l_2)),c_SetInterval_Oord__class_OgreaterThanLessThan(T_c,V_l_2,V_u_2))))) # label(fact_ivl__disj__un_I7_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3501 (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]. 5.99/6.05 3502 (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]. 5.99/6.05 3503 (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]. 5.99/6.05 3504 (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]. 5.99/6.05 3505 (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]. 5.99/6.05 3506 (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_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_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_Olinorder__class_OMin(T_c,V_A_2)),c_Big__Operators_Olinorder__class_OMin(T_c,V_B_2))))))))) # label(fact_Min_Ounion__disjoint) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3507 (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]. 5.99/6.05 3508 (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]. 5.99/6.05 3509 (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]. 5.99/6.05 3510 (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]. 5.99/6.05 3511 (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]. 5.99/6.05 3512 (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_i_2),V_k_2)) <-> hBOOL(hAPP(hAPP(c_member(T_c),V_i_2),hAPP(c_SetInterval_Oord__class_OatMost(T_c),V_k_2)))))) # label(fact_atMost__iff) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3513 (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(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) = 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)))) # label(fact_INFI__apply) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3514 (all V_Pa_2 all V_n_2 ((exists B_m (hBOOL(hAPP(V_Pa_2,B_m)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_m),V_n_2)))) <-> (exists B_x (hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B_x),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n_2))) & hBOOL(hAPP(V_Pa_2,B_x)))))) # label(fact_ex__nat__less__eq) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3515 (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 <-> hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2) = V_x_2))) # label(fact_number__of__reorient) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3516 (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]. 5.99/6.05 3517 (all V_z3 all V_z2 all V_z1 hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_z1),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_z2),V_z3)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_z1),V_z2)),V_z3)) # label(fact_zmult__assoc) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3518 (all V_y all V_x (V_y = V_x | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_y),V_x)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_x),V_y)))) # label(fact_zless__linear) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3519 (all V_x_2 all V_xs_2 all T_c all V_i_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i_2),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2))) -> (hAPP(hAPP(c_List_Onth(T_c),V_xs_2),V_i_2) = V_x_2 <-> V_xs_2 = hAPP(hAPP(c_List_Olist__update(T_c,V_xs_2),V_i_2),V_x_2)))) # label(fact_list__update__same__conv) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3520 (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]. 5.99/6.05 3521 (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]. 5.99/6.05 3522 (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]. 5.99/6.05 3523 (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]. 5.99/6.05 3524 (all V_z all V_y all V_x all T_a (class_Orderings_Olinorder(T_a) -> hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_y),V_z)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),V_y)),V_z))) # label(fact_min__max_Oinf__assoc) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3525 (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]. 5.99/6.05 3526 (all V_c_2 all V_b_2 all V_a_2 (hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),V_c_2),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool))) = c_SetInterval_Oord_OatLeastAtMost(tc_Nat_Onat,c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_a_2,V_b_2) <-> V_c_2 = V_b_2 & V_a_2 = V_b_2)) # label(fact_dvd_OatLeastAtMost__singleton__iff) # label(axiom) # label(non_clause). [assumption]. 5.99/6.05 3527 (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]. 5.99/6.05 3528 (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]. 5.99/6.06 3529 (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_Ozero__class_Ozero(tc_Nat_Onat) = c_Groups_Oone__class_Oone(tc_Nat_Onat)) -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(c_Finite__Set_Ocard(T_c),V_A_2)))) # label(fact_card_Oneutral) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3530 (all V_B_2 all V_A_2 all T_d all T_c hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_d),c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool))),V_B_2))) # label(fact_Compl__INT) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3531 (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]. 5.99/6.06 3532 (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]. 5.99/6.06 3533 (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]. 5.99/6.06 3534 (all V_l c_Int_OBit1(hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),c_Int_OMin),V_l)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),c_Int_OPls),c_Int_OBit1(V_l))) # label(fact_diff__bin__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3535 (all V_a_2 all T_c c_HOL_OThe(T_c,hAPP(hAPP(c_COMBC(T_c,T_c,tc_HOL_Obool),c_fequal),V_a_2)) = V_a_2) # label(fact_the__eq__trivial) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3536 (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]. 5.99/6.06 3537 (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]. 5.99/6.06 3538 (all V_n_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),hAPP(c_Nat_OSuc,V_n_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m_2),V_n_2)))) # label(fact_less__Suc__eq__le) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3539 (all V_N_2 all V_h_2 all T_c (class_Orderings_Olinorder(T_c) -> ((all B_x all B_y hAPP(V_h_2,hAPP(hAPP(c_Orderings_Oord__class_Omax(T_c),B_x),B_y)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(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,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]. 5.99/6.06 3540 (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]. 5.99/6.06 3541 (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]. 5.99/6.06 3542 (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]. 5.99/6.06 3543 (all V_B_2 all V_A_2 all V_a_2 all T_c all T_d 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)) = 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)) # label(fact_INT__insert) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3544 (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]. 5.99/6.06 3545 (all V_xs_2 all V_Pa_2 all T_c hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),hAPP(c_List_Ofilter(T_c,V_Pa_2),V_xs_2))),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),hAPP(c_List_Ofilter(T_c,hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_c),c_fNot),V_Pa_2)),V_xs_2))) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2)) # label(fact_sum__length__filter__compl) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3546 (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]. 5.99/6.06 3547 (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]. 5.99/6.06 3548 (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]. 5.99/6.06 3549 (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(c_List_Orev(T_a),V_xs)),V_n) = 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))))) # label(fact_rev__nth) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3550 (all V_xs all V_x all T_a all V_n ((V_n = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> hAPP(hAPP(c_List_Odrop(T_a),V_n),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)) = hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)) & (V_n != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> hAPP(hAPP(c_List_Odrop(T_a),V_n),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)) = hAPP(hAPP(c_List_Odrop(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),c_Groups_Oone__class_Oone(tc_Nat_Onat))),V_xs)))) # label(fact_drop__Cons_H) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3551 (all V_r_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_prod(T_c,T_c)),V_r_2)) -> (c_Wellfounded_Oacyclic(T_c,V_r_2) <-> c_Wellfounded_Owf(T_c,V_r_2)))) # label(fact_wf__iff__acyclic__if__finite) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3552 (all V_xs all V_x all T_b all T_a (hAPP(c_List_Orotate1(T_a),c_List_Olist_ONil(T_a)) = c_List_Olist_ONil(T_a) & hAPP(hAPP(c_List_Oappend(T_b),V_xs),hAPP(hAPP(c_List_Olist_OCons(T_b),V_x),c_List_Olist_ONil(T_b))) = hAPP(c_List_Orotate1(T_b),hAPP(hAPP(c_List_Olist_OCons(T_b),V_x),V_xs)))) # label(fact_rotate__simps) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3553 (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]. 5.99/6.06 3554 (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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_c_2),V_a_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_c_2),V_b_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_a_2),V_b_2)))))) # label(fact_mult__le__cancel__left__pos) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3555 (all V_xs all V_x all T_a all V_v ((c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v) -> hAPP(hAPP(c_List_Odrop(T_a),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)) = hAPP(hAPP(c_List_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]. 5.99/6.06 3556 (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]. 5.99/6.06 3557 (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]. 5.99/6.06 3558 (all V_C_2 all V_B_2 all V_A_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_C_2)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_C_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)),V_C_2)) # label(fact_Un__Diff) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3559 (all V_ys_2 all V_Pa_2 all V_xs_2 all T_c ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(c_List_Oset(T_c),V_xs_2))) -> hBOOL(hAPP(V_Pa_2,B_x)))) -> c_List_OdropWhile(T_c,V_Pa_2,hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_ys_2)) = c_List_OdropWhile(T_c,V_Pa_2,V_ys_2))) # label(fact_dropWhile__append2) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3560 (all V_nat_2 all V_f2_2 all V_f1_2 all T_c hAPP(c_Nat_Onat_Onat__rec(T_c,V_f1_2,V_f2_2),hAPP(c_Nat_OSuc,V_nat_2)) = hAPP(hAPP(V_f2_2,V_nat_2),hAPP(c_Nat_Onat_Onat__rec(T_c,V_f1_2,V_f2_2),V_nat_2))) # label(fact_nat__rec__Suc) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3561 (all V_B_2 all V_A_2 all T_d all V_C_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_C_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(tc_prod(T_d,T_c),tc_HOL_Obool)),hAPP(hAPP(c_Product__Type_OSigma(T_d,T_c),V_A_2),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_d),V_C_2))),hAPP(hAPP(c_Product__Type_OSigma(T_d,T_c),V_B_2),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_d),V_C_2)))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_d,tc_HOL_Obool)),V_A_2),V_B_2))))) # label(fact_Times__subset__cancel2) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3562 (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]. 5.99/6.06 3563 (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]. 5.99/6.06 3564 (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))) <-> (all B_x (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),B_x),hAPP(c_List_Oset(tc_prod(T_c,T_c)),hAPP(hAPP(c_List_Ozip(T_c,T_c),V_xs_2),V_ys_2)))) -> hBOOL(hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_c,tc_HOL_Obool),hAPP(hAPP(c_COMBC(T_c,tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),T_c),c_COMBC(T_c,tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_prod(T_c,T_c)),tc_fun(T_c,tc_fun(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_HOL_Obool)),T_c),hAPP(c_COMBB(tc_prod(T_c,T_c),tc_fun(tc_fun(tc_prod(T_c,T_c),tc_HOL_Obool),tc_HOL_Obool),T_c),c_member(tc_prod(T_c,T_c)))),c_Product__Type_OPair(T_c,T_c)))),V_r_2)),B_x)))) & hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_ys_2))) # label(fact_listrel__iff__zip) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3565 (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]. 5.99/6.06 3566 (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]. 5.99/6.06 3567 (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]. 5.99/6.06 3568 (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]. 5.99/6.06 3569 (all V_n all V_m all V_k hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n)) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_m),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k),V_n))) # label(fact_mod__mult__distrib2) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3570 (all V_f_2 all T_c all V_m_2 all V_n_2 all V_i_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n_2),V_m_2))) -> hAPP(V_f_2,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m_2),V_i_2)) = hAPP(hAPP(c_List_Onth(T_c),hAPP(hAPP(c_List_Omap(tc_Nat_Onat,T_c),V_f_2),c_List_Oupt(V_m_2,V_n_2))),V_i_2))) # label(fact_nth__map__upt) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3571 (all V_x_2 all V_A_2 all V_F_2 all V_f_2 all T_c (c_Finite__Set_Ofolding__one(T_c,V_f_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_A_2 -> 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_Oinsert) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3572 (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]. 5.99/6.06 3573 (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]. 5.99/6.06 3574 (all V_j_2 all V_i_2 c_SetInterval_Oord__class_OgreaterThanLessThan(tc_Int_Oint,V_i_2,V_j_2) = hAPP(c_List_Oset(tc_Int_Oint),c_List_Oupto(hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_i_2),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_j_2),c_Groups_Oone__class_Oone(tc_Int_Oint))))) # label(fact_greaterThanLessThan__upto) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3575 (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]. 5.99/6.06 3576 (all V_b all V_a all T_a (class_Rings_Olinordered__idom(T_a) -> hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_b)))) # label(fact_abs__mult) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3577 (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]. 5.99/6.06 3578 (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))) -> (V_b_2 = c_Groups_Ozero__class_Ozero(tc_Int_Oint) | V_a_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))) & (-(V_a_2 = c_Groups_Ozero__class_Ozero(tc_Int_Oint) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = V_b_2) -> c_Nitpick_Onorm__frac(V_a_2,V_b_2) = hAPP(hAPP(c_HOL_OLet(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint)),hAPP(hAPP(c_Nitpick_Oint__gcd,V_a_2),V_b_2)),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint)),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint)),tc_Int_Oint),c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint)),hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a_2))),hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_b_2))))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> c_Nitpick_Onorm__frac(hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a_2),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_b_2)) = c_Nitpick_Onorm__frac(V_a_2,V_b_2)))) # label(fact_norm__frac_Osimps) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3579 (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(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)),c_Groups_Oone__class_Oone(tc_Int_Oint))),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit1(V_v))),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(V_w)))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w))) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit1(V_v))),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(V_w))) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w))))) # label(fact_zdiv__number__of__Bit1) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3580 (all V_b all V_a all T_a (class_Groups_Ogroup__add(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b) -> V_b = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)))) # label(fact_minus__unique) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3581 (all V_f_2 all T_c c_Fun_Oid(T_c) = hAPP(hAPP(c_Nat_Ocompow(T_c,T_c),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_f_2)) # label(fact_funpow_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3582 (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]. 5.99/6.06 3583 (all V_xs_2 all V_Pa_2 all T_c ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(c_List_Oset(T_c),V_xs_2))) -> hBOOL(hAPP(V_Pa_2,B_x)))) <-> c_List_Olist__all(T_c,V_Pa_2,V_xs_2))) # label(fact_list__all__iff) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3584 (all V_b all V_a hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b)),V_b) = c_Groups_Ozero__class_Ozero(tc_Int_Oint)) # label(fact_zmod__zdiv__trivial) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3585 (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]. 5.99/6.06 3586 (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]. 5.99/6.06 3587 (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]. 5.99/6.06 3588 (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]. 5.99/6.06 3589 (all V_y_2 all V_xs_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_List_Omember(T_c),V_xs_2),V_y_2)) | V_x_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]. 5.99/6.06 3590 (all T_a hAPP(hAPP(c_List_Olist_OCons(tc_List_Olist(T_a)),c_List_Olist_ONil(T_a)),c_List_Olist_ONil(tc_List_Olist(T_a))) = c_Enum_Osublists(T_a,c_List_Olist_ONil(T_a))) # label(fact_sublists_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3591 (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]. 5.99/6.06 3592 (all V_f_2 all T_c all T_d (class_Orderings_Olinorder(T_d) -> c_List_Olist_ONil(T_c) = c_List_Olinorder__class_Osort__key(T_c,T_d,V_f_2,c_List_Olist_ONil(T_c)))) # label(fact_sort__key__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3593 (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]. 5.99/6.06 3594 (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)) -> 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)))))) # label(fact_inf__Sup__absorb) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3595 (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]. 5.99/6.06 3596 (all V_b all V_a (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b)) -> hAPP(c_Product__Type_Osnd(tc_Int_Oint,tc_Int_Oint),c_Divides_OposDivAlg(V_a,V_b)) = c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b)))) # label(fact_mod__pos__pos) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3597 (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]. 5.99/6.06 3598 (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]. 5.99/6.06 3599 (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]. 5.99/6.06 3600 (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]. 5.99/6.06 3601 (all V_z_2 all T_c (class_Int_Oring__char__0(T_c) -> (V_z_2 = c_Groups_Ozero__class_Ozero(tc_Int_Oint) <-> 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]. 5.99/6.06 3602 (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]. 5.99/6.06 3603 (all V_k all T_a (class_Int_Onumber__ring(T_a) -> hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_k) = hAPP(c_Int_Oring__1__class_Oof__int(T_a),V_k))) # label(fact_number__of__eq) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3604 (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]. 5.99/6.06 3605 (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]. 5.99/6.06 3606 (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]. 5.99/6.06 3607 (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]. 5.99/6.06 3608 (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]. 5.99/6.06 3609 (all V_n all T_a (class_Rings_Oring__1(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),c_Groups_Oone__class_Oone(T_a))),hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_n))) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),c_Groups_Oone__class_Oone(T_a)))) # label(fact_power__minus1__odd) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3610 (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]. 5.99/6.06 3611 (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]. 5.99/6.06 3612 (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]. 5.99/6.06 3613 (all V_B_2 all V_A_2 all T_d all T_c all V_C_2 ((V_C_2 = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool),T_c),hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_d,tc_HOL_Obool)),V_A_2)),V_B_2)) = c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool))) & (V_C_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_d,tc_HOL_Obool)),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]. 5.99/6.06 3614 (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]. 5.99/6.06 3615 (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]. 5.99/6.06 3616 (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_OatLeastLessThan(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_I8_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3617 (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]. 5.99/6.06 3618 (all V_y_2 all V_x_2 all V_S_2 all V_R_2 all T_d all T_c (hBOOL(hAPP(hAPP(hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool))),hAPP(hAPP(c_COMBC(T_c,tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_fun(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool)),T_c),c_COMBC(T_d,tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_prod(T_c,T_d)),tc_fun(T_d,tc_fun(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),tc_HOL_Obool)),T_c),hAPP(c_COMBB(tc_prod(T_c,T_d),tc_fun(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),tc_HOL_Obool),T_d),c_member(tc_prod(T_c,T_d)))),c_Product__Type_OPair(T_c,T_d)))),V_R_2)),hAPP(hAPP(c_COMBC(T_c,tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_fun(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool)),T_c),c_COMBC(T_d,tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_prod(T_c,T_d)),tc_fun(T_d,tc_fun(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),tc_HOL_Obool)),T_c),hAPP(c_COMBB(tc_prod(T_c,T_d),tc_fun(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),tc_HOL_Obool),T_d),c_member(tc_prod(T_c,T_d)))),c_Product__Type_OPair(T_c,T_d)))),V_S_2)),V_x_2),V_y_2)) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_d)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_x_2),V_y_2)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),V_R_2),V_S_2))))) # label(fact_sup__Un__eq2) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3619 (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)),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__sup__class_Osup(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool)),V_A_2),V_B_2))) # label(fact_Inter__Un__distrib) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3620 (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]. 5.99/6.06 3621 (all V_i all T_a c_List_Olist_ONil(T_a) = c_List_Oconcat(T_a,c_List_Oreplicate(tc_List_Olist(T_a),V_i,c_List_Olist_ONil(T_a)))) # label(fact_concat__replicate__trivial) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3622 (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]. 5.99/6.06 3623 (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]. 5.99/6.06 3624 (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]. 5.99/6.06 3625 (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]. 5.99/6.06 3626 (all V_a_2 all T_c (class_Rings_Olinordered__idom(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_a_2),c_Groups_Ozero__class_Ozero(T_c))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Osgn__class_Osgn(T_c,V_a_2)),c_Groups_Ozero__class_Ozero(T_c)))))) # label(fact_sgn__less) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3627 (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_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H)) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H)) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_v),c_Int_OPls)) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_v_H),c_Int_OPls)) -> hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_v),V_v_H)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_v_H),c_Int_OPls)) -> hAPP(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)))))) # label(fact_add__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3628 (all V_u_2 all V_M_2 all V_A_2 all T_d all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(T_d),B_x),V_A_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(V_M_2,B_x)),V_u_2)))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,T_c),V_A_2),V_M_2)),V_u_2))))) # label(fact_SUP__le__iff) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3629 (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]. 5.99/6.06 3630 (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]. 5.99/6.06 3631 (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]. 5.99/6.06 3632 (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]. 5.99/6.06 3633 (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]. 5.99/6.06 3634 (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]. 5.99/6.06 3635 (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,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_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)))))))) # label(fact_folding__one_Ounion__inter) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3636 (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]. 5.99/6.06 3637 (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]. 5.99/6.06 3638 (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]. 5.99/6.06 3639 (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]. 5.99/6.06 3640 (all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,c_Groups_Ozero__class_Ozero(T_a),V_a) = c_Groups_Ozero__class_Ozero(T_a))) # label(fact_mod__0) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3641 (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]. 5.99/6.06 3642 (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]. 5.99/6.06 3643 (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]. 5.99/6.06 3644 (all V_n_2 all V_Pa_2 all V_Qa_2 ((all B_x hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(V_Qa_2,B_x)),hAPP(V_Pa_2,B_x)))) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBS(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat),tc_Nat_Onat),c_Groups_Ominus__class_Ominus(tc_Nat_Onat)),V_Pa_2)),V_Qa_2)),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),V_n_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),V_Pa_2),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),V_n_2))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),V_Qa_2),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),V_n_2))))) # label(fact_sum__diff__distrib) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3645 (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]. 5.99/6.06 3646 (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_m_2 = V_n_2 <-> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n_2),V_k_2) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m_2),V_k_2))))) # label(fact_eq__diff__iff) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3647 (all V_k_2 all T_c (class_Orderings_Olinorder(T_c) -> hAPP(c_SetInterval_Oord__class_OlessThan(T_c),V_k_2) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeast(T_c),V_k_2)))) # label(fact_Compl__atLeast) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3648 (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]. 5.99/6.06 3649 (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]. 5.99/6.06 3650 (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),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_b)) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_b)))) # label(fact_minus__divide__right) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3651 (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]. 5.99/6.06 3652 (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)))) -> V_x_2 = c_Big__Operators_Olinorder__class_OMax(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)))))) # label(fact_Max_Oinsert__remove) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3653 (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]. 5.99/6.06 3654 (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]. 5.99/6.06 3655 (all V_z_2 all T_d all T_c 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) = V_z_2) # label(fact_map__pair__ident) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3656 (all V_m_2 all V_n_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n_2),V_m_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_n_2)),hAPP(c_Nat_OSuc,V_m_2))))) # label(fact_Suc__le__mono) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3657 (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]. 5.99/6.06 3658 (all T_1 class_Nat_Osize(tc_Option_Ooption(T_1))) # label(arity_Option__Ooption__Nat_Osize) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3659 (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]. 5.99/6.06 3660 (all V_A_2 all T_c all V_y_2 all V_x_2 (V_x_2 = V_y_2 -> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) <-> hBOOL(hAPP(hAPP(c_member(T_c),V_y_2),V_A_2))))) # label(fact_eqelem__imp__iff) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3661 (all V_xs_2 all V_f_2 all T_c all T_d hAPP(c_List_Otl(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_Otl(T_d),V_xs_2))) # label(fact_map__tl) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3662 (all V_n_2 all V_Pa_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_k_2)) -> (hBOOL(hAPP(hAPP(V_Pa_2,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_n_2),V_k_2)),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_n_2,V_k_2))) <-> (all B_i all B_j (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B_j)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B_j),V_k_2)) & hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_k_2),B_i)),B_j) = V_n_2 -> hBOOL(hAPP(hAPP(V_Pa_2,B_i),B_j))))))) # label(fact_split__pos__lemma) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3663 (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]. 5.99/6.06 3664 (all V_x_2 all V_A_2 all V_F_2 all V_f_2 all T_c (c_Finite__Set_Ofolding__one(T_c,V_f_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) -> hAPP(V_F_2,V_A_2) = hAPP(hAPP(V_f_2,V_x_2),hAPP(V_F_2,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))))) & (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(V_F_2,V_A_2) = V_x_2))))) # label(fact_folding__one_Oremove) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3665 (all V_r_2 all V_b_2 all V_a_2 all T_d all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_d)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),V_a_2),V_b_2)),hAPP(c_Relation_Oconverse(T_d,T_c),V_r_2))) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(T_d,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_d,T_c),V_b_2),V_a_2)),V_r_2)))) # label(fact_converse__iff) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3666 (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_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]. 5.99/6.06 3667 (all V_B_2 all V_A_2 all V_c_2 all T_d all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_d)),V_c_2),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),V_A_2),V_B_2))) -> -(all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)) -> (all B_y (hBOOL(hAPP(hAPP(c_member(T_d),B_y),hAPP(V_B_2,B_x))) -> hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),B_x),B_y) != V_c_2)))))) # label(fact_SigmaE) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3668 (all V_x_2 all V_y_2 all T_c (class_Orderings_Oorder(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_y_2),V_x_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_x_2),V_y_2)) <-> V_x_2 = V_y_2)))) # label(fact_order__antisym__conv) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3669 (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)) <-> 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__iff2) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3670 (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]. 5.99/6.06 3671 (all V_v_H_2 all V_v_2 ((-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_v_2),V_v_H_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_v_2),c_Int_OPls))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H_2))))) # label(fact_le__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3672 (all V_j all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i),V_j)) -> hAPP(hAPP(c_List_Oappend(tc_Nat_Onat),c_List_Oupt(V_i,V_j)),hAPP(hAPP(c_List_Olist_OCons(tc_Nat_Onat),V_j),c_List_Olist_ONil(tc_Nat_Onat))) = c_List_Oupt(V_i,hAPP(c_Nat_OSuc,V_j)))) # label(fact_upt__Suc__append) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3673 (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]. 5.99/6.06 3674 (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]. 5.99/6.06 3675 (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]. 5.99/6.06 3676 (all V_r_2 all T_d all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_prod(T_d,T_c)),V_r_2)) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(tc_prod(T_c,T_d)),hAPP(c_Relation_Oconverse(T_d,T_c),V_r_2))))) # label(fact_finite__converse) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3677 (all V_fun2_2 all V_com_2 all V_fun1_2 all T_c 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)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_triple_Osize_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3678 (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]. 5.99/6.06 3679 (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]. 5.99/6.06 3680 (all V_u_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_u_2)) -> hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_u_2) = hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),hAPP(c_Int_Onat,V_u_2))))) # label(fact_image__atLeastZeroLessThan__int) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3681 (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]. 5.99/6.06 3682 (all V_v_2 all V_n_2 (hBOOL(hAPP(hAPP(c_HOL_OLet(tc_Int_Oint,tc_HOL_Obool),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_Opred(V_v_2))),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Int_Oint),c_fconj),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Int_Oint),c_fNot),c_Nat__Numeral_Oneg))),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Int_Oint),c_fimplies),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Int_Oint),c_fNot),c_Nat__Numeral_Oneg))),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Nat_Onat,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_Int_Oint),c_fequal),c_Int_Onat)),V_n_2))))) <-> hAPP(c_Nat_OSuc,V_n_2) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2))) # label(fact_Suc__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3683 (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]. 5.99/6.06 3684 (all V_b all V_a all T_a (class_Lattices_Oab__semigroup__idem__mult(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)))) # label(fact_mult__left__idem) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3685 (all V_x_2 all T_c (class_Groups_Oordered__ab__group__add(T_c) -> hAPP(c_SetInterval_Oord__class_OatLeast(T_c),hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_x_2)) = hAPP(c_Set_Oimage(T_c,T_c,c_Groups_Ouminus__class_Ouminus(T_c)),hAPP(c_SetInterval_Oord__class_OatMost(T_c),V_x_2)))) # label(fact_image__uminus__atMost) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3686 (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]. 5.99/6.06 3687 (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]. 5.99/6.06 3688 (all V_B_2 all V_A_2 all T_d all T_c all V_C_2 ((V_C_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> hAPP(hAPP(c_Lattices_Osemilattice__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))) & (V_C_2 = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> V_B_2 = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_d,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),V_A_2)),V_B_2)))) # label(fact_INT__extend__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3689 (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]. 5.99/6.06 3690 (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]. 5.99/6.06 3691 (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]. 5.99/6.06 3692 (all V_b all V_a all V_c all T_a (class_Divides_Osemiring__div(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_c),c_Divides_Odiv__class_Omod(T_a,V_a,V_b)) = 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)))) # label(fact_mod__mult__mult1) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3693 (all V_z_2 all V_Pa_2 all T_c all T_e all T_d hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_d,T_e,T_c),V_Pa_2),V_z_2) = hAPP(hAPP(V_Pa_2,hAPP(c_Product__Type_Ofst(T_d,T_e),V_z_2)),hAPP(c_Product__Type_Osnd(T_d,T_e),V_z_2))) # label(fact_split__beta) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3694 (all V_k_2 (V_k_2 = c_Int_OPls <-> c_Int_OBit0(V_k_2) = c_Int_OPls)) # label(fact_rel__simps_I44_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3695 (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]. 5.99/6.06 3696 (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]. 5.99/6.06 3697 (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]. 5.99/6.06 3698 (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]. 5.99/6.06 3699 (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]. 5.99/6.06 3700 (all V_r_2 all V_b_2 all V_a_2 all T_c hAPP(c_Relation_OField(T_c),hAPP(hAPP(c_Set_Oinsert(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_b_2)),V_r_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_b_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))),hAPP(c_Relation_OField(T_c),V_r_2))) # label(fact_Field__insert) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3701 (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]. 5.99/6.06 3702 (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]. 5.99/6.06 3703 (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]. 5.99/6.06 3704 (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_zs_2) = hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_ys_2) <-> V_ys_2 = V_zs_2)) # label(fact_same__append__eq) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3705 (all V_com2_H all V_com1_H c_Com_Ocom_OSemi(V_com1_H,V_com2_H) != c_Com_Ocom_OSKIP) # label(fact_com_Osimps_I12_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3706 (all V_R_2 all T_d all T_c hAPP(hAPP(c_Relation_Orel__comp(T_c,T_c,T_d),c_Relation_OId(T_c)),V_R_2) = V_R_2) # label(fact_Id__O__R) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3707 (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]. 5.99/6.06 3708 (all V_a all V_b (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b)) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_a),V_b)) -> c_Divides_OposDivAlg(V_a,V_b) = hAPP(c_Divides_Oadjust(V_b),c_Divides_OposDivAlg(V_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_b)))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_a),V_b)) -> c_Divides_OposDivAlg(V_a,V_b) = hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a)))) # label(fact_posDivAlg__eqn) # label(axiom) # label(non_clause). [assumption]. 5.99/6.06 3709 (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))) <-> (V_k_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> hBOOL(hAPP(V_Pa_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) & (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != V_k_2 -> (all B_i all B_j (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_j),V_k_2)) -> (V_n_2 = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_k_2),B_i)),B_j) -> hBOOL(hAPP(V_Pa_2,B_i)))))))) # label(fact_split__div) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3710 (all V_rx all V_ly all V_lx all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_ly),V_rx)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_lx),V_ly)),V_rx))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3711 (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]. 5.99/6.07 3712 (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]. 5.99/6.07 3713 (all V_r_2 all V_S_2 all T_e all T_c all T_d hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_e,tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),V_S_2),hAPP(hAPP(c_COMBB(tc_fun(tc_prod(T_d,T_c),tc_HOL_Obool),tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),T_e),c_Relation_Oconverse(T_d,T_c)),V_r_2)) = hAPP(c_Relation_Oconverse(T_d,T_c),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_e,tc_fun(tc_prod(T_d,T_c),tc_HOL_Obool)),V_S_2),V_r_2))) # label(fact_converse__INTER) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3714 (all V_C_2 all V_B_2 all V_A_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_C_2)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_B_2),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_C_2))) # label(fact_Un__left__commute) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3715 (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_x_2 & c_Groups_Ozero__class_Ozero(T_c) = V_y_2 <-> 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)))))))) # label(fact_sum__power2__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3716 (all V_n c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n)) # label(fact_mult__0) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3717 (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]. 5.99/6.07 3718 (all V_x_2 all V_g_2 all V_f_2 all T_f all T_e all T_c all T_d hAPP(V_g_2,hAPP(c_Product__Type_Osnd(T_e,T_f),V_x_2)) = hAPP(c_Product__Type_Osnd(T_d,T_c),hAPP(c_Product__Type_Omap__pair(T_e,T_d,T_f,T_c,V_f_2,V_g_2),V_x_2))) # label(fact_snd__prod__fun) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3719 (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]. 5.99/6.07 3720 (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]. 5.99/6.07 3721 (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]. 5.99/6.07 3722 (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]. 5.99/6.07 3723 (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]. 5.99/6.07 3724 (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]. 5.99/6.07 3725 (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]. 5.99/6.07 3726 (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]. 5.99/6.07 3727 (all V_ys_2 all V_xs_2 all T_c (hBOOL(hAPP(c_List_Odistinct(T_c),hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_ys_2))) <-> 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)) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) & hBOOL(hAPP(c_List_Odistinct(T_c),V_ys_2)) & hBOOL(hAPP(c_List_Odistinct(T_c),V_xs_2)))) # label(fact_distinct__append) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3728 (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_member(tc_prod(T_c,T_c)),V_p_2),hAPP(hAPP(c_Nat_Ocompow(tc_prod(T_c,T_c),tc_HOL_Obool),B_n),V_R_2))) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B_n)))))) # label(fact_trancl__power) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3729 (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]. 5.99/6.07 3730 (all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool))),hAPP(hAPP(c_Product__Type_OPair(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)),c_Wellfounded_Ofinite__psubset(T_c))))) # label(fact_in__finite__psubset) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3731 (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_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)),c_Transitive__Closure_Ortrancl(T_c,V_R_2)))))) # label(fact_Not__Domain__rtrancl) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3732 (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]. 5.99/6.07 3733 (all V_z_2 all V_x_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofun__left__comm__idem(T_c,T_d,V_f_2) -> hAPP(hAPP(V_f_2,V_x_2),hAPP(hAPP(V_f_2,V_x_2),V_z_2)) = hAPP(hAPP(V_f_2,V_x_2),V_z_2))) # label(fact_fun__left__comm__idem_Ofun__left__idem) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3734 (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),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_Olist_OCons(T_c),V_x_2),V_xs_2))) & (-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_Olist_OCons(T_c),V_x_2),V_xs_2))))) # label(fact_filter_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3735 (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(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_x_2),hAPP(c_Finite__Set_Ofold1(T_c,c_Groups_Otimes__class_Otimes(T_c)),V_A_2)) = hAPP(c_Finite__Set_Ofold1(T_c,c_Groups_Otimes__class_Otimes(T_c)),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)))))) # label(fact_fold1__insert__idem) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3736 (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]. 5.99/6.07 3737 (all V_n c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),V_n),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_min__0R) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3738 (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]. 5.99/6.07 3739 (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]. 5.99/6.07 3740 (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]. 5.99/6.07 3741 (all V_f_2 all V_F_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_F_2)) -> (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) <-> (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)))))) # label(fact_setprod__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3742 (all V_m_2 all V_n_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2),V_m_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(c_Code__Numeral_Oof__nat,V_n_2)),hAPP(c_Code__Numeral_Oof__nat,V_m_2))))) # label(fact_less__code__numeral__code) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3743 (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]. 5.99/6.07 3744 (all V_a all T_a (class_Lattices_Osemilattice__sup(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_a),V_a) = V_a)) # label(fact_sup_Oidem) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3745 (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]. 5.99/6.07 3746 (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]. 5.99/6.07 3747 (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]. 5.99/6.07 3748 (all V_y all V_xs all T_a all V_k (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_k),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))) -> hAPP(hAPP(c_List_Olist__update(T_a,hAPP(c_List_Orev(T_a),V_xs)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)),V_k)),c_Groups_Oone__class_Oone(tc_Nat_Onat))),V_y) = hAPP(c_List_Orev(T_a),hAPP(hAPP(c_List_Olist__update(T_a,V_xs),V_k),V_y)))) # label(fact_rev__update) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3749 (all V_w all V_v hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(V_v))),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(V_w))) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w))) # label(fact_zdiv__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3750 (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]. 5.99/6.07 3751 (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)) -> c_Big__Operators_Olinorder__class_OMax(T_c,V_A_2) = V_x_2))))) # label(fact_Max__eqI) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3752 (all V_xs all V_x all T_a hAPP(c_List_Otl(T_a),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)) = V_xs) # label(fact_tl_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3753 (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]. 5.99/6.07 3754 (all V_b all V_a all T_a (class_Orderings_Olinorder(T_a) -> hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_b),V_a) = hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_a),V_b))) # label(fact_min__max_Oinf_Ocommute) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3755 (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]. 5.99/6.07 3756 (all V_k (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_k)) -> (exists B_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B_n)) & hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),B_n) = V_k)))) # label(fact_zero__less__imp__eq__int) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3757 (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]. 5.99/6.07 3758 (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]. 5.99/6.07 3759 (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 (hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),B_y),B_x) = V_yx_2 -> -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]. 5.99/6.07 3760 (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__eq(tc_Nat_Onat),V_m_2),V_n_2)))) # label(fact_nat__less__le) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3761 (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_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))) <-> hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),V_x_2),V_A_2)))))) # label(fact_transfer__int__nat__set__relations_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3762 (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(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__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))) # label(fact_setsum__cartesian__product) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3763 (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]. 5.99/6.07 3764 (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]. 5.99/6.07 3765 (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]. 5.99/6.07 3766 (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]. 5.99/6.07 3767 (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]. 5.99/6.07 3768 (all V_A_2 all T_c (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = V_A_2 | -hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) <-> hAPP(c_Finite__Set_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]. 5.99/6.07 3769 (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)),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))) = 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)) # label(fact_UN__le__add__shift) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3770 (all V_A_2 all T_c hAPP(c_Set_OCollect(tc_List_Olist(T_c)),hAPP(hAPP(c_COMBC(tc_List_Olist(T_c),tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),tc_List_Olist(T_c)),c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool))),c_List_Oset(T_c))),V_A_2)) = c_List_Olists(T_c,V_A_2)) # label(fact_lists__eq__set) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3771 (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]. 5.99/6.07 3772 (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]. 5.99/6.07 3773 (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),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))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Finite__Set_Ocard(T_c),V_A_2)),hAPP(c_Finite__Set_Ocard(T_c),V_B_2))))) # label(fact_card_Ounion__inter) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3774 (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]. 5.99/6.07 3775 (all V_p_2 all V_g_2 all V_f_2 all T_d all T_c hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(V_f_2,hAPP(c_Product__Type_Ofst(T_c,T_d),V_p_2))),hAPP(V_g_2,hAPP(c_Product__Type_Osnd(T_c,T_d),V_p_2)))),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) = c_Product__Type_Oprod_Oprod__size(T_c,T_d,V_f_2,V_g_2,V_p_2)) # label(fact_prod__size__simp) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3776 (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]. 5.99/6.07 3777 (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]. 5.99/6.07 3778 (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) <-> V_B_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_Plus__eq__empty__conv) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3779 (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]. 5.99/6.07 3780 (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]. 5.99/6.07 3781 (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]. 5.99/6.07 3782 (all V_code__numeral_H c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != c_Code__Numeral_OSuc__code__numeral(V_code__numeral_H)) # label(fact_code__numeral_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3783 (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]. 5.99/6.07 3784 (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]. 5.99/6.07 3785 (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]. 5.99/6.07 3786 (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]. 5.99/6.07 3787 (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]. 5.99/6.07 3788 (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))) <-> c_List_Olinorder__class_Osorted(T_c,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))))) # label(fact_sorted__many__eq) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3789 (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]. 5.99/6.07 3790 (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]. 5.99/6.07 3791 (all V_w_2 all V_z_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_z_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_w_2),V_z_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Int_Onat,V_w_2)),hAPP(c_Int_Onat,V_z_2)))))) # label(fact_nat__mono__iff) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3792 (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]. 5.99/6.07 3793 (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]. 5.99/6.07 3794 (all V_n all V_m hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),hAPP(c_Nat_OSuc,V_n))) # label(fact_diff__Suc__Suc) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3795 (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]. 5.99/6.07 3796 (all V_m all V_n hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_n),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_m),V_n)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_m),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_m,V_n))) # label(fact_zmult__div__cancel) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3797 (all V_b all V_a all T_b all T_a hAPP(c_Nat_Osize__class_Osize(tc_Nitpick_Opair__box(T_a,T_b)),c_Nitpick_Opair__box_OPairBox(T_a,T_b,V_a,V_b)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_pair__box_Osize_I2_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3798 (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]. 5.99/6.07 3799 (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]. 5.99/6.07 3800 (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]. 5.99/6.07 3801 (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]. 5.99/6.07 3802 (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]. 5.99/6.07 3803 (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]. 5.99/6.07 3804 (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]. 5.99/6.07 3805 (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_n = V_m))) # label(fact_less__SucE) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3806 (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]. 5.99/6.07 3807 (all V_w all V_z hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_w),V_z) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_z),V_w)) # label(fact_zmult__commute) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3808 (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]. 5.99/6.07 3809 (all T_c (class_Orderings_Oord(T_c) -> c_Orderings_Oord__class_Omax(T_c) = c_Orderings_Oord_Omax(T_c,c_Orderings_Oord__class_Oless__eq(T_c)))) # label(fact_max__ord__max) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3810 (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]. 5.99/6.07 3811 (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]. 5.99/6.07 3812 (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]. 5.99/6.07 3813 (all V_x_2 all V_f_2 all T_e all T_d all T_c hAPP(c_Product__Type_Ofst(T_c,T_d),hAPP(c_Product__Type_Oapsnd(T_e,T_d,T_c,V_f_2),V_x_2)) = hAPP(c_Product__Type_Ofst(T_c,T_e),V_x_2)) # label(fact_fst__apsnd) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3814 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_c (class_Orderings_Olinorder(T_c) -> c_SetInterval_Oord__class_OgreaterThanLessThan(T_c,hAPP(hAPP(c_Orderings_Oord__class_Omax(T_c),V_a_2),V_c_2),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),V_b_2),V_d_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanLessThan(T_c,V_a_2,V_b_2)),c_SetInterval_Oord__class_OgreaterThanLessThan(T_c,V_c_2,V_d_2)))) # label(fact_Int__greaterThanLessThan) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3815 (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]. 5.99/6.07 3816 (all V_A_2 all V_f_2 all T_c hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Int_Oint,T_c),c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),V_f_2)),V_A_2) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,tc_Nat_Onat),V_f_2),V_A_2))) # label(fact_int__setsum) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3817 (all V_B_2 all V_X_2 all T_d all T_c 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) = 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))) # label(fact_Sigma__Union) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3818 (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)))) -> (c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool)) != V_B_2 -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2))))) # label(fact_finite__cartesian__productD1) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3819 (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]. 5.99/6.07 3820 (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_x_2) = hAPP(c_SetInterval_Oord__class_OatMost(T_c),V_y_2) <-> V_y_2 = V_x_2))) # label(fact_atMost__eq__iff) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3821 (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]. 5.99/6.07 3822 (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]. 5.99/6.07 3823 (all V_d_2 all V_c_2 all V_a_2 all T_c (class_Orderings_Olinorder(T_c) -> c_SetInterval_Oord__class_OatLeastAtMost(T_c,hAPP(hAPP(c_Orderings_Oord__class_Omax(T_c),V_a_2),V_c_2),V_d_2) = 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)))) # label(fact_Int__atLeastAtMostR2) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3824 (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),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) = V_n)) # label(fact_Suc__pred) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3825 (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]. 5.99/6.07 3826 (all V_A_2 all V_x_2 all T_c (hBOOL(hAPP(V_A_2,V_x_2)) <-> hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)))) # label(fact_mem__def) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3827 (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_a = B_b)) -> 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_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_y_2),V_x_2)),c_List_Olexord(T_c,V_r_2))))) # label(fact_lexord__linear) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3828 (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]. 5.99/6.07 3829 (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) & (c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != V_i -> c_Code__Numeral_Onat__of__aux(hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Code__Numeral_Ocode__numeral),V_i),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),hAPP(c_Nat_OSuc,V_n)) = c_Code__Numeral_Onat__of__aux(V_i,V_n)))) # label(fact_nat__of__aux__code) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3830 (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]. 5.99/6.07 3831 (all V_n_2 all T_c hAPP(hAPP(c_Nat_Ocompow(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_n_2),c_List_Orotate1(T_c)) = c_List_Orotate(T_c,V_n_2)) # label(fact_rotate__def) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3832 (all V_x all T_a (class_Lattices_Obounded__lattice__bot(T_a) -> V_x = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),c_Orderings_Obot__class_Obot(T_a)))) # label(fact_sup__bot__right) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3833 (all V_u_2 all V_l_2 all T_c (class_Orderings_Oorder(T_c) -> c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanAtMost(T_c,V_l_2,V_u_2)),hAPP(c_SetInterval_Oord__class_OgreaterThan(T_c),V_u_2)))) # label(fact_ivl__disj__int_I5_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3834 (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]. 5.99/6.07 3835 (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]. 5.99/6.07 3836 (all V_a all V_b (V_b != c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> (c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b) = c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a)),V_b) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b))) & (c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b) != c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> hAPP(hAPP(c_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)))) # label(fact_zdiv__zminus1__eq__if) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3837 (all V_n all V_m hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m),V_n)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n))) # label(fact_zmult__int) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3838 (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]. 5.99/6.07 3839 (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]. 5.99/6.07 3840 (all V_uu all T_b all T_a c_Enum_Oproduct(T_a,T_b,c_List_Olist_ONil(T_a),V_uu) = c_List_Olist_ONil(tc_prod(T_a,T_b))) # label(fact_product_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3841 (all V_l all V_k (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_l)) -> c_Divides_OnegDivAlg(V_k,V_l) = hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_k),V_l)),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_k,V_l))))) # label(fact_negDivAlg__div__mod) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3842 (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]. 5.99/6.07 3843 (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]. 5.99/6.07 3844 (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]. 5.99/6.07 3845 (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]. 5.99/6.07 3846 (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]. 5.99/6.07 3847 (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]. 5.99/6.07 3848 (all V_l c_Int_OBit1(V_l) != c_Int_OPls) # label(fact_rel__simps_I39_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3849 (all V_r_2 all V_A_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),V_A_2),V_r_2) <-> c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = V_A_2)) # label(fact_quotient__is__empty) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3850 (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]. 5.99/6.07 3851 (all V_w all T_b (class_Groups_Omonoid__mult(T_b) & class_Int_Onumber(T_b) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_b),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w)),hAPP(c_Int_Onumber__class_Onumber__of(T_b),V_w)))) # label(fact_power2__eq__square__number__of) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3852 (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]. 5.99/6.07 3853 (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]. 5.99/6.07 3854 (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]. 5.99/6.07 3855 (all V_a all T_a (class_Groups_Ogroup__add(T_a) -> V_a = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)))) # label(fact_minus__minus) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3856 (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]. 5.99/6.07 3857 (all V_w all V_x all V_z all V_y all T_a (class_Fields_Ofield(T_a) -> (c_Groups_Ozero__class_Ozero(T_a) != V_y -> (V_z != c_Groups_Ozero__class_Ozero(T_a) -> hAPP(hAPP(c_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)) = 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)))))) # label(fact_diff__frac__eq) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3858 (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)))) # label(fact_equalityD1) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3859 (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]. 5.99/6.07 3860 (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]. 5.99/6.07 3861 (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_Osym(T_c,V_r_2) & c_Relation_Otrans(T_c,V_r_2) & c_Relation_Orefl__on(T_c,V_A_2,V_r_2))) # label(fact_equiv__def) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3862 (all V_r_H all V_q_H all V_b_H all V_r all V_q all V_b (hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),V_q)),V_r) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b_H),V_q_H)),V_r_H) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b_H),V_q_H)),V_r_H))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_r_H),V_b_H)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_r)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b_H)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_b_H),V_b)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_q),V_q_H))))))))) # label(fact_zdiv__mono2__lemma) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3863 (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]. 5.99/6.07 3864 (all V_B_2 all V_A_2 all T_d all T_c 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)) = 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))) # label(fact_Pow__INT__eq) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3865 (all V_n_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_m_2)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m_2),V_n_2))))) # label(fact_add__gr__0) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3866 (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]. 5.99/6.07 3867 (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]. 5.99/6.07 3868 (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]. 5.99/6.07 3869 (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]. 5.99/6.07 3870 (all V_b_2 all V_a_2 all T_c (class_Fields_Olinordered__field__inverse__zero(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_a_2),V_b_2)),c_Groups_Ozero__class_Ozero(T_c))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_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)),V_b_2))))) # label(fact_divide__le__0__iff) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3871 (all V_N_2 all V_h_2 all T_c (class_Orderings_Olinorder(T_c) -> ((all B_x all B_y hAPP(V_h_2,hAPP(hAPP(c_Orderings_Oord__class_Omax(T_c),B_x),B_y)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_c),hAPP(V_h_2,B_x)),hAPP(V_h_2,B_y))) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_N_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_N_2 -> c_Big__Operators_Olinorder__class_OMax(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_OMax(T_c,V_N_2))))))) # label(fact_Max_Ohom__commute) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3872 (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]. 5.99/6.07 3873 (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]. 5.99/6.07 3874 (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]. 5.99/6.07 3875 (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_d)),V_ys_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2) & (all B_x (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_d)),B_x),hAPP(c_List_Oset(tc_prod(T_c,T_d)),hAPP(hAPP(c_List_Ozip(T_c,T_d),V_xs_2),V_ys_2)))) -> hBOOL(hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,tc_HOL_Obool),V_Pa_2),B_x)))) <-> 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]. 5.99/6.07 3876 (all V_n_2 all T_c (class_Rings_Osemiring__1(T_c) -> hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_c),V_n_2) = 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)))) # label(fact_semiring__1__class_Oof__nat__code) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3877 (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]. 5.99/6.07 3878 (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]. 5.99/6.07 3879 (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]. 5.99/6.07 3880 (all V_xs_2 all V_n_2 all T_c (hBOOL(hAPP(c_List_Odistinct(T_c),V_xs_2)) <-> hBOOL(hAPP(c_List_Odistinct(T_c),hAPP(c_List_Orotate(T_c,V_n_2),V_xs_2))))) # label(fact_distinct__rotate) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3881 (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]. 5.99/6.07 3882 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]. 5.99/6.07 3883 (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]. 5.99/6.07 3884 (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]. 5.99/6.07 3885 (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]. 5.99/6.07 3886 (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]. 5.99/6.07 3887 (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]. 5.99/6.07 3888 (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_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]. 5.99/6.07 3889 (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(V_F_2,hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(V_f_2,hAPP(V_F_2,V_A_2)),hAPP(V_F_2,V_B_2)))))))) # label(fact_folding__one__idem_Ounion__idem) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3890 (all V_x all V_k all V_i all T_a hAPP(hAPP(c_List_Otake(T_a),V_i),c_List_Oreplicate(T_a,V_k,V_x)) = c_List_Oreplicate(T_a,hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),V_i),V_k),V_x)) # label(fact_take__replicate) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3891 (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]. 5.99/6.07 3892 (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]. 5.99/6.07 3893 (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]. 5.99/6.07 3894 (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_Int_Oint,tc_HOL_Obool)),V_A_2),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))))))) # label(fact_transfer__int__nat__set__relations_I4_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3895 (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]. 5.99/6.07 3896 (all V_k_2 all V_n_2 all V_Pa_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))) & 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_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 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B_j)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B_j),V_k_2)) & hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_k_2),B_i)),B_j) = V_n_2 -> hBOOL(hAPP(V_Pa_2,B_j))))) & (V_k_2 = c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> hBOOL(hAPP(V_Pa_2,V_n_2))) <-> hBOOL(hAPP(V_Pa_2,c_Divides_Odiv__class_Omod(tc_Int_Oint,V_n_2,V_k_2))))) # label(fact_split__zmod) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3897 (all V_ys_2 all V_xs_2 all T_c (V_ys_2 = hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),V_ys_2) <-> c_List_Olist_ONil(T_c) = V_xs_2)) # label(fact_append__self__conv2) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3898 (all V_Y_2 all T_c hAPP(c_Set_Ovimage(T_c,T_c,c_COMBI(T_c)),V_Y_2) = V_Y_2) # label(fact_vimage__ident) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3899 (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)) -> hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_List_Olist(T_c))),c_List_Otranspose(T_c,V_xs_2)) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),hAPP(hAPP(c_List_Onth(tc_List_Olist(T_c)),V_xs_2),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) & (V_xs_2 = c_List_Olist_ONil(tc_List_Olist(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_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_length__transpose__sorted) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3900 (all V_y all V_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_x)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_y)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Int_Onat,V_x)),hAPP(c_Int_Onat,V_y)) = hAPP(c_Int_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_x),V_y))))) # label(fact_Nat__Transfer_Otransfer__nat__int__functions_I1_J) # label(axiom) # label(non_clause). [assumption]. 5.99/6.07 3901 (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]. 5.99/6.07 3902 (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]. 5.99/6.07 3903 (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]. 5.99/6.07 3904 (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]. 5.99/6.07 3905 (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]. 6.07/6.07 3906 (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]. 6.07/6.07 3907 (all V_a_2 all T_c hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBC(T_c,T_c,tc_HOL_Obool),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__conv) # label(axiom) # label(non_clause). [assumption]. 6.07/6.07 3908 (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]. 6.07/6.07 3909 (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]. 6.07/6.07 3910 (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]. 6.07/6.07 3911 (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]. 6.07/6.08 3912 (all V_n_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m_2),V_n_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_m_2)))) # label(fact_nat__0__less__mult__iff) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 3913 (all T_c c_Lazy__Sequence_Oyieldn(T_c) = c_Lazy__Sequence_Oanamorph(tc_Lazy__Sequence_Olazy__sequence(T_c),T_c,c_Lazy__Sequence_Oyield(T_c))) # label(fact_yieldn__def) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 3914 (all T_c hAPP(c_Relation_Oinv__image(tc_Nat_Onat,T_c),c_Wellfounded_Oless__than) = c_Wellfounded_Omeasure(T_c)) # label(fact_measure__def) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 3915 (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]. 6.07/6.08 3916 (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]. 6.07/6.08 3917 (all V_r_2 all T_c ((all B_Q ((exists B_x hBOOL(hAPP(hAPP(c_member(T_c),B_x),B_Q))) -> (exists B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),B_Q)) & (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)))))))) <-> c_Wellfounded_Owf(T_c,V_r_2))) # label(fact_wf__eq__minimal) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 3918 (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]. 6.07/6.08 3919 (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]. 6.07/6.08 3920 (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]. 6.07/6.08 3921 (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]. 6.07/6.08 3922 (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]. 6.07/6.08 3923 (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]. 6.07/6.08 3924 (all V_b_2 all V_a_2 all V_r_2 all V_A_2 all T_c (c_Equiv__Relations_Oequiv(T_c,V_A_2,V_r_2) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_b_2)),V_r_2)) -> hAPP(c_Relation_OImage(T_c,T_c,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) = hAPP(c_Relation_OImage(T_c,T_c,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_b_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))))) # label(fact_equiv__class__eq) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 3925 (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]. 6.07/6.08 3926 (all V_l_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),V_l_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit0(V_k_2)),c_Int_OBit0(V_l_2))))) # label(fact_rel__simps_I14_J) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 3927 (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]. 6.07/6.08 3928 (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]. 6.07/6.08 3929 (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)) -> V_m = hAPP(c_Nat_OSuc,V_n)))) # label(fact_le__SucE) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 3930 (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]. 6.07/6.08 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)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> c_Big__Operators_Olattice_OSup__fin(T_c,c_Orderings_Oord__class_Omax(T_c),hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool,T_c),c_HOL_OEx(T_c)),hAPP(hAPP(c_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))))) = 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)))))) # label(fact_min__max_Oinf__Sup1__distrib) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 3932 (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]. 6.07/6.08 3933 (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]. 6.07/6.08 3934 (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]. 6.07/6.08 3935 (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]. 6.07/6.08 3936 (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]. 6.07/6.08 3937 (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]. 6.07/6.08 3938 (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]. 6.07/6.08 3939 (all V_Pa_2 all T_c -(exists B_x (hBOOL(hAPP(V_Pa_2,B_x)) & hBOOL(hAPP(hAPP(c_member(T_c),B_x),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))))) # label(fact_bex__empty) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 3940 (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]. 6.07/6.08 3941 (all T_a hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),c_List_Olist_ONil(T_a)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_list_Osize_I3_J) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 3942 (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]. 6.07/6.08 3943 (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]. 6.07/6.08 3944 (all V_c_2 all V_b_2 all V_w_2 all T_c (class_Int_Onumber(T_c) & class_Fields_Olinordered__field__inverse__zero(T_c) -> ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_c_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2)),V_c_2)),V_b_2))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_c_2)) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_c_2),c_Groups_Ozero__class_Ozero(T_c))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),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),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]. 6.07/6.08 3945 (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]. 6.07/6.08 3946 (all V_r_2 all V_q_2 all T_c (c_Nitpick_Oless__frac(T_c,V_q_2,V_r_2) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Nitpick_Onum(T_c),c_Nitpick_Oplus__frac(T_c,V_q_2,c_Nitpick_Ouminus__frac(T_c,V_r_2)))),c_Groups_Ozero__class_Ozero(tc_Int_Oint))))) # label(fact_Nitpick_Oless__frac__def) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 3947 (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]. 6.07/6.08 3948 (all V_b_2 all V_a_2 all V_fb_2 all V_fa_2 all T_d all T_c c_Nitpick_Opair__box_Opair__box__size(T_c,T_d,V_fa_2,V_fb_2,c_Nitpick_Opair__box_OPairBox(T_c,T_d,V_a_2,V_b_2)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(V_fa_2,V_a_2)),hAPP(V_fb_2,V_b_2))),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_pair__box_Osize_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 3949 (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]. 6.07/6.08 3950 (all V_B_2 all V_A_2 all T_d all T_c all V_C_2 ((c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = V_C_2 -> V_B_2 = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_d,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),V_A_2)),V_B_2)) & (V_C_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_d,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,tc_fun(T_d,tc_HOL_Obool)),V_C_2),V_A_2)),V_B_2) = 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))))) # label(fact_UN__extend__simps_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 3951 (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]. 6.07/6.08 3952 (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__eq(tc_Nat_Onat),V_n_2)),c_Int_Onat))))) <-> 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))))) # label(fact_le__Suc__number__of) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 3953 (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]. 6.07/6.08 3954 (all V_list_H all V_a_H all T_a c_List_Olist_ONil(T_a) != hAPP(hAPP(c_List_Olist_OCons(T_a),V_a_H),V_list_H)) # label(fact_list_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 3955 (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]. 6.07/6.08 3956 (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),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_c),V_y_2),V_z_2))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_x_2),V_z_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_x_2),V_y_2))))) # label(fact_le__inf__iff) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 3957 (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]. 6.07/6.08 3958 (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]. 6.07/6.08 3959 (all V_b all V_a ((-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b)) -> c_Divides_OnegDivAlg(V_a,V_b) = c_Divides_Odivmod__int(V_a,V_b)) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b)) -> 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_Odivmod__int(V_a,V_b))) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b)) -> c_Divides_OposDivAlg(V_a,V_b) = c_Divides_Odivmod__int(V_a,V_b)) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_b)) -> (c_Groups_Ozero__class_Ozero(tc_Int_Oint) = V_a -> c_Divides_Odivmod__int(V_a,V_b) = hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) & (V_a != c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> hAPP(c_Divides_OnegateSnd,c_Divides_OnegDivAlg(hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_b))) = c_Divides_Odivmod__int(V_a,V_b)))))) # label(fact_divmod__int__def) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 3960 (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]. 6.07/6.08 3961 (all T_a (class_Rings_Osemiring__1(T_a) -> c_Groups_Oone__class_Oone(T_a) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),c_Groups_Oone__class_Oone(T_a)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) # label(fact_one__power2) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 3962 (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]. 6.07/6.08 3963 (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,hAPP(c_List_Ofoldl(T_c,T_d,V_f_2,V_a_2),V_xs_2)),V_ys_2) = 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))) # label(fact_foldl__append) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 3964 (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]. 6.07/6.08 3965 (all V_x all T_a all V_xs hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs) != V_xs) # label(fact_not__Cons__self) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 3966 (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]. 6.07/6.08 3967 (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]. 6.07/6.08 3968 (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)) -> (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) & (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)) = 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)))))))))) # label(fact_Sup__fin_Oinsert__remove) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 3969 (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]. 6.07/6.08 3970 (all V_xs all T_a V_xs = hAPP(c_List_Orev(T_a),hAPP(c_List_Orev(T_a),V_xs))) # label(fact_rev__rev__ident) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 3971 (all V_b_2 all V_a_2 all T_c (class_Groups_Ogroup__add(T_c) -> (V_a_2 = hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_b_2) <-> 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]. 6.07/6.08 3972 (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]. 6.07/6.08 3973 (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]. 6.07/6.08 3974 (all V_x_2 all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,V_A_2) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> hAPP(c_Fun_Othe__inv__into(T_c,T_d,V_A_2,V_f_2),hAPP(V_f_2,V_x_2)) = V_x_2))) # label(fact_the__inv__into__f__f) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 3975 (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]. 6.07/6.08 3976 (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) -> 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)) & (c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != V_i_2 -> hAPP(c_New__DSequence_Oneg__decr__bind(T_d,T_c,V_x_2,V_f_2),V_i_2) = c_Lazy__Sequence_Ohb__bind(T_d,T_c,hAPP(V_x_2,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Code__Numeral_Ocode__numeral),V_i_2),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral))),hAPP(hAPP(c_COMBC(T_d,tc_Code__Numeral_Ocode__numeral,tc_Lazy__Sequence_Olazy__sequence(tc_Option_Ooption(T_c))),V_f_2),V_i_2))))) # label(fact_New__DSequence_Oneg__decr__bind__def) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 3977 (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]. 6.07/6.08 3978 (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]. 6.07/6.08 3979 (all V_xs_2 all V_x_2 all T_c (hBOOL(hAPP(c_List_Odistinct(T_c),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2))) <-> -hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_List_Oset(T_c),V_xs_2))) & hBOOL(hAPP(c_List_Odistinct(T_c),V_xs_2)))) # label(fact_distinct_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 3980 (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]. 6.07/6.08 3981 (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]. 6.07/6.08 3982 (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]. 6.07/6.08 3983 (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]. 6.07/6.08 3984 (all V_y all V_x all T_a (class_Lattices_Olattice(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y)))) # label(fact_inf__sup__aci_I4_J) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 3985 (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]. 6.07/6.08 3986 (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]. 6.07/6.08 3987 (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]. 6.07/6.08 3988 (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]. 6.07/6.08 3989 (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]. 6.07/6.08 3990 (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]. 6.07/6.08 3991 (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]. 6.07/6.08 3992 (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]. 6.07/6.08 3993 (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]. 6.07/6.08 3994 (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]. 6.07/6.08 3995 (all V_x_2 all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> hAPP(c_Nat_OSuc,hAPP(c_Finite__Set_Ocard(T_c),V_A_2)) = hAPP(c_Finite__Set_Ocard(T_c),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2))))) # label(fact_card__insert__disjoint) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 3996 (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]. 6.07/6.08 3997 (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]. 6.07/6.08 3998 (all V_A_2 all T_d all T_c all V_B_2 ((V_B_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> hAPP(c_Set_Oimage(tc_prod(T_d,T_c),T_d,c_Product__Type_Ofst(T_d,T_c)),hAPP(hAPP(c_Product__Type_OSigma(T_d,T_c),V_A_2),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_d),V_B_2))) = V_A_2) & (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = V_B_2 -> c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool)) = hAPP(c_Set_Oimage(tc_prod(T_d,T_c),T_d,c_Product__Type_Ofst(T_d,T_c)),hAPP(hAPP(c_Product__Type_OSigma(T_d,T_c),V_A_2),hAPP(c_COMBK(tc_fun(T_c,tc_HOL_Obool),T_d),V_B_2)))))) # label(fact_fst__image__times) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 3999 (all V_x_2 all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> hAPP(c_Finite__Set_Ocard(T_c),V_A_2) = hAPP(c_Finite__Set_Ocard(T_c),hAPP(hAPP(c_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(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__if) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 4000 (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_y_2),V_z_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_x_2),V_z_2))))) # label(fact_max__less__iff__conj) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 4001 (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_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)) = hAPP(c_Finite__Set_Ofold1(T_c,c_Groups_Otimes__class_Otimes(T_c)),hAPP(c_List_Oset(T_c),V_xs_2))))) # label(fact_fold1__set) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 4002 (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]. 6.07/6.08 4003 (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]. 6.07/6.08 4004 (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]. 6.07/6.08 4005 (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]. 6.07/6.08 4006 (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]. 6.07/6.08 4007 (all V_r_2 all V_ys_2 all V_y_2 all V_xs_2 all T_c (hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_xs_2),hAPP(hAPP(c_List_Olist_OCons(T_c),V_y_2),V_ys_2))),c_List_Olistrel(T_c,V_r_2))) -> -(all B_x all B_xs (hAPP(hAPP(c_List_Olist_OCons(T_c),B_x),B_xs) = V_xs_2 -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),B_x),V_y_2)),V_r_2)) -> -hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),B_xs),V_ys_2)),c_List_Olistrel(T_c,V_r_2)))))))) # label(fact_listrel__Cons2) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 4008 (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]. 6.07/6.08 4009 (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]. 6.07/6.08 4010 (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]. 6.07/6.08 4011 (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]. 6.07/6.08 4012 (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]. 6.07/6.08 4013 (all V_k ((V_k != c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) -> (c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,V_k,hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) -> hAPP(c_Code__Numeral_Oint__of,V_k) = 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_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,V_k,hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) -> hAPP(hAPP(c_Groups_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)) = hAPP(c_Code__Numeral_Oint__of,V_k))) & (c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = V_k -> c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(c_Code__Numeral_Oint__of,V_k)))) # label(fact_int__of__code) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 4014 (all V_M_2 all T_c all V_A_2 all T_d (class_Complete__Lattice_Ocomplete__lattice(T_d) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_A_2 -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,T_d),V_A_2),hAPP(c_COMBK(T_d,T_c),V_M_2)) = V_M_2))) # label(fact_INF__const) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 4015 (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]. 6.07/6.08 4016 (all V_w_2 all V_m_2 (V_m_2 = hAPP(c_Int_Onat,V_w_2) <-> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_w_2)) -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_m_2) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_w_2)) -> V_w_2 = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m_2)))) # label(fact_nat__eq__iff2) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 4017 (all V_z_2 all V_x_2 all V_A_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofun__left__comm(T_c,T_d,V_f_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_d,V_f_2),V_z_2),V_A_2) = hAPP(hAPP(V_f_2,V_x_2),hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_d,V_f_2),V_z_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))))))))) # label(fact_fun__left__comm_Ofold__rec) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 4018 (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) -> (V_a_2 != c_Groups_Ozero__class_Ozero(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),V_c_2),V_d_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) <-> hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_c),V_d_2),V_c_2) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_c),V_b_2),V_a_2)))))))) # label(fact_dvd__div__div__eq__mult) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 4019 (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]. 6.07/6.08 4020 (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 <-> hAPP(c_List_Ofoldl(tc_Nat_Onat,tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m_2),V_ns_2) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_sum__eq__0__conv) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 4021 (all V_k c_Int_Opred(c_Int_OBit1(V_k)) = c_Int_OBit0(V_k)) # label(fact_pred__Bit1) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 4022 (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]. 6.07/6.08 4023 (all V_k c_Int_OMin != c_Int_OBit0(V_k)) # label(fact_rel__simps_I45_J) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 4024 (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]. 6.07/6.08 4025 (all V_S_2 all T_c c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,tc_HOL_Obool),V_S_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),V_S_2),c_COMBI(tc_fun(T_c,tc_HOL_Obool)))) # label(fact_Union__def) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 4026 (all V_Qa_2 all V_Pa_2 (hBOOL(hAPP(hAPP(c_fimplies,V_Pa_2),V_Qa_2)) | hBOOL(V_Pa_2))) # label(help_c__fimplies__1) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 4027 (all V_ys all V_xs all T_a hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_ys))) # label(fact_length__append) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 4028 (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]. 6.07/6.08 4029 (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]. 6.07/6.08 4030 (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]. 6.07/6.08 4031 (all V_xs_2 all T_c c_List_Olist_ONil(T_c) = c_List_Osublist(T_c,V_xs_2,c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) # label(fact_sublist__empty) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 4032 (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]. 6.07/6.08 4033 (all V_k c_Int_OBit0(V_k) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_k),V_k)) # label(fact_Bit0__def) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 4034 (all V_w all V_z (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_z),V_w)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_w),V_z)) -> V_w = V_z))) # label(fact_zle__antisym) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 4035 (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]. 6.07/6.08 4036 (all V_a_2 all T_c (class_Groups_Oordered__ab__group__add__abs(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(c_Groups_Oabs__class_Oabs(T_c),V_a_2)),c_Groups_Ozero__class_Ozero(T_c))) <-> V_a_2 = c_Groups_Ozero__class_Ozero(T_c)))) # label(fact_abs__le__zero__iff) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 4037 (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]. 6.07/6.08 4038 (all V_B_2 all V_f_2 all T_d all V_a_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_d),hAPP(V_f_2,V_a_2)),V_B_2)) <-> hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),hAPP(c_Set_Ovimage(T_c,T_d,V_f_2),V_B_2))))) # label(fact_vimage__eq) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 4039 (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]. 6.07/6.08 4040 (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]. 6.07/6.08 4041 (all V_xs_2 all T_c hAPP(c_List_Oset(T_c),V_xs_2) = hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool,T_c),c_HOL_OEx(tc_Nat_Onat)),hAPP(hAPP(c_COMBC(T_c,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),T_c),c_COMBS(tc_Nat_Onat,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),T_c),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Nat_Onat),c_fconj)),hAPP(hAPP(c_COMBC(T_c,tc_fun(tc_Nat_Onat,T_c),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(tc_Nat_Onat,T_c),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),T_c),c_COMBB(T_c,tc_HOL_Obool,tc_Nat_Onat)),c_fequal)),hAPP(c_List_Onth(T_c),V_xs_2))))),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_HOL_Obool),c_Orderings_Oord__class_Oless(tc_Nat_Onat)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2)))))) # label(fact_set__conv__nth) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 4042 (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]. 6.07/6.08 4043 (all T_a all V_n ((V_n != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> c_List_Olist_ONil(tc_List_Olist(T_a)) = c_Enum_On__lists(T_a,V_n,c_List_Olist_ONil(T_a))) & (c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n -> hAPP(hAPP(c_List_Olist_OCons(tc_List_Olist(T_a)),c_List_Olist_ONil(T_a)),c_List_Olist_ONil(tc_List_Olist(T_a))) = c_Enum_On__lists(T_a,V_n,c_List_Olist_ONil(T_a))))) # label(fact_n__lists__Nil) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 4044 (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]. 6.07/6.08 4045 (all V_l all V_k (c_Divides_Odiv__class_Omod(tc_Int_Oint,V_k,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_l)) != c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> c_Groups_Ozero__class_Ozero(tc_Int_Oint) != c_Divides_Odiv__class_Omod(tc_Int_Oint,V_k,V_l))) # label(fact_zmod__zminus2__not__zero) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 4046 (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]. 6.07/6.08 4047 (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]. 6.07/6.08 4048 (all V_w hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_w) = hAPP(c_Int_Onat,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_w))) # label(fact_nat__number__of) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 4049 (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]. 6.07/6.08 4050 (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]. 6.07/6.08 4051 (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]. 6.07/6.08 4052 (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]. 6.07/6.08 4053 (all V_R_2 all T_c 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)) = 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))) # label(fact_Field__Union) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 4054 (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]. 6.07/6.08 4055 (all V_x all V_xs all T_a (hBOOL(hAPP(c_List_Odistinct(T_a),V_xs)) -> c_List_Oremove1(T_a,V_x,c_List_Oremdups(T_a,V_xs)) = c_List_Oremdups(T_a,c_List_Oremove1(T_a,V_x,V_xs)))) # label(fact_remove1__remdups) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 4056 (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]. 6.07/6.08 4057 (all V_fa_2 all T_c c_List_Olist_Olist__size(T_c,V_fa_2,c_List_Olist_ONil(T_c)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_list_Osize_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 4058 (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]. 6.07/6.08 4059 (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]. 6.07/6.08 4060 (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]. 6.07/6.08 4061 (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]. 6.07/6.08 4062 (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_Obutlast(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)))) # label(fact_takeWhile__not__last) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 4063 (all V_B_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_A_2 -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_B_2 -> 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]. 6.07/6.08 4064 (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]. 6.07/6.08 4065 (all V_x_2 all V_n_2 all T_c (c_List_Olist_ONil(T_c) = c_List_Oreplicate(T_c,V_n_2,V_x_2) <-> V_n_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_replicate__empty) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 4066 (all V_xs_2 all V_Pa_2 all T_c (hAPP(c_List_Ofilter(T_c,V_Pa_2),V_xs_2) = V_xs_2 <-> (all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(c_List_Oset(T_c),V_xs_2))) -> hBOOL(hAPP(V_Pa_2,B_x)))))) # label(fact_filter__id__conv) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 4067 (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]. 6.07/6.08 4068 (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]. 6.07/6.08 4069 (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]. 6.07/6.08 4070 (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]. 6.07/6.08 4071 (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]. 6.07/6.08 4072 (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)),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_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))) & hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)))) # label(fact_Cons__in__lists__iff) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 4073 (all V_y_2 all V_x_2 (hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),V_x_2),c_Orderings_Otop__class_Otop(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) -> (hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),V_y_2),c_Orderings_Otop__class_Otop(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) -> (hAPP(c_Code__Numeral_Oof__nat,V_y_2) = hAPP(c_Code__Numeral_Oof__nat,V_x_2) <-> V_y_2 = V_x_2)))) # label(fact_code__numeral_Oof__nat__inject) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 4074 (all V_xs_2 all T_c (class_Groups_Ocomm__monoid__add(T_c) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_c),hAPP(c_List_Onth(T_c),V_xs_2)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2))) = hAPP(c_List_Omonoid__add__class_Olistsum(T_c),V_xs_2))) # label(fact_listsum__setsum__nth) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 4075 (all V_n all T_a all V_xs (c_List_Olist_ONil(T_a) != V_xs -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))) -> hAPP(c_List_Ohd(T_a),hAPP(hAPP(c_List_Odrop(T_a),V_n),V_xs)) = hAPP(hAPP(c_List_Onth(T_a),V_xs),V_n)))) # label(fact_hd__drop__conv__nth) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 4076 (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]. 6.07/6.08 4077 (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(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)) = hAPP(hAPP(c_List_Omap(T_c,T_d),V_f_2),hAPP(c_List_OremoveAll(T_c,V_x_2),V_xs_2)))) # label(fact_map__removeAll__inj__on) # label(axiom) # label(non_clause). [assumption]. 6.07/6.08 4078 (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]. 6.07/6.08 4079 (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]. 6.07/6.08 4080 (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]. 6.07/6.08 4081 (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]. 6.07/6.08 4082 (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]. 6.07/6.08 4083 (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]. 6.07/6.08 4084 (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__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]. 6.07/6.08 4085 (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]. 6.07/6.08 4086 (all V_xs all T_a all V_ys ((V_ys = c_List_Olist_ONil(T_a) -> c_List_Obutlast(T_a,hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys)) = c_List_Obutlast(T_a,V_xs)) & (c_List_Olist_ONil(T_a) != V_ys -> c_List_Obutlast(T_a,hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys)) = hAPP(hAPP(c_List_Oappend(T_a),V_xs),c_List_Obutlast(T_a,V_ys))))) # label(fact_butlast__append) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4087 (all V_j_2 all V_i_2 all V_k_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)) = 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))) # label(fact_image__add__atLeastAtMost) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4088 (all V_B_2 all V_C_2 all T_d all V_A_2 all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_d),hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2)),V_B_2)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_d,tc_fun(T_c,tc_HOL_Obool)),V_C_2),V_B_2))) # label(fact_UN__extend__simps_I7_J) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4089 (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_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_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]. 6.07/6.09 4090 (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]. 6.07/6.09 4091 (all V_a all V_b all T_a (class_Divides_Osemiring__div(T_a) -> (V_b != c_Groups_Ozero__class_Ozero(T_a) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),V_a)),V_b) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_a),V_b)),c_Groups_Oone__class_Oone(T_a))))) # label(fact_div__add__self1) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4092 (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]. 6.07/6.09 4093 (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]. 6.07/6.09 4094 (all V_xs_2 all V_Pa_2 all T_c (c_List_OdropWhile(T_c,V_Pa_2,V_xs_2) != c_List_Olist_ONil(T_c) -> -hBOOL(hAPP(V_Pa_2,hAPP(c_List_Ohd(T_c),c_List_OdropWhile(T_c,V_Pa_2,V_xs_2)))))) # label(fact_hd__dropWhile) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4095 (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]. 6.07/6.09 4096 (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]. 6.07/6.09 4097 (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]. 6.07/6.09 4098 (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]. 6.07/6.09 4099 (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]. 6.07/6.09 4100 (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]. 6.07/6.09 4101 (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]. 6.07/6.09 4102 (all T_c (class_Groups_Omonoid__add(T_c) -> c_List_Ofoldl(T_c,T_c,c_Groups_Oplus__class_Oplus(T_c),c_Groups_Ozero__class_Ozero(T_c)) = c_List_Omonoid__add__class_Olistsum(T_c))) # label(fact_listsum__foldl) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4103 (all V_r_2 all V_g_2 all V_f_2 all T_e all T_c all T_d hAPP(c_Set_Oimage(T_e,T_c,V_f_2),hAPP(c_Set_Oimage(T_d,T_e,V_g_2),V_r_2)) = hAPP(c_Set_Oimage(T_d,T_c,hAPP(c_Fun_Ocomp(T_e,T_c,T_d,V_f_2),V_g_2)),V_r_2)) # label(fact_image__compose) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4104 (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]. 6.07/6.09 4105 (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]. 6.07/6.09 4106 (all V_k2_2 all V_k1_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k1_2),V_k2_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit1(V_k1_2)),c_Int_OBit1(V_k2_2))))) # label(fact_less__int__code_I16_J) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4107 (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]. 6.07/6.09 4108 (all V_z_2 all V_w_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_w_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_z_2),c_Groups_Oone__class_Oone(tc_Int_Oint)))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_w_2),V_z_2)))) # label(fact_zle__diff1__eq) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4109 (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]. 6.07/6.09 4110 (all V_n all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_n),V_m)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m),V_n)))) # label(fact_nat__le__linear) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4111 (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_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)))))) # label(fact_refl__on__def) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4112 (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]. 6.07/6.09 4113 (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]. 6.07/6.09 4114 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_c (class_Groups_Oab__group__add(T_c) -> (hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_c),V_a_2),V_b_2) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_c),V_c_2),V_d_2) -> (V_d_2 = V_c_2 <-> V_a_2 = V_b_2)))) # label(fact_diff__eq__diff__eq) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4115 (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) <-> c_List_Olist__all(tc_Int_Oint,V_Pa_2,c_List_Oupto(V_i_2,V_j_2)))) # label(fact_list__all__iff__all__interval__int) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4116 (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(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_folding__idem_Oidem__comp) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4117 (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(c_Groups_Ouminus__class_Ouminus(T_c),hAPP(c_Int_Oring__1__class_Oof__int(T_c),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_k_2))) = hAPP(c_Int_Oring__1__class_Oof__int(T_c),V_k_2)) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> hAPP(c_Int_Oring__1__class_Oof__int(T_c),V_k_2) = 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))))))) & (V_k_2 = c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> 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]. 6.07/6.09 4118 (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]. 6.07/6.09 4119 (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]. 6.07/6.09 4120 (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]. 6.07/6.09 4121 (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]. 6.07/6.09 4122 (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]. 6.07/6.09 4123 (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]. 6.07/6.09 4124 (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]. 6.07/6.09 4125 (all V_k_2 all V_less_2 all V_i_2 all T_c (hBOOL(hAPP(hAPP(V_less_2,V_k_2),V_i_2)) <-> hBOOL(hAPP(hAPP(c_member(T_c),V_i_2),c_SetInterval_Oord_OgreaterThan(T_c,V_less_2,V_k_2))))) # label(fact_ord_OgreaterThan__iff) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4126 (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]. 6.07/6.09 4127 (all V_a3_2 all V_a2_2 all V_a1_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_a1_2),V_a2_2)),V_a3_2)) <-> (exists B_a exists 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(hAPP(c_member(T_c),B_a),B_A)) & hAPP(hAPP(c_Set_Oinsert(T_c),V_a1_2),V_a2_2) = hAPP(hAPP(c_Set_Oinsert(T_c),B_a),B_A))))) # label(fact_fold1Set_Oequations) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4128 (all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit1(V_k_2)),c_Int_OMin)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),c_Int_OMin)))) # label(fact_rel__simps_I13_J) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4129 (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]. 6.07/6.09 4130 (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]. 6.07/6.09 4131 (all V_u_2 all V_l_2 all T_c (class_Orderings_Oorder(T_c) -> c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatMost(T_c),V_l_2)),c_SetInterval_Oord__class_OgreaterThanAtMost(T_c,V_l_2,V_u_2)))) # label(fact_ivl__disj__int_I3_J) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4132 (all V_a all V_b all T_a (class_Divides_Osemiring__div(T_a) -> (V_b != c_Groups_Ozero__class_Ozero(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),V_a),V_b)),c_Groups_Oone__class_Oone(T_a)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),V_b)))) # label(fact_div__add__self2) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4133 (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),hAPP(c_Finite__Set_Ofold1(T_c,c_Orderings_Oord__class_Omin(T_c)),V_A_2)),V_x_2)) <-> (exists B_x (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),B_x),V_x_2)) & hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2))))))))) # label(fact_fold1__strict__below__iff) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4134 (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(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)))) & (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))))) # label(fact_power__nat__number__of__number__of) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4135 (all V_xs_2 all T_c hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_List_Olist(T_c),tc_fun(T_c,tc_HOL_Obool)),hAPP(c_List_Oset(tc_List_Olist(T_c)),V_xs_2)),c_List_Oset(T_c)) = hAPP(c_List_Oset(T_c),c_List_Oconcat(T_c,V_xs_2))) # label(fact_set__concat) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4136 (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_OSUPR(T_c,tc_fun(T_d,tc_HOL_Obool)),V_A_2),V_B_2),V_b_2)) <-> (exists B_x (hBOOL(hAPP(hAPP(V_B_2,B_x),V_b_2)) & hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)))))) # label(fact_SUP1__iff) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4137 (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]. 6.07/6.09 4138 (all T_c (class_Orderings_Olinorder(T_c) & class_Lattices_Osemilattice__sup(T_c) -> c_Lattices_Osemilattice__sup__class_Osup(T_c) = c_Orderings_Oord__class_Omax(T_c))) # label(fact_sup__max) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4139 (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_OdropWhile(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),c_List_OdropWhile(T_c,V_Pa_2,V_xs_2)),V_ys_2)))) # label(fact_dropWhile__append1) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4140 (all V_q all V_p hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_p),V_q)),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_p,V_q)) = c_Divides_Odivmod__int(V_p,V_q)) # label(fact_divmod__int__mod__div) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4141 (all V_B_2 all V_A_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,tc_HOL_Obool),V_B_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),V_B_2),hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2))) # label(fact_Int__Union) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4142 (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_H),V_b_H) = hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_a),V_b) -> -(V_a = V_a_H -> V_b != V_b_H))) # label(fact_Pair__inject) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4143 (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]. 6.07/6.09 4144 (all V_k all V_x all V_n all T_a hAPP(hAPP(c_List_Oappend(T_a),c_List_Oreplicate(T_a,V_n,V_x)),c_List_Oreplicate(T_a,V_k,V_x)) = hAPP(hAPP(c_List_Oappend(T_a),c_List_Oreplicate(T_a,V_k,V_x)),c_List_Oreplicate(T_a,V_n,V_x))) # label(fact_append__replicate__commute) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4145 (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]. 6.07/6.09 4146 (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]. 6.07/6.09 4147 (all V_xs_2 all V_x_2 all V_Qa_2 all T_c c_List_Oremove1(T_c,V_x_2,hAPP(c_List_Ofilter(T_c,V_Qa_2),V_xs_2)) = hAPP(c_List_Ofilter(T_c,V_Qa_2),c_List_Oremove1(T_c,V_x_2,V_xs_2))) # label(fact_filter__remove1) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4148 (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]. 6.07/6.09 4149 (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]. 6.07/6.09 4150 (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]. 6.07/6.09 4151 (all V_n_2 all V_m_2 all V_k_2 (V_n_2 = V_m_2 | V_k_2 = c_Groups_Ozero__class_Ozero(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__eq__cancel__disj) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4152 (all V_nat_H c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(c_Nat_OSuc,V_nat_H)) # label(fact_nat_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4153 (all V_C_2 all V_B_2 all V_A_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_C_2)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_B_2),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_C_2))) # label(fact_Int__left__commute) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4154 (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]. 6.07/6.09 4155 (all V_n_2 all V_m_2 (V_m_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) | (exists B_j (V_m_2 = hAPP(c_Nat_OSuc,B_j) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_j),V_n_2)))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),hAPP(c_Nat_OSuc,V_n_2))))) # label(fact_less__Suc__eq__0__disj) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4156 (all V_z 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_z),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_x)) & hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_x),V_z))))) # label(fact_dvd_Oless__trans) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4157 (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]. 6.07/6.09 4158 (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]. 6.07/6.09 4159 (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]. 6.07/6.09 4160 (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]. 6.07/6.09 4161 (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]. 6.07/6.09 4162 (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]. 6.07/6.09 4163 (all V_xss all T_a c_List_Otranspose(T_a,V_xss) = c_List_Otranspose(T_a,hAPP(hAPP(c_List_Olist_OCons(tc_List_Olist(T_a)),c_List_Olist_ONil(T_a)),V_xss))) # label(fact_transpose_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4164 (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]. 6.07/6.09 4165 (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]. 6.07/6.09 4166 (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]. 6.07/6.09 4167 (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]. 6.07/6.09 4168 (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_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)) = 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)))))))) # label(fact_min__max_OInf__fin_Ounion__inter) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4169 (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]. 6.07/6.09 4170 (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]. 6.07/6.09 4171 (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)) & (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__def) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4172 (all V_c all V_b all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),V_c) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_c)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_c)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4173 (all V_xs all T_a c_Enum_On__lists(T_a,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),V_xs) = hAPP(hAPP(c_List_Olist_OCons(tc_List_Olist(T_a)),c_List_Olist_ONil(T_a)),c_List_Olist_ONil(tc_List_Olist(T_a)))) # label(fact_n__lists_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4174 (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]. 6.07/6.09 4175 (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]. 6.07/6.09 4176 (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]. 6.07/6.09 4177 (all V_a all T_a (class_Groups_Omonoid__mult(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_a))) # label(fact_power2__eq__square) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4178 (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]. 6.07/6.09 4179 (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]. 6.07/6.09 4180 (all V_v_2 all V_i_2 all V_xs_2 all V_x_2 all T_c c_Nat_Onat_Onat__case(tc_List_Olist(T_c),hAPP(hAPP(c_List_Olist_OCons(T_c),V_v_2),V_xs_2),hAPP(hAPP(c_COMBB(tc_List_Olist(T_c),tc_List_Olist(T_c),tc_Nat_Onat),hAPP(c_List_Olist_OCons(T_c),V_x_2)),hAPP(hAPP(c_COMBC(tc_Nat_Onat,T_c,tc_List_Olist(T_c)),c_List_Olist__update(T_c,V_xs_2)),V_v_2)),V_i_2) = hAPP(hAPP(c_List_Olist__update(T_c,hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2)),V_i_2),V_v_2)) # label(fact_list__update_Osimps_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4181 (all V_b all V_a all T_a (class_Rings_Oring(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)),V_b) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)))) # label(fact_minus__mult__left) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4182 (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]. 6.07/6.09 4183 (all V_ys all V_xs all T_a hAPP(c_List_Orev(T_a),hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys)) = hAPP(hAPP(c_List_Oappend(T_a),hAPP(c_List_Orev(T_a),V_ys)),hAPP(c_List_Orev(T_a),V_xs))) # label(fact_rev__append) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4184 (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]. 6.07/6.09 4185 (all V_xs_2 all V_x_2 all V_zs_2 all V_ys_2 all T_c (hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2) = hAPP(hAPP(c_List_Oappend(T_c),V_ys_2),V_zs_2) <-> (exists B_ys_H (hAPP(hAPP(c_List_Oappend(T_c),B_ys_H),V_zs_2) = V_xs_2 & hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),B_ys_H) = V_ys_2)) | 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]. 6.07/6.09 4186 (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]. 6.07/6.09 4187 (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_B_2)),hAPP(V_F_2,V_A_2)) = hAPP(c_Fun_Ocomp(T_d,T_d,T_d,hAPP(V_F_2,V_A_2)),hAPP(V_F_2,V_B_2)))))) # label(fact_folding_Ocommute__comp_H_H) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4188 (all V_y_2 (hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),V_y_2),c_Orderings_Otop__class_Otop(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))) -> V_y_2 = hAPP(c_Code__Numeral_Onat__of,hAPP(c_Code__Numeral_Oof__nat,V_y_2)))) # label(fact_of__nat__inverse) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4189 (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]. 6.07/6.09 4190 (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]. 6.07/6.09 4191 (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]. 6.07/6.09 4192 (all V_l_2 all V_Pa_2 all T_c (class_Rings_Osemiring__0(T_c) & class_Rings_Odvd(T_c) -> ((exists B_x hBOOL(hAPP(V_Pa_2,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_l_2),B_x)))) <-> (exists B_x (hBOOL(hAPP(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]. 6.07/6.09 4193 (all V_xs_2 all V_x_2 all V_f_2 all T_c all T_d hAPP(hAPP(c_List_Oappend(T_c),hAPP(V_f_2,V_x_2)),c_List_Omaps(T_d,T_c,V_f_2,V_xs_2)) = c_List_Omaps(T_d,T_c,V_f_2,hAPP(hAPP(c_List_Olist_OCons(T_d),V_x_2),V_xs_2))) # label(fact_maps__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4194 (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]. 6.07/6.09 4195 (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]. 6.07/6.09 4196 (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_g_2),V_x_2) = hAPP(c_Product__Type_Oapfst(T_e,T_c,T_d,V_f_2),V_x_2) <-> hAPP(V_g_2,hAPP(c_Product__Type_Ofst(T_e,T_d),V_x_2)) = hAPP(V_f_2,hAPP(c_Product__Type_Ofst(T_e,T_d),V_x_2)))) # label(fact_apfst__eq__conv) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4197 (all V_a_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> V_a_2 = hAPP(hAPP(c_Orderings_Oord__class_Omax(T_c),V_a_2),c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),V_A_2)))))) # label(fact_min__max_Osup__Inf__absorb) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4198 (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]. 6.07/6.09 4199 (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]. 6.07/6.09 4200 (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]. 6.07/6.09 4201 (all V_a_2 all V_w_2 all V_b_2 all T_c (class_Int_Onumber(T_c) & class_Fields_Olinordered__field__inverse__zero(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_b_2),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2))),V_a_2)) <-> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2)),c_Groups_Ozero__class_Ozero(T_c))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(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)))))))) # label(fact_divide__less__eq__number__of1) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4202 (all V_j all V_i ((-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),V_j)) -> c_List_Olist_ONil(tc_Nat_Onat) = c_List_Oupt(V_i,V_j)) & (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]. 6.07/6.09 4203 (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]. 6.07/6.09 4204 (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]. 6.07/6.09 4205 (all V_S_2 all V_R_2 all T_d all T_c (hAPP(hAPP(c_COMBC(T_c,tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_fun(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),tc_HOL_Obool)),tc_fun(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool)),T_c),c_COMBC(T_d,tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_prod(T_c,T_d)),tc_fun(T_d,tc_fun(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),tc_HOL_Obool)),T_c),hAPP(c_COMBB(tc_prod(T_c,T_d),tc_fun(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool),tc_HOL_Obool),T_d),c_member(tc_prod(T_c,T_d)))),c_Product__Type_OPair(T_c,T_d)))),V_S_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) <-> V_S_2 = V_R_2)) # label(fact_pred__equals__eq2) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4206 (all V_v_2 all V_v_H_2 ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_v_H_2),c_Int_OPls)) -> hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H_2))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_v_H_2),c_Int_OPls)) -> hAPP(hAPP(c_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)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H_2))))) # label(fact_diff__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4207 (all V_xs_2 all T_c hAPP(c_List_Orev(T_c),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_Orev(T_c)),hAPP(c_List_Orev(tc_List_Olist(T_c)),V_xs_2)))) # label(fact_rev__concat) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4208 (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]. 6.07/6.09 4209 (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]. 6.07/6.09 4210 (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]. 6.07/6.09 4211 (all V_u_2 all V_l_2 all V_less_2 all V_less__eq_2 all V_i_2 all T_c (hBOOL(hAPP(hAPP(V_less__eq_2,V_i_2),V_u_2)) & hBOOL(hAPP(hAPP(V_less_2,V_l_2),V_i_2)) <-> hBOOL(hAPP(hAPP(c_member(T_c),V_i_2),c_SetInterval_Oord_OgreaterThanAtMost(T_c,V_less__eq_2,V_less_2,V_l_2,V_u_2))))) # label(fact_ord_OgreaterThanAtMost__iff) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4212 (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]. 6.07/6.09 4213 (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_z_2 = V_w_2)) # label(fact_zless__add1__eq) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4214 (all V_b all V_a ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_b),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_a),V_b)) -> hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_a) = c_Divides_OposDivAlg(V_a,V_b)) & (-(hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_b),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_a),V_b))) -> c_Divides_OposDivAlg(V_a,V_b) = hAPP(c_Divides_Oadjust(V_b),c_Divides_OposDivAlg(V_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_b)))))) # label(fact_posDivAlg_Osimps) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4215 (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]. 6.07/6.09 4216 (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]. 6.07/6.09 4217 (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]. 6.07/6.09 4218 (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]. 6.07/6.09 4219 (all V_f2_2 all V_f1_2 all T_d all T_c V_f1_2 = 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))) # label(fact_lazy__sequence_Orecs_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4220 (all T_c c_List_Oset(T_c) = c_List_Omember(T_c)) # label(fact_member__set) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4221 (all V_A_2 all T_d all T_c hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))),hAPP(c_COMBK(tc_fun(T_d,tc_HOL_Obool),T_c),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_d,tc_HOL_Obool)),V_A_2))) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))),hAPP(c_COMBK(tc_fun(T_d,tc_HOL_Obool),T_c),V_A_2)))) # label(fact_Compl__Times__UNIV1) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4222 (all V_l_2 all V_k_2 (V_k_2 = V_l_2 <-> hAPP(c_Code__Numeral_Onat__of,V_l_2) = hAPP(c_Code__Numeral_Onat__of,V_k_2))) # label(fact_Code__Numeral_Onat__of__inject) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4223 (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(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_b)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)),hAPP(c_Groups_Oabs__class_Oabs(T_a),V_b))))) # label(fact_abs__eq__mult) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4224 (all V_xs_2 all V_x_2 all T_c (class_Lattices_Olattice(T_c) -> hAPP(c_List_Ofoldl(T_c,T_c,c_Lattices_Osemilattice__inf__class_Oinf(T_c),V_x_2),V_xs_2) = c_Big__Operators_Olattice__class_OInf__fin(T_c,hAPP(c_List_Oset(T_c),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2))))) # label(fact_Inf__fin__set__fold) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4225 (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]. 6.07/6.09 4226 (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]. 6.07/6.09 4227 (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]. 6.07/6.09 4228 (all V_n_2 all V_m_2 (c_Groups_Oone__class_Oone(tc_Nat_Onat) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m_2),V_n_2) <-> c_Groups_Oone__class_Oone(tc_Nat_Onat) = V_m_2 & V_n_2 = c_Groups_Oone__class_Oone(tc_Nat_Onat))) # label(fact_nat__1__eq__mult__iff) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4229 (all V_z all V_w all T_a (class_Rings_Oring__1(T_a) -> hAPP(c_Int_Oring__1__class_Oof__int(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_w),V_z)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Int_Oring__1__class_Oof__int(T_a),V_w)),hAPP(c_Int_Oring__1__class_Oof__int(T_a),V_z)))) # label(fact_of__int__add) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4230 (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))) -> hAPP(c_Product__Type_OPair(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_HOL_Obool,tc_Option_Ooption(tc_Lazy__Sequence_Olazy__sequence(T_c)))),tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral)),c_DSequence_Oempty(T_c)) = c_Random__Sequence_ORandom(T_c,V_g_2,V_nrandom_2,V_size_2)) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),V_nrandom_2),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral))) -> hAPP(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))))))) = c_Random__Sequence_ORandom(T_c,V_g_2,V_nrandom_2,V_size_2)))) # label(fact_Random_Osimps) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4231 (all V_c_2 all V_b_2 all V_w_2 all T_c (class_Fields_Ofield__inverse__zero(T_c) & class_Int_Onumber(T_c) -> ((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) -> 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(c_Int_Onumber__class_Onumber__of(T_c),V_w_2) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),V_b_2),V_c_2)))) # label(fact_eq__divide__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4232 (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]. 6.07/6.09 4233 (all V_m_2 all V_w_2 ((hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_w_2)) -> V_w_2 = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_m_2)) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_w_2)) -> 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]. 6.07/6.09 4234 (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]. 6.07/6.09 4235 (all V_n all V_m ((hAPP(c_Nat_OSuc,c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,V_n)) != V_n -> c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_m),V_n) = 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)) = 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]. 6.07/6.09 4236 (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]. 6.07/6.09 4237 (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]. 6.07/6.09 4238 (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(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]. 6.07/6.09 4239 (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]. 6.07/6.09 4240 (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]. 6.07/6.09 4241 (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) -> 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)) = 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))))))))) # label(fact_Inf__fin_Ounion__disjoint) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4242 (all V_z all T_a (class_Int_Onumber__ring(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_z),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OMin)) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_z))) # label(fact_mult__minus1__right) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4243 (all V_ys_2 all V_xs_2 all V_Pa_2 all T_d all T_c (c_List_Olist__all2(T_c,T_d,V_Pa_2,V_xs_2,V_ys_2) <-> c_List_Olist__all2(T_c,T_d,V_Pa_2,hAPP(c_List_Orev(T_c),V_xs_2),hAPP(c_List_Orev(T_d),V_ys_2)))) # label(fact_list__all2__rev) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4244 (all V_ys all V_xs all V_n all T_a hAPP(hAPP(c_List_Odrop(T_a),V_n),hAPP(hAPP(c_List_Oappend(T_a),V_xs),V_ys)) = hAPP(hAPP(c_List_Oappend(T_a),hAPP(hAPP(c_List_Odrop(T_a),V_n),V_xs)),hAPP(hAPP(c_List_Odrop(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))),V_ys))) # label(fact_drop__append) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4245 (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]. 6.07/6.09 4246 (all V_a 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),V_a),c_Groups_Oone__class_Oone(T_a))),V_m) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_m),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),V_m)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I3_J) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4247 (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,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_ys_2 = V_xs_2))) # label(fact_map__inj__on) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4248 (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_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))) = 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__inter) # label(axiom) # label(non_clause). [assumption]. 6.07/6.09 4249 (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]. 6.07/6.10 4250 (all V_xs_2 all T_c c_List_Ofoldr(tc_List_Olist(T_c),tc_Nat_Onat,hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat),tc_List_Olist(T_c)),c_Orderings_Oord__class_Omax(tc_Nat_Onat)),c_Nat_Osize__class_Osize(tc_List_Olist(T_c))),V_xs_2,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_List_Olist(T_c))),c_List_Otranspose(T_c,V_xs_2))) # label(fact_length__transpose) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4251 (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]. 6.07/6.10 4252 (all V_k V_k = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_k),c_Int_OPls)) # label(fact_diff__bin__simps_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4253 (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(c_Nat_OSuc,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n))) # label(fact_add__Suc__right) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4254 (all V_y_2 all V_x_2 all V_f_2 all T_d all T_c all T_e hAPP(c_Product__Type_Oapfst(T_e,T_c,T_d,V_f_2),hAPP(hAPP(c_Product__Type_OPair(T_e,T_d),V_x_2),V_y_2)) = hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),hAPP(V_f_2,V_x_2)),V_y_2)) # label(fact_apfst__conv) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4255 (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]. 6.07/6.10 4256 (all V_a_2 all T_c all V_A_2 (V_A_2 = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> -hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)))) # label(fact_equals0D) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4257 (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]. 6.07/6.10 4258 (all T_c c_Relation_OId(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)))))) # label(fact_Id__def) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4259 (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]. 6.07/6.10 4260 (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]. 6.07/6.10 4261 (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),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)))) <-> (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(hAPP(c_member(T_c),B_x),V_A_2)) & hBOOL(hAPP(V_Pa_2,B_x)))))) # label(fact_bex__Un) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4262 (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]. 6.07/6.10 4263 (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]. 6.07/6.10 4264 (all V_A_2 all V_t_2 all V_D_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_D_2)) -> (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),V_t_2),V_A_2)) -> (all B_x ((all B_xa (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xa),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),V_D_2))) -> (all B_xb (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xb),V_A_2)) -> B_x != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B_xb),B_xa))))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B_x),V_t_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B_x),V_D_2)),V_t_2)))))))) # label(fact_aset_I5_J) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4265 (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]. 6.07/6.10 4266 (all V_P all T_a hAPP(c_COMBI(T_a),V_P) = V_P) # label(help_c__COMBI__1) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4267 (all V_y all V_x all V_z all T_a (class_Fields_Ofield(T_a) -> (V_z != c_Groups_Ozero__class_Ozero(T_a) -> hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),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]. 6.07/6.10 4268 (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]. 6.07/6.10 4269 (all V_xs all V_n all T_a c_List_Obutlast(T_a,hAPP(hAPP(c_List_Odrop(T_a),V_n),V_xs)) = hAPP(hAPP(c_List_Odrop(T_a),V_n),c_List_Obutlast(T_a,V_xs))) # label(fact_drop__butlast) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4270 (all V_A_2 all T_d all T_c hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),V_A_2),hAPP(c_COMBK(tc_fun(T_d,tc_HOL_Obool),T_c),c_Orderings_Obot__class_Obot(tc_fun(T_d,tc_HOL_Obool)))) = c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool))) # label(fact_Sigma__empty2) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4271 (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]. 6.07/6.10 4272 (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]. 6.07/6.10 4273 (all V_g_2 all T_e all V_B_2 all V_A_2 all V_F_2 all V_f_2 all T_d all T_c (c_Finite__Set_Ofolding(T_c,T_d,V_f_2,V_F_2) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> hAPP(c_Fun_Ocomp(T_d,T_d,T_e,hAPP(V_F_2,V_A_2)),hAPP(c_Fun_Ocomp(T_d,T_d,T_e,hAPP(V_F_2,V_B_2)),V_g_2)) = hAPP(c_Fun_Ocomp(T_d,T_d,T_e,hAPP(V_F_2,V_B_2)),hAPP(c_Fun_Ocomp(T_d,T_d,T_e,hAPP(V_F_2,V_A_2)),V_g_2)))))) # label(fact_folding_Ocommute__left__comp_H_H) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4274 (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]. 6.07/6.10 4275 (all V_k hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Int_OBit1(V_k)) = c_Int_OBit1(c_Int_Opred(hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_k)))) # label(fact_minus__Bit1) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4276 (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]. 6.07/6.10 4277 (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]. 6.07/6.10 4278 (all V_m hAPP(c_Nat_OSuc,V_m) != c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) # label(fact_Suc__not__Zero) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4279 (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]. 6.07/6.10 4280 (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]. 6.07/6.10 4281 (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_z_2 = V_y_2)))) # label(fact_single__valuedD) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4282 (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]. 6.07/6.10 4283 (all V_B_2 all V_A_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)),V_B_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) & hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_B_2)))) # label(fact_insert__subset) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4284 (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]. 6.07/6.10 4285 (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]. 6.07/6.10 4286 (all V_s_2 all V_r_2 all T_c all T_d 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)) = 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))) # label(fact_converse__meet) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4287 (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]. 6.07/6.10 4288 (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_equal__neg__zero) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4289 (all V_A_2 all T_d all T_c hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(T_c,tc_HOL_Obool)),V_A_2)),hAPP(c_COMBK(tc_fun(T_d,tc_HOL_Obool),T_c),c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool)))) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(tc_prod(T_c,T_d),tc_HOL_Obool)),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),V_A_2),hAPP(c_COMBK(tc_fun(T_d,tc_HOL_Obool),T_c),c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool)))))) # label(fact_Compl__Times__UNIV2) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4290 (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]. 6.07/6.10 4291 (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]. 6.07/6.10 4292 (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]. 6.07/6.10 4293 (all V_n all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_m),V_n)) -> (V_n != V_m -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n))))) # label(fact_le__neq__implies__less) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4294 (all V_n hAPP(c_Nat_OSuc,V_n) != V_n) # label(fact_Suc__n__not__n) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4295 (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)) -> 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(c_fequal,V_a_2))),V_b_2)),c_Groups_Ozero__class_Ozero(T_d))),V_S_2))))) # label(fact_setsum__delta_H) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4296 (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]. 6.07/6.10 4297 (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]. 6.07/6.10 4298 (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]. 6.07/6.10 4299 (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]. 6.07/6.10 4300 (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]. 6.07/6.10 4301 (all V_w_2 hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(V_w_2)) = hAPP(hAPP(c_HOL_OLet(tc_Nat_Onat,tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_w_2)),hAPP(hAPP(c_COMBS(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),c_COMBI(tc_Nat_Onat)))) # label(fact_nat__number__of__Bit0) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4302 (all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> V_a = hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a)))) # label(fact_abs__of__pos) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4303 (all V_y all V_x all T_a (class_Orderings_Opreorder(T_a) -> (V_y = V_x -> 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]. 6.07/6.10 4304 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_c (class_Groups_Oordered__ab__group__add(T_c) -> (hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_c),V_c_2),V_d_2) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_c),V_a_2),V_b_2) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_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]. 6.07/6.10 4305 (all V_ys_2 all V_y_2 all V_xs_2 all 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) <-> 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))))) # label(fact_rev__eq__Cons__iff) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4306 (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_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(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__Int) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4307 (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]. 6.07/6.10 4308 (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]. 6.07/6.10 4309 (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]. 6.07/6.10 4310 (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))) -> 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(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))) -> hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)) = hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_v)))) # label(fact_of__nat__number__of__eq) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4311 (all V_b_2 all V_a_2 (hBOOL(hAPP(c_Wellfounded_Oaccp(tc_prod(tc_Int_Oint,tc_Int_Oint),c_Nitpick_Onorm__frac__rel),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_a_2),V_b_2))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> c_Nitpick_Onorm__frac(V_a_2,V_b_2) = c_Nitpick_Onorm__frac(hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a_2),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_b_2))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_b_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> (-(V_b_2 = c_Groups_Ozero__class_Ozero(tc_Int_Oint) | V_a_2 = c_Groups_Ozero__class_Ozero(tc_Int_Oint)) -> c_Nitpick_Onorm__frac(V_a_2,V_b_2) = hAPP(hAPP(c_HOL_OLet(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint)),hAPP(hAPP(c_Nitpick_Oint__gcd,V_a_2),V_b_2)),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint)),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint)),tc_Int_Oint),c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint)),hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a_2))),hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_b_2)))) & (c_Groups_Ozero__class_Ozero(tc_Int_Oint) = V_b_2 | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = V_a_2 -> hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Groups_Oone__class_Oone(tc_Int_Oint)) = c_Nitpick_Onorm__frac(V_a_2,V_b_2))))) # label(fact_norm__frac_Opsimps) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4312 (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]. 6.07/6.10 4313 (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) -> V_x = 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))) & (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)) = hAPP(hAPP(c_List_Onth(T_a),V_xs),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),c_Groups_Oone__class_Oone(tc_Nat_Onat)))))) # label(fact_nth__Cons__number__of) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4314 (all V_y all V_x all T_a (class_Orderings_Oorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y)) -> V_y != V_x))) # label(fact_order__less__imp__not__eq2) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4315 (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]. 6.07/6.10 4316 (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 (hAPP(hAPP(c_Product__Type_OPair(T_c,T_d),hAPP(V_f_2,B_x)),hAPP(V_g_2,B_y)) = V_c_2 -> -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]. 6.07/6.10 4317 (all V_xss_2 all T_c 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)) = c_List_Oconcat(tc_List_Olist(T_c),hAPP(hAPP(c_List_Omap(tc_List_Olist(T_c),tc_List_Olist(tc_List_Olist(T_c))),c_List_Olist_Olist__case(tc_List_Olist(tc_List_Olist(T_c)),T_c,c_List_Olist_ONil(tc_List_Olist(T_c)),hAPP(c_COMBK(tc_fun(tc_List_Olist(T_c),tc_List_Olist(tc_List_Olist(T_c))),T_c),hAPP(hAPP(c_COMBC(tc_List_Olist(T_c),tc_List_Olist(tc_List_Olist(T_c)),tc_List_Olist(tc_List_Olist(T_c))),c_List_Olist_OCons(tc_List_Olist(T_c))),c_List_Olist_ONil(tc_List_Olist(T_c)))))),V_xss_2))) # label(fact_transpose__aux__filter__tail) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4318 (all V_xs_2 all V_r_2 all T_d all T_c (class_Rings_Osemiring__1(T_c) -> hAPP(c_List_Omonoid__add__class_Olistsum(T_c),hAPP(hAPP(c_List_Omap(T_d,T_c),hAPP(c_COMBK(T_c,T_d),V_r_2)),V_xs_2)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_c),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_d)),V_xs_2))),V_r_2))) # label(fact_listsum__triv) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4319 (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]. 6.07/6.10 4320 (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]. 6.07/6.10 4321 (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]. 6.07/6.10 4322 (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]. 6.07/6.10 4323 (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]. 6.07/6.10 4324 (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]. 6.07/6.10 4325 (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]. 6.07/6.10 4326 (all V_i_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),V_i_2)) <-> hAPP(c_Code__Numeral_Oof__nat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) != V_i_2)) # label(fact_code__numeral__not__eq__zero) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4327 (all T_c all V_xs_2 (c_List_Olist_ONil(T_c) != V_xs_2 -> hBOOL(hAPP(hAPP(c_member(T_c),hAPP(c_List_Ohd(T_c),V_xs_2)),hAPP(c_List_Oset(T_c),V_xs_2))))) # label(fact_hd__in__set) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4328 (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]. 6.07/6.10 4329 (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]. 6.07/6.10 4330 (all V_xs_2 all V_Pa_2 all T_c ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(c_List_Oset(T_c),V_xs_2))) -> hBOOL(hAPP(V_Pa_2,B_x)))) <-> c_List_Olist_ONil(T_c) = c_List_OdropWhile(T_c,V_Pa_2,V_xs_2))) # label(fact_dropWhile__eq__Nil__conv) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4331 (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_OatLeastAtMost(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_I16_J) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4332 (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]. 6.07/6.10 4333 (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]. 6.07/6.10 4334 (all V_z_2 all V_w_2 all T_c (class_Int_Oring__char__0(T_c) -> (V_z_2 = V_w_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]. 6.07/6.10 4335 (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]. 6.07/6.10 4336 (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]. 6.07/6.10 4337 (all V_z all V_y all V_x all T_a (class_Lattices_Osemilattice__inf(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),V_y)),V_z) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),V_y),V_z)))) # label(fact_inf__assoc) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4338 (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]. 6.07/6.10 4339 (all V_a_2 all V_b_2 (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_b_2),V_a_2)) & -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_a_2),V_b_2)) -> c_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]. 6.07/6.10 4340 (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]. 6.07/6.10 4341 (all V_xs_2 all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> c_Complete__Lattice_OInf__class_OInf(T_c,hAPP(c_List_Oset(T_c),V_xs_2)) = hAPP(c_List_Ofoldl(T_c,T_c,c_Lattices_Osemilattice__inf__class_Oinf(T_c),c_Orderings_Otop__class_Otop(T_c)),V_xs_2))) # label(fact_Inf__set__fold) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4342 (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]. 6.07/6.10 4343 (all V_k V_k = hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_k)) # label(fact_number__of__is__id) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4344 (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]. 6.07/6.10 4345 (all V_b_2 all V_a_2 all T_c (class_Groups_Oordered__ab__group__add(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_a_2),V_b_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_c),V_a_2),V_b_2)),c_Groups_Ozero__class_Ozero(T_c)))))) # label(fact_le__iff__diff__le__0) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4346 (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]. 6.07/6.10 4347 (all V_n_2 all V_k_2 all V_m_2 all V_f_2 all T_c (class_Groups_Ocomm__monoid__add(T_c) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_c),hAPP(hAPP(c_COMBB(tc_Nat_Onat,T_c,tc_Nat_Onat),V_f_2),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),V_k_2))),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,V_m_2,V_n_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_c),V_f_2),c_SetInterval_Oord__class_OatLeastAtMost(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m_2),V_k_2),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n_2),V_k_2))))) # label(fact_setsum__shift__bounds__cl__nat__ivl) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4348 (all V_n all V_v ((hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))) -> hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),V_n) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n)) & (-hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))) -> hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),V_n) = hAPP(c_Int_Onat,hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v)),V_n))))) # label(fact_power__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4349 (all T_d all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(T_c,T_d)),c_Orderings_Otop__class_Otop(tc_fun(tc_fun(T_c,T_d),tc_HOL_Obool)))) -> (hAPP(c_Finite__Set_Ocard(T_d),c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool))) != hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))))))) # label(fact_finite__fun__UNIVD1) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4350 (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]. 6.07/6.10 4351 (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]. 6.07/6.10 4352 (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_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__simps_I5_J) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4353 (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]. 6.07/6.10 4354 (all V_l_2 all V_k_2 ((V_k_2 != c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> (c_Groups_Ozero__class_Ozero(tc_Int_Oint) != V_l_2 -> c_Divides_Odivmod__int(V_k_2,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_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_k_2 = c_Groups_Ozero__class_Ozero(tc_Int_Oint) -> hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),c_Groups_Ozero__class_Ozero(tc_Int_Oint)) = c_Divides_Odivmod__int(V_k_2,V_l_2)))) # label(fact_divmod__int__pdivmod) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4355 (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_a_2 = V_c_2 -> (V_b_2 = V_d_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]. 6.07/6.10 4356 (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]. 6.07/6.10 4357 (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]. 6.07/6.10 4358 (all V_a_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> c_Big__Operators_Olinorder__class_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_Max_Oeq__fold__idem_H) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4359 (all V_A_2 all V_a_2 all T_c c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_A_2)) # label(fact_insert__not__empty) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4360 (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]. 6.07/6.10 4361 (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]. 6.07/6.10 4362 (all V_c_2 all V_b_2 all V_B_2 all V_A_2 all T_e all T_d all T_c ((all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_A_2)) -> hBOOL(hAPP(hAPP(hAPP(V_B_2,B_x),V_b_2),V_c_2)))) <-> hBOOL(hAPP(hAPP(hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_c,tc_fun(T_d,tc_fun(T_e,tc_HOL_Obool))),V_A_2),V_B_2),V_b_2),V_c_2)))) # label(fact_INF2__iff) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4363 (all V_n_2 all V_m_2 (V_n_2 = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) & V_m_2 = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) <-> hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m_2),V_n_2) = hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_mult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4364 (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]. 6.07/6.10 4365 (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]. 6.07/6.10 4366 (all V_l_2 (c_Int_OBit0(V_l_2) = c_Int_OPls <-> V_l_2 = c_Int_OPls)) # label(fact_rel__simps_I38_J) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4367 (all V_n_2 all V_Pa_2 all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) -> (hBOOL(hAPP(hAPP(V_Pa_2,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_n_2),V_k_2)),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_n_2,V_k_2))) <-> (all B_i all B_j (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))))))) # label(fact_split__neg__lemma) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4368 (all V_X_2 all T_c c_Set_Othe__elem(T_c,V_X_2) = 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)))))) # label(fact_the__elem__def) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4369 (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]. 6.07/6.10 4370 (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_n_2) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_c),V_m_2) <-> V_n_2 = V_m_2))) # label(fact_of__nat__eq__iff) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4371 (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]. 6.07/6.10 4372 (all V_n all V_m (V_m = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n) -> V_n = c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_add__eq__self__zero) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4373 (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]. 6.07/6.10 4374 (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]. 6.07/6.10 4375 (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]. 6.07/6.10 4376 (all V_b all V_a all T_a (class_Groups_Ogroup__add(T_a) -> V_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))) # label(fact_add__diff__cancel) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4377 (all V_B_2 all V_J_2 all T_e all V_A_2 all V_I_2 all T_d all T_c hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool)),V_I_2),V_A_2)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_e,tc_fun(T_c,tc_HOL_Obool)),V_J_2),V_B_2)) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool)),V_I_2),hAPP(hAPP(c_COMBB(tc_fun(T_e,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_c,tc_HOL_Obool),T_d),hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_e,tc_fun(T_c,tc_HOL_Obool)),V_J_2)),hAPP(hAPP(c_COMBC(T_d,tc_fun(T_e,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_e,tc_fun(T_c,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),tc_fun(tc_fun(T_e,tc_fun(T_c,tc_HOL_Obool)),tc_fun(T_e,tc_fun(T_c,tc_HOL_Obool))),T_d),c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_e)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),T_d),c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool))),V_A_2))),V_B_2)))) # label(fact_Int__UN__distrib2) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4378 (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]. 6.07/6.10 4379 (all V_u_2 all V_l_2 all V_less__eq_2 all V_i_2 all T_c (hBOOL(hAPP(hAPP(V_less__eq_2,V_i_2),V_u_2)) & hBOOL(hAPP(hAPP(V_less__eq_2,V_l_2),V_i_2)) <-> 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))))) # label(fact_ord_OatLeastAtMost__iff) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4380 (all V_y all V_x all T_a (class_Orderings_Oorder(T_a) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_x),V_y)) -> V_y != V_x))) # label(fact_order__less__imp__not__eq) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4381 (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_X_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) -> (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]. 6.07/6.10 4382 (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]. 6.07/6.10 4383 (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]. 6.07/6.10 4384 (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]. 6.07/6.10 4385 (all V_m_2 hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool))) # label(fact_atLeastLessThan0) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4386 (all V_r_2 all T_c (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))) <-> c_Relation_Oantisym(T_c,V_r_2))) # label(fact_antisym__reflcl) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4387 (all V_x all V_j all V_xs all T_a all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_i),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))) -> (V_i != V_j -> hAPP(hAPP(c_List_Onth(T_a),V_xs),V_j) = hAPP(hAPP(c_List_Onth(T_a),hAPP(hAPP(c_List_Olist__update(T_a,V_xs),V_i),V_x)),V_j)) & (V_i = V_j -> 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]. 6.07/6.10 4388 (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]. 6.07/6.10 4389 (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]. 6.07/6.10 4390 (all V_z hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),V_z) = V_z) # label(fact_zmult__1) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4391 (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]. 6.07/6.10 4392 (all V_y_2 all V_x_2 all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,V_A_2) -> (V_y_2 = hAPP(V_f_2,V_x_2) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> hAPP(c_Fun_Othe__inv__into(T_c,T_d,V_A_2,V_f_2),V_y_2) = V_x_2)))) # label(fact_the__inv__into__f__eq) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4393 (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]. 6.07/6.10 4394 (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]. 6.07/6.10 4395 (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]. 6.07/6.10 4396 (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]. 6.07/6.10 4397 (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]. 6.07/6.10 4398 (all V_A_2 all V_t_2 all V_D_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_D_2)) -> (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_t_2),c_Groups_Oone__class_Oone(tc_Int_Oint))),V_A_2)) -> ((all B_x (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_x),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),V_D_2))) -> (all B_xa (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xa),V_A_2)) -> V_t_2 != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B_xa),B_x))))) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_t_2),V_D_2) = V_t_2)))) # label(fact_aset_I3_J) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4399 (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_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_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__Un__disjoint) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4400 (all V_y_2 all T_c (class_Int_Onumber__ring(T_c) -> (c_Groups_Ozero__class_Ozero(T_c) = hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_y_2) <-> c_Int_Oiszero(T_c,hAPP(c_Int_Onumber__class_Onumber__of(T_c),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Int_OPls),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_y_2))))))) # label(fact_eq__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4401 (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]. 6.07/6.10 4402 (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]. 6.07/6.10 4403 (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]. 6.07/6.10 4404 (all V_no_2 all V_yes_2 all V_xs_2 all V_Pa_2 all T_c (c_List_Opartition(T_c,V_Pa_2,V_xs_2) = hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_yes_2),V_no_2) -> (all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(c_List_Oset(T_c),V_no_2))) -> -hBOOL(hAPP(V_Pa_2,B_x)))) & (all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(c_List_Oset(T_c),V_yes_2))) -> hBOOL(hAPP(V_Pa_2,B_x)))))) # label(fact_partition__P) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4405 (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]. 6.07/6.10 4406 (all V_y all V_x all T_a (class_Lattices_Olattice(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_a),V_x),V_y))) # label(fact_inf__sup__aci_I8_J) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4407 (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]. 6.07/6.10 4408 (all V_A_2 all V_r_2 all T_c (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 -> (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]. 6.07/6.10 4409 (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]. 6.07/6.10 4410 (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]. 6.07/6.10 4411 (all V_xs_2 all V_f_2 all T_d all T_c (c_List_Olist_ONil(T_c) = hAPP(hAPP(c_List_Omap(T_d,T_c),V_f_2),V_xs_2) <-> V_xs_2 = c_List_Olist_ONil(T_d))) # label(fact_Nil__is__map__conv) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4412 (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]. 6.07/6.10 4413 (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]. 6.07/6.10 4414 (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]. 6.07/6.10 4415 (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]. 6.07/6.10 4416 (all V_xs_2 all V_x_2 all V_n_2 all T_c c_Nat_Onat_Onat__case(tc_List_Olist(T_c),c_List_Olist_ONil(T_c),hAPP(hAPP(c_COMBB(tc_List_Olist(T_c),tc_List_Olist(T_c),tc_Nat_Onat),hAPP(c_List_Olist_OCons(T_c),V_x_2)),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_List_Olist(T_c),tc_List_Olist(T_c)),c_List_Otake(T_c)),V_xs_2)),V_n_2) = hAPP(hAPP(c_List_Otake(T_c),V_n_2),hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),V_xs_2))) # label(fact_take__Cons) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4417 (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]. 6.07/6.10 4418 (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]. 6.07/6.10 4419 (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]. 6.07/6.10 4420 (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]. 6.07/6.10 4421 (all V_s_2 all V_r_2 all T_c all T_d 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)) = 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))) # label(fact_converse__join) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4422 (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]. 6.07/6.10 4423 (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]. 6.07/6.10 4424 (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]. 6.07/6.10 4425 (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]. 6.07/6.10 4426 (all V_ys all V_xs all T_a c_List_Oconcat(T_a,hAPP(hAPP(c_List_Oappend(tc_List_Olist(T_a)),V_xs),V_ys)) = hAPP(hAPP(c_List_Oappend(T_a),c_List_Oconcat(T_a,V_xs)),c_List_Oconcat(T_a,V_ys))) # label(fact_concat__append) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4427 (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]. 6.07/6.10 4428 (all V_x all T_a (class_Fields_Ofield__inverse__zero(T_a) & class_Int_Onumber__ring(T_a) -> hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_x),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OMin)) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_x))) # label(fact_divide__minus1) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4429 (all V_m all V_n hAPP(c_Code__Numeral_Oof__nat,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(c_Code__Numeral_Onat__of,V_n)),hAPP(c_Code__Numeral_Onat__of,V_m))) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),V_n),V_m)) # label(fact_div__code__numeral__def) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4430 (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)) -> ((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)))) <-> 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)))))))) # label(fact_strict__below__fold1__iff) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4431 (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]. 6.07/6.10 4432 (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]. 6.07/6.10 4433 (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 hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),B_x),B_x) != V_p_2))) # label(fact_IdE) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4434 (all V_d_2 all V_c_2 all V_b_2 all V_a_2 all T_c (class_Orderings_Olinorder(T_c) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_c,V_a_2),V_b_2)),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_c,V_c_2),V_d_2)) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(T_c,hAPP(hAPP(c_Orderings_Oord__class_Omax(T_c),V_a_2),V_c_2)),hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),V_b_2),V_d_2)))) # label(fact_Int__atLeastLessThan) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4435 (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]. 6.07/6.10 4436 (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]. 6.07/6.10 4437 (all V_y all V_ys all V_x all V_i all V_xs all T_b all T_a hAPP(hAPP(c_List_Ozip(T_a,T_b),hAPP(hAPP(c_List_Olist__update(T_a,V_xs),V_i),V_x)),hAPP(hAPP(c_List_Olist__update(T_b,V_ys),V_i),V_y)) = hAPP(hAPP(c_List_Olist__update(tc_prod(T_a,T_b),hAPP(hAPP(c_List_Ozip(T_a,T_b),V_xs),V_ys)),V_i),hAPP(hAPP(c_Product__Type_OPair(T_a,T_b),V_x),V_y))) # label(fact_zip__update) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4438 (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]. 6.07/6.10 4439 (all V_w all T_a (class_Int_Onumber__ring(T_a) -> hAPP(c_Int_Onumber__class_Onumber__of(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_w))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),c_Groups_Oone__class_Oone(T_a)),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_w)))) # label(fact_diff__special_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4440 (all T_a (class_Rings_Oring__1(T_a) -> c_Groups_Ozero__class_Ozero(T_a) = hAPP(c_Int_Oring__1__class_Oof__int(T_a),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))) # label(fact_of__int__0) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4441 (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]. 6.07/6.10 4442 (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]. 6.07/6.10 4443 (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(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)) = 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))))))) # label(fact_card__Diff__insert) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4444 (all V_n_2 all V_k_2 all V_m_2 (V_k_2 = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) | V_m_2 = V_n_2 <-> hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m_2),V_k_2) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_n_2),V_k_2))) # label(fact_mult__cancel2) # label(axiom) # label(non_clause). [assumption]. 6.07/6.10 4445 (all V_A_2 all V_a_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_A_2)) <-> 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))))))))) # label(fact_inj__on__insert) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4446 (all V_f_2 all V_B_2 all V_A_2 all T_c all T_d (class_Complete__Lattice_Ocomplete__lattice(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_d),V_B_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_c,T_d),V_A_2),V_f_2)) = hAPP(hAPP(c_Finite__Set_Ofold(T_c,T_d,hAPP(hAPP(c_COMBB(T_d,tc_fun(T_d,T_d),T_c),c_Lattices_Osemilattice__sup__class_Osup(T_d)),V_f_2)),V_B_2),V_A_2)))) # label(fact_sup__SUPR__fold__sup) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4447 (all V_X_2 all V_f_2 all T_d all T_c (c_Fun_Oinj__on(T_c,T_d,V_f_2,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> hAPP(c_Set_Oimage(T_d,T_c,c_Hilbert__Choice_Oinv__into(T_c,T_d,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)),V_f_2)),hAPP(c_Set_Oimage(T_c,T_d,V_f_2),V_X_2)) = V_X_2)) # label(fact_inv__image__comp) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4448 (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]. 6.07/6.11 4449 (all V_x_2 all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),hAPP(c_Finite__Set_Ocard(T_c),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))))) = hAPP(c_Finite__Set_Ocard(T_c),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2)))) # label(fact_card_Oinsert__remove) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4450 (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]. 6.07/6.11 4451 (all V_a_2 all V_xs_2 all V_f_2 all T_c all T_d hAPP(c_List_Ofoldl(T_c,T_d,hAPP(c_COMBC(T_d,T_c,T_c),V_f_2),V_a_2),hAPP(c_List_Orev(T_d),V_xs_2)) = c_List_Ofoldr(T_d,T_c,V_f_2,V_xs_2,V_a_2)) # label(fact_foldr__foldl) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4452 (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]. 6.07/6.11 4453 (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_ys_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2))) # label(fact_listrel1__eq__len) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4454 (all V_a_2 all V_A_H_2 all V_A_2 all V_f_2 all T_d all T_c (c_Fun_Obij__betw(T_c,T_d,V_f_2,V_A_2,V_A_H_2) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_A_2)) -> hAPP(V_f_2,V_a_2) = hAPP(c_Hilbert__Choice_Oinv__into(T_d,T_c,V_A_H_2,c_Hilbert__Choice_Oinv__into(T_c,T_d,V_A_2,V_f_2)),V_a_2)))) # label(fact_inv__into__inv__into__eq) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4455 (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_Divides_Odiv__class_Omod(tc_Int_Oint,V_k,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_l)) = c_SMT_Oz3mod(V_k,V_l)) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_l)) -> c_SMT_Oz3mod(V_k,V_l) = c_Divides_Odiv__class_Omod(tc_Int_Oint,V_k,V_l)))) # label(fact_z3mod__def) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4456 (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]. 6.07/6.11 4457 (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]. 6.07/6.11 4458 (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]. 6.07/6.11 4459 (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]. 6.07/6.11 4460 (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)) -> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),V_n)) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) -> hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),V_n) = hAPP(c_Nat_OSuc,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)),V_n))))) # label(fact_div__if) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4461 (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]. 6.07/6.11 4462 (all V_x all T_a (class_Groups_Osgn__if(T_a) -> (V_x = c_Groups_Ozero__class_Ozero(T_a) -> c_Groups_Osgn__class_Osgn(T_a,V_x) = c_Groups_Ozero__class_Ozero(T_a)) & (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)) -> c_Groups_Osgn__class_Osgn(T_a,V_x) = c_Groups_Oone__class_Oone(T_a)) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_x)) -> c_Groups_Osgn__class_Osgn(T_a,V_x) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),c_Groups_Oone__class_Oone(T_a)))))) # label(fact_sgn__if) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4463 (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]. 6.07/6.11 4464 (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]. 6.07/6.11 4465 (all T_a (class_Int_Onumber__ring(T_a) -> hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OMin) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),c_Groups_Oone__class_Oone(T_a)))) # label(fact_arith__simps_I31_J) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4466 (all V_a_2 all T_c (class_Groups_Ogroup__add(T_c) -> (c_Groups_Ozero__class_Ozero(T_c) = V_a_2 <-> hAPP(c_Groups_Ouminus__class_Ouminus(T_c),V_a_2) = c_Groups_Ozero__class_Ozero(T_c)))) # label(fact_neg__0__equal__iff__equal) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4467 (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]. 6.07/6.11 4468 (all V_n all V_m (hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> (hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),V_m) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> V_m = V_n))) # label(fact_diffs0__imp__equal) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4469 (all V_r_2 all V_ys_2 all V_xs_2 all T_c ((exists B_y exists B_n (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B_n),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2))) & hAPP(hAPP(c_List_Olist__update(T_c,V_xs_2),B_n),B_y) = V_ys_2 & hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),hAPP(hAPP(c_List_Onth(T_c),V_xs_2),B_n)),B_y)),V_r_2)))) <-> hBOOL(hAPP(hAPP(c_member(tc_prod(tc_List_Olist(T_c),tc_List_Olist(T_c))),hAPP(hAPP(c_Product__Type_OPair(tc_List_Olist(T_c),tc_List_Olist(T_c)),V_xs_2),V_ys_2)),c_List_Olistrel1(T_c,V_r_2))))) # label(fact_listrel1__iff__update) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4470 (all V_B_2 all T_e all V_A_2 all V_r_2 all T_d all T_c (c_Relation_Osingle__valued(T_c,T_d,hAPP(c_Relation_Oconverse(T_d,T_c),V_r_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_e,tc_HOL_Obool)) -> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_e,tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool),T_e),c_Relation_OImage(T_d,T_c,V_r_2)),V_B_2)) = hAPP(c_Relation_OImage(T_d,T_c,V_r_2),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(T_e,tc_fun(T_d,tc_HOL_Obool)),V_A_2),V_B_2))))) # label(fact_Image__INT__eq) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4471 (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]. 6.07/6.11 4472 (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_OInf__class_OInf(tc_fun(T_c,tc_HOL_Obool),V_B_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_B_2),hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2))) # label(fact_Un__Inter) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4473 (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]. 6.07/6.11 4474 (all V_r_2 all V_x_2 all V_y_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)),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)))))) = 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))) # label(fact_trancl__insert) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4475 (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]. 6.07/6.11 4476 (all V_x all T_a all V_xs (c_List_Olist_ONil(T_a) = V_xs -> V_x = c_List_Olast(T_a,hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)))) # label(fact_last__ConsL) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4477 (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]. 6.07/6.11 4478 (all T_c (class_Orderings_Olinorder(T_c) -> c_Orderings_Oord_Omax(T_c,hAPP(c_COMBC(T_c,T_c,tc_HOL_Obool),c_Orderings_Oord__class_Oless__eq(T_c))) = c_Orderings_Oord__class_Omin(T_c))) # label(fact_dual__max) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4479 (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]. 6.07/6.11 4480 (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]. 6.07/6.11 4481 (all V_B_2 all V_A_2 all V_C_2 all T_c all T_d hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool)),V_C_2),V_A_2)),V_B_2) = hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool)),V_C_2),hAPP(hAPP(c_COMBC(T_d,tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_fun(tc_fun(T_c,tc_HOL_Obool),tc_fun(T_c,tc_HOL_Obool)),T_d),c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool))),V_A_2)),V_B_2))) # label(fact_UN__simps_I6_J) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4482 (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 = hAPP(c_Groups_Ouminus__class_Ouminus(T_c),c_Groups_Oone__class_Oone(T_c)) | c_Groups_Oone__class_Oone(T_c) = V_a_2))) # label(fact_power2__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4483 (all V_xs_2 all V_p_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_COMBB(T_c,tc_HOL_Obool,tc_Nat_Onat),V_p_2),hAPP(c_List_Onth(T_c),V_xs_2))))) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),hAPP(c_List_Ofilter(T_c,V_p_2),V_xs_2))) # label(fact_length__filter__conv__card) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4484 (all V_f_2 all V_m_2 all V_n_2 all T_c (class_Groups_Ocomm__monoid__add(T_c) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2),V_m_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_c),V_f_2),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2),hAPP(c_Nat_OSuc,V_n_2))) = c_Groups_Ozero__class_Ozero(T_c)) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_n_2),V_m_2)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_c),V_f_2),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2),V_n_2))),hAPP(V_f_2,V_n_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,T_c),V_f_2),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_m_2),hAPP(c_Nat_OSuc,V_n_2)))))) # label(fact_setsum__op__ivl__Suc) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4485 (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_member(T_d),B_x),V_A_2)) & hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(V_f_2,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]. 6.07/6.11 4486 (all V_f_2 all T_d all T_c all V_xs_2 (V_xs_2 != c_List_Olist_ONil(T_c) -> 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]. 6.07/6.11 4487 (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]. 6.07/6.11 4488 (all V_seed_2 all V_size_2 all V_nrandom_2 all V_f_2 all V_R_2 all T_c all T_d hAPP(hAPP(hAPP(c_New__Random__Sequence_Oneg__bind(T_d,T_c,V_R_2,V_f_2),V_nrandom_2),V_size_2),V_seed_2) = c_New__DSequence_Oneg__bind(T_d,T_c,hAPP(hAPP(hAPP(V_R_2,V_nrandom_2),V_size_2),V_seed_2),hAPP(hAPP(c_COMBC(T_d,tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Lazy__Sequence_Olazy__sequence(tc_Option_Ooption(T_c)))),hAPP(hAPP(c_COMBC(T_d,tc_Code__Numeral_Ocode__numeral,tc_fun(tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Lazy__Sequence_Olazy__sequence(tc_Option_Ooption(T_c))))),hAPP(hAPP(c_COMBC(T_d,tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Lazy__Sequence_Olazy__sequence(tc_Option_Ooption(T_c)))))),V_f_2),V_nrandom_2)),V_size_2)),V_seed_2))) # label(fact_neg__bind__def) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4489 (all V_k_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OBit1(V_k_2)),c_Int_OMin)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_k_2),c_Int_OMin)))) # label(fact_rel__simps_I30_J) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4490 (all V_n_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n_2)) <-> (exists B_m hAPP(c_Nat_OSuc,B_m) = V_n_2))) # label(fact_gr0__conv__Suc) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4491 (all V_C_2 all V_B_2 all V_A_2 all T_c hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)),V_C_2) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_C_2)),hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_C_2))) # label(fact_Diff__Int__distrib2) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4492 (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]. 6.07/6.11 4493 (all V_xs all V_n all T_a hAPP(hAPP(c_List_Oappend(T_a),hAPP(hAPP(c_List_Odrop(T_a),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_n,hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))),V_xs)),hAPP(hAPP(c_List_Otake(T_a),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_n,hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs))),V_xs)) = hAPP(c_List_Orotate(T_a,V_n),V_xs)) # label(fact_rotate__drop__take) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4494 (all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)))) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Option_Ooption(T_c)),c_Orderings_Otop__class_Otop(tc_fun(tc_Option_Ooption(T_c),tc_HOL_Obool)))))) # label(fact_finite__option__UNIV) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4495 (all V_n all V_m hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),c_Groups_Oone__class_Oone(tc_Nat_Onat))),V_n) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),hAPP(c_Nat_OSuc,V_n))) # label(fact_diff__Suc__eq__diff__pred) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4496 (all V_y all V_x hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_x)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_y)) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_x),V_y))) # label(fact_Nat__Transfer_Otransfer__int__nat__functions_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4497 (all V_v all V_i all T_a hAPP(hAPP(c_List_Olist__update(T_a,c_List_Olist_ONil(T_a)),V_i),V_v) = c_List_Olist_ONil(T_a)) # label(fact_list__update_Osimps_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4498 (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]. 6.07/6.11 4499 (all V_ys all T_b all V_xs all T_a (hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_b)),V_ys) -> hAPP(c_List_Orev(tc_prod(T_a,T_b)),hAPP(hAPP(c_List_Ozip(T_a,T_b),V_xs),V_ys)) = hAPP(hAPP(c_List_Ozip(T_a,T_b),hAPP(c_List_Orev(T_a),V_xs)),hAPP(c_List_Orev(T_b),V_ys)))) # label(fact_zip__rev) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4500 (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_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) = 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))))) # label(fact_inv__into__comp) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4501 (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]. 6.07/6.11 4502 (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]. 6.07/6.11 4503 (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]. 6.07/6.11 4504 (all V_B_2 all V_A_2 all T_c (class_Lattices_Oab__semigroup__idem__mult(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_A_2 -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_B_2 -> hAPP(c_Finite__Set_Ofold1(T_c,c_Groups_Otimes__class_Otimes(T_c)),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(c_Finite__Set_Ofold1(T_c,c_Groups_Otimes__class_Otimes(T_c)),V_A_2)),hAPP(c_Finite__Set_Ofold1(T_c,c_Groups_Otimes__class_Otimes(T_c)),V_B_2)))))))) # label(fact_fold1__Un2) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4505 (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]. 6.07/6.11 4506 (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_Olinorder__class_Oinsort__key(T_d,T_c,V_f_2),V_x_2),hAPP(hAPP(c_List_Olist_OCons(T_d),V_y_2),V_ys_2)) = hAPP(hAPP(c_List_Olist_OCons(T_d),V_y_2),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_d,T_c,V_f_2),V_x_2),V_ys_2))) & (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]. 6.07/6.11 4507 (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]. 6.07/6.11 4508 (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]. 6.07/6.11 4509 (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) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_x_2),V_x_2)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_y_2),V_y_2)) <-> c_Groups_Ozero__class_Ozero(T_c) = V_x_2 & c_Groups_Ozero__class_Ozero(T_c) = V_y_2))) # label(fact_sum__squares__eq__zero__iff) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4510 (all V_b_2 all V_a_2 all V_S_2 all T_c all T_d (class_Groups_Ocomm__monoid__mult(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_S_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_S_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),hAPP(hAPP(c_COMBC(T_c,T_d,T_d),hAPP(hAPP(c_COMBS(T_c,T_d,tc_fun(T_d,T_d)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(T_d,tc_fun(T_d,T_d)),T_c),c_If(T_d)),hAPP(c_fequal,V_a_2))),V_b_2)),c_Groups_Oone__class_Oone(T_d))),V_S_2) = c_Groups_Oone__class_Oone(T_d)) & (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_S_2)) -> hAPP(V_b_2,V_a_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),hAPP(hAPP(c_COMBC(T_c,T_d,T_d),hAPP(hAPP(c_COMBS(T_c,T_d,tc_fun(T_d,T_d)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(T_d,tc_fun(T_d,T_d)),T_c),c_If(T_d)),hAPP(c_fequal,V_a_2))),V_b_2)),c_Groups_Oone__class_Oone(T_d))),V_S_2))))) # label(fact_setprod__delta_H) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4511 (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]. 6.07/6.11 4512 (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]. 6.07/6.11 4513 (all V_f_2 all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),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(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) <-> (exists B_x ((all B_xa (hBOOL(hAPP(hAPP(c_member(T_c),B_xa),V_A_2)) -> (B_x != B_xa -> hAPP(V_f_2,B_xa) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) & hAPP(c_Nat_OSuc,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__eq__Suc0__iff) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4514 (all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> c_Orderings_Obot__class_Obot(T_c) = c_Complete__Lattice_OInf__class_OInf(T_c,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))))) # label(fact_Inf__UNIV) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4515 (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]. 6.07/6.11 4516 (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]. 6.07/6.11 4517 (all T_c all V_a_2 all V_Pa_2 ((-hBOOL(hAPP(V_Pa_2,V_a_2)) -> hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBS(T_c,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_c),c_fconj),hAPP(c_fequal,V_a_2))),V_Pa_2)) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) & (hBOOL(hAPP(V_Pa_2,V_a_2)) -> hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBS(T_c,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_c),c_fconj),hAPP(c_fequal,V_a_2))),V_Pa_2)) = 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]. 6.07/6.11 4518 (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(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)) = 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)))) # label(fact_listsum__update__nat) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4519 (all V_i_1 all V_x all T_a hAPP(c_New__DSequence_Opos__single(T_a,V_x),V_i_1) = hAPP(c_Lazy__Sequence_Osingle(T_a),V_x)) # label(fact_New__DSequence_Opos__single__def) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4520 (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_S_2))) | 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,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]. 6.07/6.11 4521 (all V_m_2 all V_n_2 ((c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n_2 | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)) -> 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)) & (-(c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = V_n_2 | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2))) -> c_Divides_Odivmod__nat(V_m_2,V_n_2) = hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(tc_Nat_Onat,tc_Nat_Onat,tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_prod(tc_Nat_Onat,tc_Nat_Onat)),tc_Nat_Onat),c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat)),c_Nat_OSuc)),c_Divides_Odivmod__nat(hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m_2),V_n_2),V_n_2))))) # label(fact_divmod__nat__if) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4522 (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)),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)),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)))) # label(fact_wf__union__merge) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4523 (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))) <-> c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(c_Int_Onat,V_i_2))) # label(fact_nat__0__iff) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4524 (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]. 6.07/6.11 4525 (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]. 6.07/6.11 4526 (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]. 6.07/6.11 4527 (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(hAPP(V_f_2,V_x_2),hAPP(V_F_2,V_A_2)) = hAPP(V_F_2,V_A_2))))) # label(fact_folding__one__idem_Oin__idem) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4528 (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]. 6.07/6.11 4529 (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]. 6.07/6.11 4530 (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]. 6.07/6.11 4531 (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),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__Diff2) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4532 (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]. 6.07/6.11 4533 (all V_x_2 all V_R_2 all T_c (hBOOL(hAPP(c_Quotient_ORespects(T_c,V_R_2),V_x_2)) <-> hBOOL(hAPP(hAPP(V_R_2,V_x_2),V_x_2)))) # label(fact_Respects__def) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4534 (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]. 6.07/6.11 4535 (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]. 6.07/6.11 4536 (all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> c_Complete__Lattice_OSup__class_OSup(T_c,c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) = c_Orderings_Otop__class_Otop(T_c))) # label(fact_Sup__UNIV) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4537 (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]. 6.07/6.11 4538 (all V_a_2 all T_c (class_Complete__Lattice_Ocomplete__lattice(T_c) -> V_a_2 = c_Complete__Lattice_OInf__class_OInf(T_c,hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))))) # label(fact_Inf__singleton) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4539 (all V_A_2 all T_c c_List_Olist_ONil(T_c) = c_List_Osublist(T_c,c_List_Olist_ONil(T_c),V_A_2)) # label(fact_sublist__nil) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4540 (all V_a all T_a (class_Groups_Oabs__if(T_a) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a))) -> hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a) = V_a) & (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_a),V_a),c_Groups_Ozero__class_Ozero(T_a))) -> hAPP(c_Groups_Oabs__class_Oabs(T_a),V_a) = hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_a)))) # label(fact_abs__if) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4541 (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]. 6.07/6.11 4542 (all V_y all V_n all V_x (c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_y,V_n) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_x,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]. 6.07/6.11 4543 (all V_y_2 all V_x_2 all T_c (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),hAPP(c_fequal,V_y_2))) <-> V_y_2 = V_x_2)) # label(fact_eq__mem) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4544 (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]. 6.07/6.11 4545 (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]. 6.07/6.11 4546 (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_I3_J) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4547 (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]. 6.07/6.11 4548 (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]. 6.07/6.11 4549 (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]. 6.07/6.11 4550 (all V_com_2 all V_fun_2 all V_loc_2 c_Com_Ocom_Ocom__size(c_Com_Ocom_OLocal(V_loc_2,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_I3_J) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4551 (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]. 6.07/6.11 4552 (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]. 6.07/6.11 4553 (all V_r_2 all V_A_2 all V_f_2 all T_d all T_c (class_Fields_Ofield(T_c) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_c),hAPP(hAPP(c_COMBC(T_d,T_c,T_c),hAPP(hAPP(c_COMBB(T_c,tc_fun(T_c,T_c),T_d),c_Rings_Oinverse__class_Odivide(T_c)),V_f_2)),V_r_2)),V_A_2) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_c),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_c),V_f_2),V_A_2)),V_r_2))) # label(fact_setsum__divide__distrib) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4554 (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_c,T_e),V_g_2),V_A_2) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_d,T_e),V_h_2),V_B_2)))))) # label(fact_strong__setprod__reindex__cong) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4555 (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]. 6.07/6.11 4556 (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]. 6.07/6.11 4557 (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]. 6.07/6.11 4558 (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]. 6.07/6.11 4559 (all V_n all V_m (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) -> V_n != c_Groups_Ozero__class_Ozero(tc_Nat_Onat))) # label(fact_gr__implies__not0) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4560 (all V_n all T_a (class_Rings_Oring__1(T_a) -> hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),V_n) = hAPP(c_Int_Oring__1__class_Oof__int(T_a),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),V_n)))) # label(fact_of__int__of__nat__eq) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4561 (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]. 6.07/6.11 4562 (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]. 6.07/6.11 4563 (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]. 6.07/6.11 4564 (all V_j_2 all V_i_2 hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,hAPP(c_Nat_OSuc,V_i_2)),hAPP(c_Nat_OSuc,V_j_2)) = hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Nat_Onat,c_Nat_OSuc),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,V_i_2),V_j_2))) # label(fact_image__Suc__atLeastLessThan) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4565 (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]. 6.07/6.11 4566 (all V_y_2 all V_x_2 all T_c (class_Lattices_Obounded__lattice__top(T_c) -> (c_Orderings_Otop__class_Otop(T_c) = V_x_2 & V_y_2 = c_Orderings_Otop__class_Otop(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)))) # label(fact_inf__eq__top__iff) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4567 (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]. 6.07/6.11 4568 (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(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))) = 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))))) # label(fact_range__composition) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4569 (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]. 6.07/6.11 4570 (all V_x_2 all V_g_2 all V_A_2 all T_c all T_d (class_Groups_Ocomm__monoid__add(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_d),hAPP(V_g_2,V_x_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_g_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))))) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_g_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),V_A_2))))) # label(fact_setsum_Oinsert__remove) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4571 (all V_r_2 all T_c 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)))) = c_List_Olenlex(T_c,V_r_2)) # label(fact_lenlex__def) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4572 (all V_B_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_B_2 -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_B_2),V_A_2)) -> hAPP(hAPP(c_Orderings_Oord__class_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)) = 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]. 6.07/6.11 4573 (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]. 6.07/6.11 4574 (all V_x_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> hAPP(hAPP(c_Orderings_Oord__class_Omin(T_c),V_x_2),c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),V_A_2)) = c_Big__Operators_Olattice_OInf__fin(T_c,c_Orderings_Oord__class_Omin(T_c),V_A_2))))) # label(fact_min__max_OInf__fin_Oin__idem) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4575 (all V_com_2 all V_fun_2 all V_loc_2 all V_com2_H_2 all V_com1_H_2 c_Com_Ocom_OLocal(V_loc_2,V_fun_2,V_com_2) != c_Com_Ocom_OSemi(V_com1_H_2,V_com2_H_2)) # label(fact_com_Osimps_I35_J) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4576 (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]. 6.07/6.11 4577 (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]. 6.07/6.11 4578 (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)) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_H)) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_mult__nat__number__of) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4579 (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]. 6.07/6.11 4580 (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_OWhile(V_fun_H_2,V_com_H_2) != c_Com_Ocom_OCond(V_fun_2,V_com1_2,V_com2_2)) # label(fact_com_Osimps_I52_J) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4581 (all V_r_2 all V_b_2 all V_a_2 all T_c all T_d hAPP(hAPP(c_Set_Oinsert(T_c),V_b_2),hAPP(c_Relation_ORange(T_d,T_c),V_r_2)) = hAPP(c_Relation_ORange(T_d,T_c),hAPP(hAPP(c_Set_Oinsert(tc_prod(T_d,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_d,T_c),V_a_2),V_b_2)),V_r_2))) # label(fact_Range__insert) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4582 (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(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]. 6.07/6.11 4583 (all V_a all T_a (class_Rings_Olinordered__idom(T_a) -> c_Groups_Osgn__class_Osgn(T_a,V_a) = c_Groups_Osgn__class_Osgn(T_a,c_Groups_Osgn__class_Osgn(T_a,V_a)))) # label(fact_sgn__sgn) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4584 (all V_xs all V_n all T_a hAPP(c_List_Otl(T_a),hAPP(hAPP(c_List_Otake(T_a),hAPP(c_Nat_OSuc,V_n)),V_xs)) = hAPP(hAPP(c_List_Otake(T_a),V_n),hAPP(c_List_Otl(T_a),V_xs))) # label(fact_take__tl) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4585 (all V_b all V_n all V_a all T_a (class_Rings_Olinordered__semidom(T_a) -> (hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_b),hAPP(c_Nat_OSuc,V_n)) = hAPP(hAPP(c_Power_Opower__class_Opower(T_a),V_a),hAPP(c_Nat_OSuc,V_n)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_a)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_a),c_Groups_Ozero__class_Ozero(T_a)),V_b)) -> V_a = V_b))))) # label(fact_power__inject__base) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4586 (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))) -> c_Divides_OnegDivAlg(V_a,V_b) = hAPP(c_Divides_Oadjust(V_b),c_Divides_OnegDivAlg(V_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),V_b)))))) # label(fact_negDivAlg__eqn) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4587 (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]. 6.07/6.11 4588 (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]. 6.07/6.11 4589 (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]. 6.07/6.11 4590 (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]. 6.07/6.11 4591 (all V_g_2 all V_h_2 all V_A_2 all T_c all T_d (class_Groups_Ocomm__monoid__add(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (c_Fun_Oinj__on(T_c,T_c,V_h_2,V_A_2) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),V_g_2),hAPP(c_Set_Oimage(T_c,T_c,V_h_2),V_A_2)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_d),hAPP(c_Fun_Ocomp(T_c,T_d,T_c,V_g_2),V_h_2)),V_A_2))))) # label(fact_setsum_Oreindex) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4592 (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)))))) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_x_2),c_Complete__Lattice_OSup__class_OSup(tc_fun(T_c,tc_HOL_Obool),V_F_2)) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))) # label(fact_insert__partition) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4593 (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_m_2 = V_n_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]. 6.07/6.11 4594 (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]. 6.07/6.11 4595 (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]. 6.07/6.11 4596 (all V_A_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool)))) -> (hAPP(c_Finite__Set_Ocard(T_c),V_A_2) = hAPP(c_Finite__Set_Ocard(T_c),c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))) -> V_A_2 = c_Orderings_Otop__class_Otop(tc_fun(T_c,tc_HOL_Obool))))) # label(fact_card__eq__UNIV__imp__eq__UNIV) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4597 (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]. 6.07/6.11 4598 (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) -> (V_d_2 = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_c),V_a_2),V_b_2)),V_e_2)),V_c_2) <-> 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__iff1) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4599 (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]. 6.07/6.11 4600 (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)) & (c_Groups_Ozero__class_Ozero(tc_Int_Oint) != V_l_2 -> c_Divides_Odivmod__int(V_k_2,V_l_2) = hAPP(c_Product__Type_Oapsnd(tc_Int_Oint,tc_Int_Oint,tc_Int_Oint,hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),c_Groups_Osgn__class_Osgn(tc_Int_Oint,V_l_2))),hAPP(hAPP(hAPP(c_If(tc_prod(tc_Int_Oint,tc_Int_Oint)),hAPP(hAPP(c_fequal,c_Groups_Osgn__class_Osgn(tc_Int_Oint,V_k_2)),c_Groups_Osgn__class_Osgn(tc_Int_Oint,V_l_2))),c_Divides_Opdivmod(V_k_2,V_l_2)),hAPP(hAPP(c_Product__Type_Oprod_Oprod__case(tc_Int_Oint,tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint)),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint)),tc_fun(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint))),hAPP(hAPP(c_COMBB(tc_fun(tc_Int_Oint,tc_fun(tc_prod(tc_Int_Oint,tc_Int_Oint),tc_prod(tc_Int_Oint,tc_Int_Oint))),tc_fun(tc_fun(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint)),tc_fun(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint))),tc_Int_Oint),c_COMBS(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint),tc_prod(tc_Int_Oint,tc_Int_Oint))),hAPP(hAPP(c_COMBB(tc_prod(tc_Int_Oint,tc_Int_Oint),tc_fun(tc_Int_Oint,tc_fun(tc_prod(tc_Int_Oint,tc_Int_Oint),tc_prod(tc_Int_Oint,tc_Int_Oint))),tc_Int_Oint),hAPP(c_COMBC(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint),tc_fun(tc_prod(tc_Int_Oint,tc_Int_Oint),tc_prod(tc_Int_Oint,tc_Int_Oint))),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_prod(tc_Int_Oint,tc_Int_Oint),tc_fun(tc_prod(tc_Int_Oint,tc_Int_Oint),tc_prod(tc_Int_Oint,tc_Int_Oint))),tc_Int_Oint),c_If(tc_prod(tc_Int_Oint,tc_Int_Oint))),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Int_Oint,tc_HOL_Obool),c_fequal),c_Groups_Ozero__class_Ozero(tc_Int_Oint))))),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint)),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint)),tc_Int_Oint),c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint)),c_Groups_Ouminus__class_Ouminus(tc_Int_Oint))),c_Groups_Ozero__class_Ozero(tc_Int_Oint))))),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint),tc_fun(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint))),hAPP(hAPP(c_COMBB(tc_fun(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint)),tc_fun(tc_fun(tc_Int_Oint,tc_Int_Oint),tc_fun(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint))),tc_Int_Oint),c_COMBB(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint),tc_Int_Oint)),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_prod(tc_Int_Oint,tc_Int_Oint)),tc_Int_Oint),c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint)),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint),tc_Int_Oint),c_Groups_Ominus__class_Ominus(tc_Int_Oint)),c_Groups_Ouminus__class_Ouminus(tc_Int_Oint))),c_Groups_Oone__class_Oone(tc_Int_Oint))))),hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),V_l_2))))),c_Divides_Opdivmod(V_k_2,V_l_2)))))) & (c_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]. 6.07/6.11 4601 (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_a = V_b & V_c != V_d -> 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]. 6.07/6.11 4602 (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]. 6.07/6.11 4603 (all V_f2_2 all V_f1_2 all T_c c_Code__Numeral_Ocode__numeral_Ocode__numeral__rec(T_c,V_f1_2,V_f2_2,c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)) = V_f1_2) # label(fact_code__numeral_Orecs_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4604 (all V_k all V_n all V_m 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)) = 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)) # label(fact_nat__add__assoc) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4605 (all V_d_2 hAPP(c_Set_OCollect(tc_prod(tc_Int_Oint,tc_Int_Oint)),hAPP(c_Product__Type_Oprod_Oprod__case(tc_Int_Oint,tc_Int_Oint,tc_HOL_Obool),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_HOL_Obool),tc_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_fun(tc_fun(tc_Int_Oint,tc_HOL_Obool),tc_fun(tc_Int_Oint,tc_HOL_Obool)),tc_Int_Oint),c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Int_Oint)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Int_Oint),c_fconj),hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_d_2)))),c_Orderings_Oord__class_Oless(tc_Int_Oint)))) = c_Int_Oint__ge__less__than(V_d_2)) # label(fact_int__ge__less__than__def) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4606 (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]. 6.07/6.11 4607 (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]. 6.07/6.11 4608 (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]. 6.07/6.11 4609 (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]. 6.07/6.11 4610 (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]. 6.07/6.11 4611 (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]. 6.07/6.11 4612 (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_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]. 6.07/6.11 4613 (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]. 6.07/6.11 4614 (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(hAPP(c_COMBC(T_c,T_c,tc_HOL_Obool),c_fequal),V_a_2))),V_b_2)),c_Groups_Ozero__class_Ozero(T_d))),V_S_2) = c_Groups_Ozero__class_Ozero(T_d)) & (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))))) # label(fact_setsum__delta) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4615 (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]. 6.07/6.11 4616 (all V_zs_2 all V_xs_2 all V_ys_2 all T_c (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) <-> V_ys_2 = V_zs_2)) # label(fact_append__same__eq) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4617 (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]. 6.07/6.11 4618 (all V_a all T_a (class_Groups_Ocomm__monoid__mult(T_a) -> hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Groups_Oone__class_Oone(T_a)),V_a) = V_a)) # label(fact_mult__1) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4619 (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]. 6.07/6.11 4620 (all V_n all V_i (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),V_i),V_n)) -> hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),V_i)) = V_i)) # label(fact_diff__diff__cancel) # label(axiom) # label(non_clause). [assumption]. 6.07/6.11 4621 (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]. 6.07/6.11 4622 (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]. 6.07/6.11 4623 (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]. 6.07/6.11 4624 (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]. 6.07/6.11 4625 (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]. 6.07/6.11 4626 (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]. 6.07/6.11 4627 (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]. 6.07/6.11 4628 (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]. 6.07/6.11 4629 (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]. 6.07/6.11 4630 (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]. 6.07/6.11 4631 (all V_R_2 all T_c (c_Equiv__Relations_Opart__equivp(T_c,V_R_2) <-> (all B_x all B_y (hBOOL(hAPP(hAPP(V_R_2,B_y),B_y)) & hAPP(V_R_2,B_y) = hAPP(V_R_2,B_x) & hBOOL(hAPP(hAPP(V_R_2,B_x),B_x)) <-> hBOOL(hAPP(hAPP(V_R_2,B_x),B_y)))) & (exists B_x hBOOL(hAPP(hAPP(V_R_2,B_x),B_x))))) # label(fact_part__equivp__def) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4632 (all V_k_2 all V_N_2 all T_c (class_Groups_Olinordered__ab__semigroup__add(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_N_2)) -> (V_N_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> c_Big__Operators_Olinorder__class_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))))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_c),V_k_2),c_Big__Operators_Olinorder__class_OMax(T_c,V_N_2)))))) # label(fact_add__Max__commute) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4633 (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]. 6.07/6.12 4634 (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]. 6.07/6.12 4635 (all V_B_2 all V_A_2 all V_g_2 all V_f_2 all T_d all T_c all T_e (class_Rings_Osemiring__0(T_e) -> (c_Fun_Oinj__on(tc_prod(T_c,T_d),T_e,hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,T_e),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_d,T_e),tc_fun(T_d,T_e)),hAPP(hAPP(c_COMBB(tc_fun(T_e,T_e),tc_fun(tc_fun(T_d,T_e),tc_fun(T_d,T_e)),T_c),c_COMBB(T_e,T_e,T_d)),hAPP(hAPP(c_COMBB(T_e,tc_fun(T_e,T_e),T_c),c_Groups_Otimes__class_Otimes(T_e)),V_f_2))),V_g_2)),hAPP(hAPP(c_Product__Type_OSigma(T_c,T_d),V_A_2),hAPP(c_COMBK(tc_fun(T_d,tc_HOL_Obool),T_c),V_B_2))) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_e,T_e),c_Fun_Oid(T_e)),hAPP(c_Set_OCollect(T_e),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool,T_e),c_HOL_OEx(T_c)),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool)),tc_fun(T_c,tc_HOL_Obool),T_e),hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_HOL_Obool,T_c),c_HOL_OEx(T_d))),hAPP(hAPP(c_COMBC(T_e,tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool)),tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool))),tc_fun(tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool)),tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool))),T_e),c_COMBS(T_c,tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(T_d,tc_fun(tc_HOL_Obool,tc_HOL_Obool))),tc_fun(T_c,tc_fun(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool))),T_e),hAPP(c_COMBB(tc_fun(T_d,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool)),T_c),c_COMBS(T_d,tc_HOL_Obool,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool)),tc_fun(T_c,tc_fun(T_d,tc_fun(tc_HOL_Obool,tc_HOL_Obool))),T_e),hAPP(c_COMBB(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),T_c),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_d),c_fconj))),hAPP(hAPP(c_COMBC(T_e,tc_fun(T_c,tc_fun(T_d,T_e)),tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool))),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(T_d,T_e),tc_fun(T_d,tc_HOL_Obool)),tc_fun(tc_fun(T_c,tc_fun(T_d,T_e)),tc_fun(T_c,tc_fun(T_d,tc_HOL_Obool))),T_e),c_COMBB(tc_fun(T_d,T_e),tc_fun(T_d,tc_HOL_Obool),T_c)),hAPP(hAPP(c_COMBB(tc_fun(T_e,tc_HOL_Obool),tc_fun(tc_fun(T_d,T_e),tc_fun(T_d,tc_HOL_Obool)),T_e),c_COMBB(T_e,tc_HOL_Obool,T_d)),c_fequal))),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_d,T_e),tc_fun(T_d,T_e)),hAPP(hAPP(c_COMBB(tc_fun(T_e,T_e),tc_fun(tc_fun(T_d,T_e),tc_fun(T_d,T_e)),T_c),c_COMBB(T_e,T_e,T_d)),hAPP(hAPP(c_COMBB(T_e,tc_fun(T_e,T_e),T_c),c_Groups_Otimes__class_Otimes(T_e)),V_f_2))),V_g_2)))))),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_fun(tc_fun(T_d,tc_HOL_Obool),tc_fun(T_d,tc_HOL_Obool)),T_c),c_COMBB(tc_HOL_Obool,tc_HOL_Obool,T_d)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_c),c_fconj),hAPP(hAPP(c_COMBC(T_c,tc_fun(T_c,tc_HOL_Obool),tc_HOL_Obool),c_member(T_c)),V_A_2)))),hAPP(hAPP(c_COMBC(T_d,tc_fun(T_d,tc_HOL_Obool),tc_HOL_Obool),c_member(T_d)),V_B_2))))))) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_e),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_c,T_e),V_f_2),V_A_2)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(T_d,T_e),V_g_2),V_B_2))))) # label(fact_setsum__mult__setsum__if__inj) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4636 (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]. 6.07/6.12 4637 (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]. 6.07/6.12 4638 (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_Otake(T_a),V_n),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)) = c_List_Olist_ONil(T_a)) & (V_n != c_Groups_Ozero__class_Ozero(tc_Nat_Onat) -> hAPP(hAPP(c_List_Otake(T_a),V_n),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs)) = hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),hAPP(hAPP(c_List_Otake(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_n),c_Groups_Oone__class_Oone(tc_Nat_Onat))),V_xs))))) # label(fact_take__Cons_H) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4639 (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]. 6.07/6.12 4640 (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]. 6.07/6.12 4641 (all V_xs_2 all T_c V_xs_2 = 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)))) # label(fact_map__nth) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4642 (all V_y_2 all V_x_2 (V_x_2 = V_y_2 <-> hAPP(c_Code__Numeral_Onat__of,V_x_2) = hAPP(c_Code__Numeral_Onat__of,V_y_2))) # label(fact_code__numeral_Onat__of__inject) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4643 (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]. 6.07/6.12 4644 (all V_a_2 all T_c hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) = hAPP(c_Set_OCollect(T_c),hAPP(c_fequal,V_a_2))) # label(fact_singleton__conv2) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4645 (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]. 6.07/6.12 4646 (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]. 6.07/6.12 4647 (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]. 6.07/6.12 4648 (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]. 6.07/6.12 4649 (all V_c all V_b all V_a all T_a (class_Divides_Osemiring__div(T_a) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),c_Divides_Odiv__class_Omod(T_a,V_a,V_c)),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__eq) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4650 (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]. 6.07/6.12 4651 (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]. 6.07/6.12 4652 (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]. 6.07/6.12 4653 (all V_B_2 all V_A_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_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_Lattices_Osemilattice__sup__class_Osup(tc_fun(T_d,tc_HOL_Obool))),V_A_2)),V_B_2))) # label(fact_Sigma__Un__distrib2) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4654 (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]. 6.07/6.12 4655 (all V_k all V_b all V_a all T_a (class_Rings_Odvd(T_a) -> (V_a = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_b),V_k) -> hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_a),V_b),V_a))))) # label(fact_dvdI) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4656 (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]. 6.07/6.12 4657 (all V_list_H_2 all V_a_H_2 all V_list_2 all V_a_2 all T_c (V_list_2 = V_list_H_2 & V_a_2 = V_a_H_2 <-> 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))) # label(fact_list_Oinject) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4658 (all V_i_2 all V_k_2 all V_j_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)) <-> 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))))) # label(fact_le__diff__conv) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4659 (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]. 6.07/6.12 4660 (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]. 6.07/6.12 4661 (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]. 6.07/6.12 4662 (all V_b_2 all V_a_2 all V_S_2 all T_c all T_d (class_Groups_Ocomm__monoid__mult(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_S_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_S_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),hAPP(hAPP(c_COMBC(T_c,T_d,T_d),hAPP(hAPP(c_COMBS(T_c,T_d,tc_fun(T_d,T_d)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(T_d,tc_fun(T_d,T_d)),T_c),c_If(T_d)),hAPP(hAPP(c_COMBC(T_c,T_c,tc_HOL_Obool),c_fequal),V_a_2))),V_b_2)),c_Groups_Oone__class_Oone(T_d))),V_S_2) = c_Groups_Oone__class_Oone(T_d)) & (hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),V_S_2)) -> hAPP(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))))) # label(fact_setprod__delta) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4663 (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]. 6.07/6.12 4664 (all V_n all V_m (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m),V_n)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_n),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),V_m),V_n)) = V_m)) # label(fact_add__diff__inverse) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4665 (all V_r_2 all T_c (c_Wellfounded_Owf(T_c,V_r_2) <-> -(exists B_f all B_i hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),hAPP(B_f,hAPP(c_Nat_OSuc,B_i))),hAPP(B_f,B_i))),V_r_2))))) # label(fact_wf__iff__no__infinite__down__chain) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4666 (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]. 6.07/6.12 4667 (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]. 6.07/6.12 4668 (all V_xs all V_x all T_a V_x = hAPP(c_List_Ohd(T_a),hAPP(hAPP(c_List_Olist_OCons(T_a),V_x),V_xs))) # label(fact_hd_Osimps) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4669 (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]. 6.07/6.12 4670 (all V_B_2 all V_A_2 all T_d all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_d),V_B_2)) & hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) <-> hBOOL(hAPP(c_Finite__Set_Ofinite(tc_sum(T_c,T_d)),c_Sum__Type_OPlus(T_c,T_d,V_A_2,V_B_2))))) # label(fact_finite__Plus__iff) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4671 (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]. 6.07/6.12 4672 (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]. 6.07/6.12 4673 (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]. 6.07/6.12 4674 (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]. 6.07/6.12 4675 (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]. 6.07/6.12 4676 (all V_x_2 all V_A_2 all T_c (class_Groups_Oab__semigroup__mult(T_c) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (-hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> hAPP(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]. 6.07/6.12 4677 (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]. 6.07/6.12 4678 (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]. 6.07/6.12 4679 (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]. 6.07/6.12 4680 (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]. 6.07/6.12 4681 (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]. 6.07/6.12 4682 (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]. 6.07/6.12 4683 (all V_b all V_a all T_a (class_Groups_Oordered__ab__group__add__abs(T_a) -> hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_b),V_a)) = hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_b)))) # label(fact_abs__minus__commute) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4684 (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]. 6.07/6.12 4685 (all V_Pa_2 all V_f_2 all T_c all T_d c_Random__Sequence_Omap(T_d,T_c,V_f_2,V_Pa_2) = c_Random__Sequence_Obind(T_d,T_c,V_Pa_2,hAPP(c_Fun_Ocomp(T_c,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_prod(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_HOL_Obool,tc_Option_Ooption(tc_Lazy__Sequence_Olazy__sequence(T_c)))),tc_prod(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral))))),T_d,c_Random__Sequence_Osingle(T_c)),V_f_2))) # label(fact_Random__Sequence_Omap__def) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4686 (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]. 6.07/6.12 4687 (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)) <-> (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)))))) # label(fact_SUP2__iff) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4688 (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]. 6.07/6.12 4689 (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]. 6.07/6.12 4690 (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]. 6.07/6.12 4691 (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]. 6.07/6.12 4692 (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]. 6.07/6.12 4693 (all V_xs_2 all V_f_2 all T_d all V_Pa_2 all T_c c_List_OdropWhile(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),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))) # label(fact_dropWhile__map) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4694 (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]. 6.07/6.12 4695 (all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(c_Predicate_OPowp(T_c,V_A_2),V_B_2)) <-> (all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),V_B_2)) -> hBOOL(hAPP(V_A_2,B_x)))))) # label(fact_Powp__def) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4696 (all V_x all T_a (class_Lattices_Obounded__lattice__top(T_a) -> hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(T_a),c_Orderings_Otop__class_Otop(T_a)),V_x) = V_x)) # label(fact_inf__top__left) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4697 (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]. 6.07/6.12 4698 (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)) -> 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_Groups_Ouminus__class_Ouminus(tc_fun(T_d,tc_HOL_Obool)),V_B_2) = hAPP(c_Set_Ovimage(T_d,T_c,hAPP(hAPP(c_COMBC(T_d,T_c,T_c),hAPP(hAPP(c_COMBC(T_d,T_c,tc_fun(T_c,T_c)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(T_c,tc_fun(T_c,T_c)),T_d),c_If(T_c)),hAPP(hAPP(c_COMBC(T_d,tc_fun(T_d,tc_HOL_Obool),tc_HOL_Obool),c_member(T_d)),V_B_2))),V_c_2)),V_d_2)),V_A_2))) & (hBOOL(hAPP(hAPP(c_member(T_c),V_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) = c_Orderings_Otop__class_Otop(tc_fun(T_d,tc_HOL_Obool))) & (-hBOOL(hAPP(hAPP(c_member(T_c),V_d_2),V_A_2)) -> V_B_2 = hAPP(c_Set_Ovimage(T_d,T_c,hAPP(hAPP(c_COMBC(T_d,T_c,T_c),hAPP(hAPP(c_COMBC(T_d,T_c,tc_fun(T_c,T_c)),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(T_c,tc_fun(T_c,T_c)),T_d),c_If(T_c)),hAPP(hAPP(c_COMBC(T_d,tc_fun(T_d,tc_HOL_Obool),tc_HOL_Obool),c_member(T_d)),V_B_2))),V_c_2)),V_d_2)),V_A_2))))) # label(fact_vimage__if) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4699 (all V_r_2 all T_c all T_d V_r_2 = hAPP(c_Relation_Oconverse(T_d,T_c),hAPP(c_Relation_Oconverse(T_c,T_d),V_r_2))) # label(fact_converse__converse) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4700 (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]. 6.07/6.12 4701 (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]. 6.07/6.12 4702 (all V_n_2 all V_v_2 (hBOOL(hAPP(hAPP(c_HOL_OLet(tc_Int_Oint,tc_HOL_Obool),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_Opred(V_v_2))),hAPP(hAPP(c_COMBS(tc_Int_Oint,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Int_Oint),c_fimplies),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_HOL_Obool,tc_Int_Oint),c_fNot),c_Nat__Numeral_Oneg))),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_Nat_Onat,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_Int_Oint),c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat)),c_Int_Onat)),V_n_2)))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),V_v_2)),hAPP(c_Nat_OSuc,V_n_2))))) # label(fact_le__number__of__Suc) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4703 (all V_w_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit1(V_w_2)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_w_2),c_Groups_Ozero__class_Ozero(tc_Int_Oint))))) # label(fact_bin__less__0__simps_I4_J) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4704 (all V_k_2 hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_k_2)) = hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),V_k_2)) # label(fact_lessThan__Suc__atMost) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4705 (all V_xs all V_x all T_a c_List_Oinsert(T_a,V_x,c_List_Oremdups(T_a,V_xs)) = c_List_Oremdups(T_a,c_List_Oinsert(T_a,V_x,V_xs))) # label(fact_insert__remdups) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4706 (all V_n_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),V_m_2),V_n_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m_2)),V_n_2)))) # label(fact_Suc__le__eq) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4707 (all V_n all V_m hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),hAPP(c_Nat_OSuc,V_m)),hAPP(c_Nat_OSuc,V_n)) = hAPP(c_Nat_OSuc,hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),V_m),V_n))) # label(fact_min__Suc__Suc) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4708 (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]. 6.07/6.12 4709 (all V_x_2 all T_c all V_A_2 ((hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_A_2)) -> c_List_Osublist(T_c,hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),c_List_Olist_ONil(T_c)),V_A_2) = hAPP(hAPP(c_List_Olist_OCons(T_c),V_x_2),c_List_Olist_ONil(T_c))) & (-hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_A_2)) -> c_List_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]. 6.07/6.12 4710 (all V_x_2 all V_X_2 all V_a_2 all V_f_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofold1Set(T_c,V_f_2,hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_X_2)),V_x_2)) -> -(all B_a all B_A (hAPP(hAPP(c_Set_Oinsert(T_c),V_a_2),V_X_2) = hAPP(hAPP(c_Set_Oinsert(T_c),B_a),B_A) -> (hBOOL(hAPP(c_Finite__Set_Ofold__graph(T_c,T_c,V_f_2,B_a,B_A),V_x_2)) -> hBOOL(hAPP(hAPP(c_member(T_c),B_a),B_A))))))) # label(fact_insert__fold1SetE) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4711 (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]. 6.07/6.12 4712 (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]. 6.07/6.12 4713 (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]. 6.07/6.12 4714 (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]. 6.07/6.12 4715 (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]. 6.07/6.12 4716 (all V_n hAPP(c_Code__Numeral_Onat__of,hAPP(c_Code__Numeral_Oof__nat,V_n)) = V_n) # label(fact_nat__of__of__nat) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4717 (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))) -> c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = V_A_2)) # label(fact_bij__betw__empty2) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4718 (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]. 6.07/6.12 4719 (all V_b_2 all V_a_2 all T_c (class_Orderings_Oorder(T_c) -> (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_a_2),V_b_2)) <-> c_SetInterval_Oord__class_OatLeastAtMost(T_c,V_a_2,V_b_2) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))) # label(fact_atLeastatMost__empty__iff) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4720 (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]. 6.07/6.12 4721 (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]. 6.07/6.12 4722 (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]. 6.07/6.12 4723 (all V_n hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),V_n) = V_n) # label(fact_plus__nat_Oadd__0) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4724 (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))) -> hAPP(hAPP(c_List_Oappend(T_c),V_xs_2),c_List_Oremove1(T_c,V_x_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))))) # label(fact_remove1__append) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4725 (all V_x_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))) -> V_A_2 = 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)))) # label(fact_subset__singletonD) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4726 (all V_a all T_a (class_Int_Onumber__ring(T_a) -> V_a = hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_a),V_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),c_Int_OBit1(c_Int_OPls))))) # label(fact_mult__numeral__1__right) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4727 (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]. 6.07/6.12 4728 (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]. 6.07/6.12 4729 (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]. 6.07/6.12 4730 (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]. 6.07/6.12 4731 (all V_B_2 all V_A_2 (V_A_2 = V_B_2 <-> hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),V_B_2) = hAPP(c_Set_Oimage(tc_Nat_Onat,tc_Int_Oint,c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),V_A_2))) # label(fact_transfer__nat__int__set__relations_I3_J) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4732 (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]. 6.07/6.12 4733 (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_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]. 6.07/6.12 4734 (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]. 6.07/6.12 4735 (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]. 6.07/6.12 4736 (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_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))) # label(fact_rtrancl__unfold) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4737 (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]. 6.07/6.12 4738 (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]. 6.07/6.12 4739 (all V_B_2 all V_A_2 all T_c (class_Orderings_Olinorder(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_A_2 -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_B_2 -> 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]. 6.07/6.12 4740 (all V_a1_2 all V_a2_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_a2_2),V_a1_2)) <-> c_List_Olinorder__class_Osorted(T_c,V_a1_2) & (all B_x (hBOOL(hAPP(hAPP(c_member(T_c),B_x),hAPP(c_List_Oset(T_c),V_a1_2))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_a2_2),B_x))))))) # label(fact_sorted_Oequations_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4741 (all V_b all V_a all T_a (class_Groups_Oab__group__add(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),hAPP(c_Groups_Ouminus__class_Ouminus(T_a),V_b)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_b))) # label(fact_ab__diff__minus) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4742 (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_com2_H_2 = V_com2_2 & V_com1_2 = V_com1_H_2 & V_fun_2 = V_fun_H_2 <-> c_Com_Ocom_OCond(V_fun_H_2,V_com1_H_2,V_com2_H_2) = c_Com_Ocom_OCond(V_fun_2,V_com1_2,V_com2_2))) # label(fact_com_Osimps_I4_J) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4743 (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_h,T_f,T_e,hAPP(c_Fun_Ocomp(T_g,T_f,T_h,V_f_2),V_g_2)),V_h_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)))) # label(fact_folding_Ocommute__comps_I1_J) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4744 (all V_s_2 all V_r_2 all T_e all T_c all T_d hAPP(c_Relation_Oconverse(T_d,T_c),hAPP(hAPP(c_Relation_Orel__comp(T_d,T_e,T_c),V_r_2),V_s_2)) = hAPP(hAPP(c_Relation_Orel__comp(T_c,T_e,T_d),hAPP(c_Relation_Oconverse(T_e,T_c),V_s_2)),hAPP(c_Relation_Oconverse(T_d,T_e),V_r_2))) # label(fact_converse__rel__comp) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4745 (all V_nat_H_2 all V_nat_2 (hAPP(c_Nat_OSuc,V_nat_H_2) = hAPP(c_Nat_OSuc,V_nat_2) <-> V_nat_2 = V_nat_H_2)) # label(fact_nat_Oinject) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4746 (all T_c all T_d c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = hAPP(c_Relation_ORange(T_d,T_c),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(T_d,T_c),tc_HOL_Obool)))) # label(fact_Range__empty) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4747 (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]. 6.07/6.12 4748 (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]. 6.07/6.12 4749 (all V_k2_2 all V_k1_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),V_k1_2),V_k2_2)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit1(V_k1_2)),c_Int_OBit0(V_k2_2))))) # label(fact_less__int__code_I15_J) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4750 (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]. 6.07/6.12 4751 (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]. 6.07/6.12 4752 (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_ys_2) & V_n_2 = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_c)),V_xs_2))) # label(fact_lexn__length) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4753 (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_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2))) <-> (exists B_x (hBOOL(hAPP(hAPP(c_member(T_c),V_b_2),hAPP(V_B_2,B_x))) & hBOOL(hAPP(hAPP(c_member(T_d),B_x),V_A_2)))))) # label(fact_UN__iff) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4754 (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(c_Fun_Ooverride__on(T_c,T_d,V_f_2,V_g_2,V_A_2),V_a_2) = hAPP(V_f_2,V_a_2))) # label(fact_override__on__apply__notin) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4755 (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_c,tc_HOL_Obool)),hAPP(c_Set_Ovimage(T_c,T_d,V_f_2),V_B_2)),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)))))) # label(fact_vimage__subset__eq) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4756 (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]. 6.07/6.12 4757 (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]. 6.07/6.12 4758 (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]. 6.07/6.12 4759 (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_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)) & (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))))) # label(fact_min__max_OSup__fin_Oremove) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4760 (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]. 6.07/6.12 4761 (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]. 6.07/6.12 4762 (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]. 6.07/6.12 4763 (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_c)),V_a_2) = hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_d)),V_b_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]. 6.07/6.12 4764 (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]. 6.07/6.12 4765 (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]. 6.07/6.12 4766 (all V_B_2 all V_A_2 all T_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)) & V_A_2 != V_B_2 <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2)))) # label(fact_psubset__eq) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4767 (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]. 6.07/6.12 4768 (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]. 6.07/6.12 4769 (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]. 6.07/6.12 4770 (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]. 6.07/6.12 4771 (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]. 6.07/6.12 4772 (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]. 6.07/6.12 4773 (all V_xs all V_n all T_a hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_List_Olist(T_a))),c_Enum_On__lists(T_a,V_n,V_xs)) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)),V_n)) # label(fact_length__n__lists) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4774 (all V_B_2 all V_A_2 all T_c all T_d ((all B_x (hBOOL(hAPP(hAPP(c_member(T_d),B_x),V_A_2)) -> c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = hAPP(V_B_2,B_x))) <-> hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(T_d,tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2) = c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) # label(fact_UNION__empty__conv_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4775 (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]. 6.07/6.12 4776 (all V_k c_Int_Osucc(c_Int_OBit0(V_k)) = c_Int_OBit1(V_k)) # label(fact_succ__Bit0) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4777 (all V_p all T_b all T_a V_p = 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))) # label(fact_pair__collapse) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4778 (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]. 6.07/6.12 4779 (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))) <-> c_Nitpick_Orefl_H(T_c,V_r_2))) # label(fact_Nitpick_Orefl_H__def) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4780 (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]. 6.07/6.12 4781 (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]. 6.07/6.12 4782 (all V_u_2 all V_m_2 all V_l_2 all T_c (class_Orderings_Oorder(T_c) -> c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),c_SetInterval_Oord__class_OgreaterThanAtMost(T_c,V_l_2,V_m_2)),c_SetInterval_Oord__class_OgreaterThanAtMost(T_c,V_m_2,V_u_2)))) # label(fact_ivl__disj__int_I14_J) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4783 (all V_c_2 all V_b_2 all V_a_2 all T_c (class_Orderings_Oorder(T_c) -> (c_SetInterval_Oord__class_OatLeastAtMost(T_c,V_a_2,V_b_2) = hAPP(hAPP(c_Set_Oinsert(T_c),V_c_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))) <-> V_b_2 = V_a_2 & V_c_2 = V_b_2))) # label(fact_atLeastAtMost__singleton__iff) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4784 (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]. 6.07/6.12 4785 (all V_n all V_m all T_a (class_Rings_Osemiring__1(T_a) -> hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),V_m),V_n)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),V_m)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_a),V_n)))) # label(fact_of__nat__add) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4786 (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]. 6.07/6.12 4787 (all V_b all V_a all V_c (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_c)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b),V_c))),c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,V_b)) = c_Divides_Odiv__class_Omod(tc_Int_Oint,V_a,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),V_b),V_c)))) # label(fact_zmod__zmult2__eq) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4788 (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]. 6.07/6.12 4789 (all V_w all V_z hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_w),V_z) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_z),V_w)) # label(fact_zadd__commute) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4790 (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]. 6.07/6.12 4791 (all V_y_2 all V_x_2 all V_r_2 all V_A_2 all T_c (c_Equiv__Relations_Oequiv(T_c,V_A_2,V_r_2) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_x_2),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(T_c),V_y_2),V_A_2)) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_x_2),V_y_2)),V_r_2)) <-> hAPP(c_Relation_OImage(T_c,T_c,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_y_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)))) = hAPP(c_Relation_OImage(T_c,T_c,V_r_2),hAPP(hAPP(c_Set_Oinsert(T_c),V_x_2),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))))))) # label(fact_eq__equiv__class__iff) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4792 (all V_b_2 all V_a_2 all V_r_2 all T_c (c_Relation_Oantisym(T_c,V_r_2) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_a_2),V_b_2)),V_r_2)) -> (hBOOL(hAPP(hAPP(c_member(tc_prod(T_c,T_c)),hAPP(hAPP(c_Product__Type_OPair(T_c,T_c),V_b_2),V_a_2)),V_r_2)) -> V_b_2 = V_a_2)))) # label(fact_antisymD) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4793 (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]. 6.07/6.12 4794 (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]. 6.07/6.12 4795 (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]. 6.07/6.12 4796 (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]. 6.07/6.12 4797 (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),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_double__add__less__zero__iff__single__add__less__zero) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4798 (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]. 6.07/6.12 4799 (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_Ozero__class_Ozero(tc_Nat_Onat)) = c_Groups_Oone__class_Oone(T_a))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J) # label(axiom) # label(non_clause). [assumption]. 6.07/6.12 4800 (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]. 6.07/6.12 4801 (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]. 6.07/6.12 4802 (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]. 6.07/6.12 4803 (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]. 6.07/6.12 4804 (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]. 6.07/6.12 4805 (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]. 6.07/6.12 4806 (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_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)))) = 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)))))) # label(fact_setsum__ub__add__nat) # label(axiom) # label(non_clause). [assumption]. 6.07/6.13 4807 (all V_b_2 all V_f_2 all T_d all V_a_2 all T_c (V_b_2 = hAPP(V_f_2,V_a_2) <-> 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)))))))) # label(fact_vimage__singleton__eq) # label(axiom) # label(non_clause). [assumption]. 6.07/6.13 4808 (all V_R_2 all T_d all T_c hAPP(c_Relation_ODomain(T_c,T_d),V_R_2) = hAPP(c_Set_Oimage(tc_prod(T_c,T_d),T_c,c_Product__Type_Ofst(T_c,T_d)),V_R_2)) # label(fact_fst__eq__Domain) # label(axiom) # label(non_clause). [assumption]. 6.07/6.13 4809 (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]. 6.07/6.13 4810 (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]. 6.07/6.13 4811 (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]. 6.07/6.13 4812 (all V_A_2 all T_c (class_Lattices_Olattice(T_c) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> (V_A_2 != c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) -> ((all B_x all B_y hBOOL(hAPP(hAPP(c_member(T_c),hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_c),B_x),B_y)),hAPP(hAPP(c_Set_Oinsert(T_c),B_x),hAPP(hAPP(c_Set_Oinsert(T_c),B_y),c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool))))))) -> hBOOL(hAPP(hAPP(c_member(T_c),c_Big__Operators_Olattice__class_OSup__fin(T_c,V_A_2)),V_A_2))))))) # label(fact_Sup__fin_Oclosed) # label(axiom) # label(non_clause). [assumption]. 6.07/6.13 4813 (all V_xs_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)),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)))) <-> hBOOL(hAPP(hAPP(c_member(T_c),V_a_2),hAPP(c_List_Oset(T_c),V_xs_2))) & V_b_2 = V_a_2)) # label(fact_zip__same) # label(axiom) # label(non_clause). [assumption]. 6.07/6.13 4814 (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]. 6.07/6.13 4815 (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]. 6.07/6.13 4816 (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]. 6.07/6.13 4817 (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]. 6.07/6.13 4818 (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)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_A_2 -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_B_2)) -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) != V_B_2 -> (c_Orderings_Obot__class_Obot(tc_fun(T_c,tc_HOL_Obool)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(T_c,tc_HOL_Obool)),V_A_2),V_B_2) -> hAPP(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]. 6.07/6.13 4819 (all V_r_2 all V_A_2 all T_c (c_Relation_Ototal__on(T_c,V_A_2,V_r_2) <-> c_Relation_Ototal__on(T_c,V_A_2,hAPP(c_Relation_Oconverse(T_c,T_c),V_r_2)))) # label(fact_total__on__converse) # label(axiom) # label(non_clause). [assumption]. 6.07/6.13 4820 (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]. 6.07/6.13 4821 (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]. 6.07/6.13 4822 (all V_j_2 all V_i_2 all T_c (class_Rings_Oring__1(T_c) -> hAPP(c_Int_Oring__1__class_Oof__int(T_c),hAPP(c_Int_OAbs__Integ,hAPP(c_Relation_OImage(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Int_Ointrel),hAPP(hAPP(c_Set_Oinsert(tc_prod(tc_Nat_Onat,tc_Nat_Onat)),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_i_2),V_j_2)),c_Orderings_Obot__class_Obot(tc_fun(tc_prod(tc_Nat_Onat,tc_Nat_Onat),tc_HOL_Obool)))))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_c),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_c),V_i_2)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(T_c),V_j_2)))) # label(fact_of__int) # label(axiom) # label(non_clause). [assumption]. 6.07/6.13 4823 (all V_a all V_b all V_c all T_a (class_Divides_Osemiring__div(T_a) -> (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(T_a),V_c),V_b)) -> c_Divides_Odiv__class_Omod(T_a,V_a,V_c) = c_Divides_Odiv__class_Omod(T_a,c_Divides_Odiv__class_Omod(T_a,V_a,V_b),V_c)))) # label(fact_mod__mod__cancel) # label(axiom) # label(non_clause). [assumption]. 6.07/6.13 4824 (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]. 6.07/6.13 4825 (all V_Pa_2 all V_R_2 all T_c (c_Equiv__Relations_Oequivp(T_c,V_R_2) -> ((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)))) <-> (exists B_x1 hBOOL(hAPP(V_Pa_2,B_x1)))))) # label(fact_bex__reg__eqv) # label(axiom) # label(non_clause). [assumption]. 6.07/6.13 4826 (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]. 6.07/6.13 4827 (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]. 6.07/6.13 4828 (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]. 6.07/6.13 4829 (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_ys_2),hAPP(hAPP(c_List_Olist_OCons(T_c),V_y_2),c_List_Olist_ONil(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))))) # label(fact_append1__eq__conv) # label(axiom) # label(non_clause). [assumption]. 6.07/6.13 4830 (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]. 6.07/6.13 4831 (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]. 6.07/6.13 4832 (all V_k_2 all V_l_2 (hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),V_l_2),V_k_2)) -> c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = c_SetInterval_Oord_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) # label(axiom) # label(non_clause). [assumption]. 6.07/6.13 4833 (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),V_s_2) = hAPP(c_Set_OCollect(tc_prod(T_c,T_d)),hAPP(c_Product__Type_Oprod_Oprod__case(T_c,T_d,tc_HOL_Obool),hAPP(hAPP(c_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)))))) # label(fact_rel__comp__def) # label(axiom) # label(non_clause). [assumption]. 6.07/6.13 4834 (all V_v ((-hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),V_v))) -> hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_Osucc(V_v)) = hAPP(c_Nat_OSuc,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_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]. 6.07/6.13 4835 (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]. 6.07/6.13 4836 (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]. 6.07/6.13 4837 (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]. 6.07/6.13 4838 (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]. 6.07/6.13 4839 (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]. 6.07/6.13 4840 (all V_y_2 all V_x_2 all T_c (class_Lattices_Obounded__lattice__bot(T_c) -> (hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(T_c),V_x_2),V_y_2) = c_Orderings_Obot__class_Obot(T_c) <-> c_Orderings_Obot__class_Obot(T_c) = V_y_2 & V_x_2 = c_Orderings_Obot__class_Obot(T_c)))) # label(fact_sup__eq__bot__iff) # label(axiom) # label(non_clause). [assumption]. 6.07/6.13 4841 (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]. 6.07/6.13 4842 (all V_xq_2 all V_x_2 all V_f_2 all T_c all T_d 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)) = 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))) # label(fact__06) # label(axiom) # label(non_clause). [assumption]. 6.07/6.13 4843 (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]. 6.07/6.13 4844 (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]. 6.07/6.13 4845 (all V_a all V_b all T_a (class_Fields_Olinordered__field(T_a) -> (c_Groups_Ozero__class_Ozero(T_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)) = hAPP(c_Groups_Oabs__class_Oabs(T_a),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(T_a),V_a),V_b))))) # label(fact_nonzero__abs__divide) # label(axiom) # label(non_clause). [assumption]. 6.07/6.13 4846 (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]. 6.07/6.13 4847 (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]. 6.07/6.13 4848 (all V_n_2 all V_m_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),V_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)) <-> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),V_m_2),V_n_2))))) # label(fact_one__le__mult__iff) # label(axiom) # label(non_clause). [assumption]. 6.07/6.13 4849 (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]. 6.07/6.13 4850 (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]. 6.07/6.13 4851 (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]. 6.07/6.13 4852 (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]. 6.07/6.13 4853 (all V_Pa_2 all V_a1_2 all V_a0_2 (hBOOL(hAPP(c_Wellfounded_Oaccp(tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Nitpick_Onat__gcd__rel),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),V_a0_2),V_a1_2))) -> ((all B_x all B_y (hBOOL(hAPP(c_Wellfounded_Oaccp(tc_prod(tc_Nat_Onat,tc_Nat_Onat),c_Nitpick_Onat__gcd__rel),hAPP(hAPP(c_Product__Type_OPair(tc_Nat_Onat,tc_Nat_Onat),B_x),B_y))) -> ((c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != B_y -> hBOOL(hAPP(hAPP(V_Pa_2,B_y),c_Divides_Odiv__class_Omod(tc_Nat_Onat,B_x,B_y)))) -> hBOOL(hAPP(hAPP(V_Pa_2,B_x),B_y))))) -> hBOOL(hAPP(hAPP(V_Pa_2,V_a0_2),V_a1_2))))) # label(fact_nat__gcd_Opinduct) # label(axiom) # label(non_clause). [assumption]. 6.07/6.13 4854 (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_f,T_d,T_c,hAPP(c_Fun_Ocomp(T_e,T_d,T_f,V_f_2),V_g_2)),V_x_2) = 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))) # label(fact_apsnd__compose) # label(axiom) # label(non_clause). [assumption]. 6.07/6.13 4855 (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]. 6.07/6.13 4856 (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]. 6.07/6.13 4857 (all V_B_2 all V_t_2 all V_D_2 (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),V_D_2)) -> (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_t_2),c_Groups_Oone__class_Oone(tc_Int_Oint))),V_B_2)) -> (all B_x ((all B_xa (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xa),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint),V_D_2))) -> (all B_xb (hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B_xb),V_B_2)) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B_xb),B_xa) != B_x)))) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_t_2),B_x)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),V_t_2),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B_x),V_D_2))))))))) # label(fact_bset_I8_J) # label(axiom) # label(non_clause). [assumption]. 6.07/6.13 4858 (all V_xs all V_n all T_a hAPP(c_List_Orotate(T_a,V_n),hAPP(c_List_Orev(T_a),V_xs)) = hAPP(c_List_Orev(T_a),hAPP(c_List_Orotate(T_a,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)),c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_n,hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(T_a)),V_xs)))),V_xs))) # label(fact_rotate__rev) # label(axiom) # label(non_clause). [assumption]. 6.07/6.13 4859 (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]. 6.07/6.13 4860 (all V_M_2 all T_c ((exists B_x (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),B_x)) & hBOOL(hAPP(hAPP(c_member(tc_fun(T_c,tc_HOL_Obool)),B_x),V_M_2)))) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),c_Complete__Lattice_OInf__class_OInf(tc_fun(T_c,tc_HOL_Obool),V_M_2))))) # label(fact_finite__Inter) # label(axiom) # label(non_clause). [assumption]. 6.07/6.13 4861 (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]. 6.07/6.13 4862 (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]. 6.07/6.13 4863 (all V_Qa_2 all V_Pa_2 all T_c (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(c_Set_OCollect(T_c),V_Pa_2))) | hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(c_Set_OCollect(T_c),V_Qa_2))) -> hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),hAPP(c_Set_OCollect(T_c),hAPP(hAPP(c_COMBS(T_c,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),T_c),c_fconj),V_Pa_2)),V_Qa_2)))))) # label(fact_finite__Collect__conjI) # label(axiom) # label(non_clause). [assumption]. 6.07/6.13 4864 (all V_xs_2 all V_x_2 all T_c (class_Orderings_Olinorder(T_c) -> c_List_Oremove1(T_c,V_x_2,hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(T_c,T_c,c_COMBI(T_c)),V_x_2),V_xs_2)) = V_xs_2)) # label(fact_remove1__insort) # label(axiom) # label(non_clause). [assumption]. 6.07/6.13 4865 (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]. 6.07/6.13 4866 (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_x_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_y_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]. 6.07/6.13 4867 (all V_y all V_x all T_a (class_Int_Onumber__ring(T_a) -> hAPP(hAPP(c_Power_Opower__class_Opower(T_a),hAPP(hAPP(c_Groups_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)))) = 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)))) # label(fact_power2__sum) # label(axiom) # label(non_clause). [assumption]. 6.07/6.13 4868 (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]. 6.07/6.13 4869 (all V_r all V_q all V_b all V_a (hBOOL(hAPP(c_Divides_Odivmod__int__rel(V_a,V_b),hAPP(hAPP(c_Product__Type_OPair(tc_Int_Oint,tc_Int_Oint),V_q),V_r))) -> (c_Groups_Ozero__class_Ozero(tc_Int_Oint) != V_b -> 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]. 6.07/6.13 4870 (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]. 6.07/6.13 4871 (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]. 6.07/6.13 4872 (all V_yq all T_a c_Lazy__Sequence_Oappend(T_a,c_Lazy__Sequence_Olazy__sequence_OEmpty(T_a),V_yq) = V_yq) # label(fact__02) # label(axiom) # label(non_clause). [assumption]. 6.07/6.13 4873 (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]. 6.07/6.13 4874 (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]. 6.07/6.13 4875 (all V_m V_m = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),V_m),hAPP(c_Nat_OSuc,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))) # label(fact_div__1) # label(axiom) # label(non_clause). [assumption]. 6.07/6.13 4876 (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),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))) <-> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_c_2)) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2)),V_c_2)))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),V_c_2)) -> (hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_c_2),c_Groups_Ozero__class_Ozero(T_c))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2)),V_c_2)),V_b_2))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),V_c_2),c_Groups_Ozero__class_Ozero(T_c))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),c_Groups_Ozero__class_Ozero(T_c)),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2)))))))) # label(fact_divide__less__eq__number__of) # label(axiom) # label(non_clause). [assumption]. 6.07/6.13 4877 (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]. 6.07/6.13 4878 (all V_x all T_a (class_Orderings_Olinorder(T_a) -> hAPP(hAPP(c_Orderings_Oord__class_Omin(T_a),V_x),V_x) = V_x)) # label(fact_Min_Oidem) # label(axiom) # label(non_clause). [assumption]. 6.07/6.13 4879 (all T_c hAPP(c_Fun_Ocomp(tc_prod(tc_Int_Oint,tc_Int_Oint),tc_Int_Oint,T_c,c_Product__Type_Ofst(tc_Int_Oint,tc_Int_Oint)),c_Nitpick_ORep__Frac(T_c)) = c_Nitpick_Onum(T_c)) # label(fact_Nitpick_Onum__def) # label(axiom) # label(non_clause). [assumption]. 6.07/6.13 4880 (all V_y all V_x 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)) = 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))) # label(fact_Nat__Transfer_Otransfer__int__nat__functions_I3_J) # label(axiom) # label(non_clause). [assumption]. 6.07/6.13 4881 (all V_v all T_a (class_Int_Onumber__ring(T_a) -> hAPP(c_Int_Onumber__class_Onumber__of(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_v),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),hAPP(c_Int_Onumber__class_Onumber__of(T_a),V_v)),c_Groups_Oone__class_Oone(T_a)))) # label(fact_diff__special_I2_J) # label(axiom) # label(non_clause). [assumption]. 6.07/6.13 4882 (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]. 6.07/6.13 4883 (all V_a hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin))) # label(fact_zdiv__minus1__right) # label(axiom) # label(non_clause). [assumption]. 6.07/6.13 4884 (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]. 6.07/6.13 4885 (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_a_2 = V_a_H_2 & V_lazy__sequence_H_2 = V_lazy__sequence_2)) # label(fact_lazy__sequence_Oinject) # label(axiom) # label(non_clause). [assumption]. 6.07/6.13 4886 (all V_y_2 all V_A_2 all T_c all T_d (class_Groups_Ocomm__monoid__mult(T_d) -> (hBOOL(hAPP(c_Finite__Set_Ofinite(T_c),V_A_2)) -> hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(T_c,T_d),hAPP(c_COMBK(T_d,T_c),V_y_2)),V_A_2) = hAPP(hAPP(c_Power_Opower__class_Opower(T_d),V_y_2),hAPP(c_Finite__Set_Ocard(T_c),V_A_2))))) # label(fact_setprod__constant) # label(axiom) # label(non_clause). [assumption]. 6.07/6.13 4887 (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]. 6.07/6.13 4888 (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]. 6.07/6.13 4889 (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,V_c) = c_Divides_Odiv__class_Omod(T_a,V_b_H,V_c) -> c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a_H),V_b_H),V_c) = c_Divides_Odiv__class_Omod(T_a,hAPP(hAPP(c_Groups_Ominus__class_Ominus(T_a),V_a),V_b),V_c))))) # label(fact_mod__diff__cong) # label(axiom) # label(non_clause). [assumption]. 6.07/6.13 4890 (all V_w all V_z hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),V_z),V_w) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),V_z),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_w))) # label(fact_diff__int__def__symmetric) # label(axiom) # label(non_clause). [assumption]. 6.07/6.13 4891 (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]. 6.07/6.13 4892 (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 & V_m_2 = c_Groups_Oone__class_Oone(tc_Int_Oint)))) # label(fact_pos__zmult__eq__1__iff) # label(axiom) # label(non_clause). [assumption]. 6.07/6.13 4893 (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__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(T_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2)),c_Groups_Ozero__class_Ozero(T_c))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_b_2),hAPP(hAPP(c_Groups_Otimes__class_Otimes(T_c),V_a_2),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2))))) & (-hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(T_c),hAPP(c_Int_Onumber__class_Onumber__of(T_c),V_w_2)),c_Groups_Ozero__class_Ozero(T_c))) -> hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(T_c),V_a_2),c_Groups_Ozero__class_Ozero(T_c)))))))) # label(fact_le__divide__eq__number__of1) # label(axiom) # label(non_clause). [assumption]. 6.07/6.13 4894 (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]. 6.07/6.13 4895 (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]. 6.07/6.13 4896 (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]. 6.07/6.13 4897 (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(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_posDivAlg_Opinduct) # label(axiom) # label(non_clause). [assumption]. 6.07/6.13 4898 (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]. 6.07/6.13 4899 (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_x),V_y)) = 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]. 6.07/6.13 4900 (all V_m c_Divides_Odiv__class_Omod(tc_Nat_Onat,V_m,hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(c_Nat_OSuc,hAPP(c_Nat_OSuc,V_m)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) # label(fact_mod2__Suc__Suc) # label(axiom) # label(non_clause). [assumption]. 6.07/6.13 4901 (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]. 6.07/6.13 4902 (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]. 6.07/6.13 4903 (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]. 6.07/6.13 4904 (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]. 6.07/6.13 4905 (all V_b all V_a hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),V_a),V_b) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_a)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),V_b))) # label(fact_zdiv__zminus__zminus) # label(axiom) # label(non_clause). [assumption]. 6.21/6.21 4906 (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]. 6.21/6.21 4907 (all V_c all V_b all V_a all T_a (class_Rings_Ocomm__semiring__1(T_a) -> hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),V_b)),V_c) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_a),hAPP(hAPP(c_Groups_Oplus__class_Oplus(T_a),V_b),V_c)))) # label(fact_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J) # label(axiom) # label(non_clause). [assumption]. 6.21/6.21 6.21/6.21 ============================== end of process non-clausal formulas === 6.21/6.21 6.21/6.21 ============================== PROCESS INITIAL CLAUSES =============== 6.21/6.21 6.21/6.21 ============================== PREDICATE ELIMINATION ================= 6.21/6.21 4908 class_Orderings_Olinorder(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Orderings_Olinorder) # label(axiom). [assumption]. 6.21/6.21 4909 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f1(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(9)]. 6.21/6.21 4910 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),f1(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(9)]. 6.21/6.21 4911 -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(15)]. 6.21/6.21 4912 -class_Orderings_Olinorder(A) | c_List_Olinorder__class_Osorted(A,c_List_Olist_ONil(A)) # label(fact_sorted_Oequations_I1_J) # label(axiom). [clausify(18)]. 6.21/6.21 4913 -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(36)]. 6.21/6.21 4914 -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(53)]. 6.21/6.21 4915 -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(57)]. 6.21/6.21 4916 -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(57)]. 6.21/6.21 4917 -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)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),D)) # label(fact_min__max_Ole__sup__iff) # label(axiom). [clausify(57)]. 6.21/6.21 4918 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(hAPP(c_Orderings_Oord__class_Omin(A),C),hAPP(hAPP(c_Finite__Set_Ofold(A,A,c_Orderings_Oord__class_Omin(A)),D),B)) = hAPP(hAPP(c_Finite__Set_Ofold(A,A,c_Orderings_Oord__class_Omin(A)),D),hAPP(hAPP(c_Set_Oinsert(A),C),B)) # label(fact_min__max_Ofold__inf__insert) # label(axiom). [clausify(61)]. 6.21/6.21 4919 -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(64)]. 6.21/6.21 4920 -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__eq(A),B),C)) # label(fact_linorder__antisym__conv1) # label(axiom). [clausify(77)]. 6.21/6.21 4921 -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__eq(A),B),C)) # label(fact_linorder__antisym__conv1) # label(axiom). [clausify(77)]. 6.21/6.21 4922 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f13(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(81)]. 6.21/6.21 4923 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),f13(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(81)]. 6.21/6.21 4924 -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(85)]. 6.21/6.21 4925 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(c_List_Oset(A),f15(B,A)) = B # label(fact_finite__sorted__distinct__unique) # label(axiom). [clausify(86)]. 6.21/6.21 4926 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_List_Olinorder__class_Osorted(A,f15(B,A)) # label(fact_finite__sorted__distinct__unique) # label(axiom). [clausify(86)]. 6.21/6.21 4927 -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 = f15(B,A) # label(fact_finite__sorted__distinct__unique) # label(axiom). [clausify(86)]. 6.21/6.21 4928 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(c_List_Odistinct(A),f15(B,A))) # label(fact_finite__sorted__distinct__unique) # label(axiom). [clausify(86)]. 6.21/6.21 4929 -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(90)]. 6.21/6.21 4930 -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(90)]. 6.21/6.21 4931 -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(90)]. 6.21/6.21 4932 -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(114)]. 6.21/6.21 4933 -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(118)]. 6.21/6.21 4934 -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(119)]. 6.21/6.21 4935 -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(126)]. 6.21/6.21 4936 -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(182)]. 6.21/6.21 4937 -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(205)]. 6.21/6.21 4938 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),hAPP(c_Finite__Set_Ofold1(A,c_Orderings_Oord__class_Omin(A)),B))) | -hBOOL(hAPP(hAPP(c_member(A),D),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),D)) # label(fact_min__max_Obelow__fold1__iff) # label(axiom). [clausify(270)]. 6.21/6.21 4939 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = B | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),hAPP(c_Finite__Set_Ofold1(A,c_Orderings_Oord__class_Omin(A)),B))) | hBOOL(hAPP(hAPP(c_member(A),f34(C,B,A)),B)) # label(fact_min__max_Obelow__fold1__iff) # label(axiom). [clausify(270)]. 6.21/6.21 4940 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = B | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),hAPP(c_Finite__Set_Ofold1(A,c_Orderings_Oord__class_Omin(A)),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),f34(C,B,A))) # label(fact_min__max_Obelow__fold1__iff) # label(axiom). [clausify(270)]. 6.21/6.21 4941 -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(296)]. 6.21/6.21 4942 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omin(A),hAPP(B,f41(C,B,A))),hAPP(B,f42(C,B,A))) != hAPP(B,hAPP(hAPP(c_Orderings_Oord__class_Omin(A),f41(C,B,A)),f42(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(305)]. 6.21/6.21 4943 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = B | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = C | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) != hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,tc_HOL_Obool)),B),C) | c_Big__Operators_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(317)]. 6.21/6.21 4944 -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(324)]. 6.21/6.21 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),f1(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(4908,a,4909,a)]. 6.21/6.21 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),f1(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(4908,a,4910,a)]. 6.21/6.21 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(4908,a,4911,a)]. 6.21/6.23 Derived: c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,c_List_Olist_ONil(tc_Code__Numeral_Ocode__numeral)). [resolve(4908,a,4912,a)]. 6.21/6.23 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(4908,a,4913,a)]. 6.21/6.23 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),B),C))). [resolve(4908,a,4914,a)]. 6.21/6.23 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(4908,a,4915,a)]. 6.21/6.23 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(4908,a,4916,a)]. 6.21/6.23 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)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),B),C)). [resolve(4908,a,4917,a)]. 6.21/6.23 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),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)),C),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)),C),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),B),A)). [resolve(4908,a,4918,a)]. 6.21/6.23 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(4908,a,4919,a)]. 6.21/6.23 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),B)) | B != A | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),B)). [resolve(4908,a,4920,a)]. 6.21/6.23 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),B)) | B = A | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),B)). [resolve(4908,a,4921,a)]. 6.21/6.23 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),f13(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(4908,a,4922,a)]. 6.21/6.23 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),f13(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(4908,a,4923,a)]. 6.21/6.23 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(4908,a,4924,a)]. 6.25/6.24 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | hAPP(c_List_Oset(tc_Code__Numeral_Ocode__numeral),f15(A,tc_Code__Numeral_Ocode__numeral)) = A. [resolve(4908,a,4925,a)]. 6.25/6.24 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,f15(A,tc_Code__Numeral_Ocode__numeral)). [resolve(4908,a,4926,a)]. 6.25/6.24 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 = f15(A,tc_Code__Numeral_Ocode__numeral). [resolve(4908,a,4927,a)]. 6.25/6.24 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | hBOOL(hAPP(c_List_Odistinct(tc_Code__Numeral_Ocode__numeral),f15(A,tc_Code__Numeral_Ocode__numeral))). [resolve(4908,a,4928,a)]. 6.25/6.24 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(4908,a,4929,a)]. 6.25/6.24 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(4908,a,4930,a)]. 6.25/6.24 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(4908,a,4931,a)]. 6.25/6.24 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(4908,a,4932,a)]. 6.25/6.24 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(4908,a,4933,a)]. 6.25/6.25 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(4908,a,4934,a)]. 6.25/6.25 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(4908,a,4935,a)]. 6.25/6.26 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(4908,a,4936,a)]. 6.25/6.26 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(4908,a,4937,a)]. 6.25/6.26 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)) = A | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),B),hAPP(c_Finite__Set_Ofold1(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral)),A))) | -hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),C),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),B),C)). [resolve(4908,a,4938,a)]. 6.25/6.26 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)) = A | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),B),hAPP(c_Finite__Set_Ofold1(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral)),A))) | hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),f34(B,A,tc_Code__Numeral_Ocode__numeral)),A)). [resolve(4908,a,4939,a)]. 6.25/6.26 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)) = A | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),B),hAPP(c_Finite__Set_Ofold1(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral)),A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),B),f34(B,A,tc_Code__Numeral_Ocode__numeral))). [resolve(4908,a,4940,a)]. 6.25/6.26 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(4908,a,4941,a)]. 6.25/6.26 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),hAPP(A,f41(B,A,tc_Code__Numeral_Ocode__numeral))),hAPP(A,f42(B,A,tc_Code__Numeral_Ocode__numeral))) != hAPP(A,hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),f41(B,A,tc_Code__Numeral_Ocode__numeral)),f42(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(4908,a,4942,a)]. 6.25/6.26 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_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(4908,a,4943,a)]. 6.25/6.27 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(4908,a,4944,a)]. 6.25/6.27 4945 -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(349)]. 6.25/6.27 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(4945,a,4908,a)]. 6.25/6.27 4946 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_member(B),f47(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(368)]. 6.25/6.27 Derived: hBOOL(hAPP(hAPP(c_member(A),f47(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(4946,a,4908,a)]. 6.25/6.27 4947 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(B,C)),hAPP(B,f47(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(368)]. 6.25/6.27 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(A,B)),hAPP(A,f47(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(4947,a,4908,a)]. 6.25/6.27 4948 -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(371)]. 6.25/6.27 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(4948,a,4908,a)]. 6.25/6.27 4949 -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(383)]. 6.25/6.28 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(4949,a,4908,a)]. 6.25/6.28 4950 -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(400)]. 6.25/6.28 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(4950,a,4908,a)]. 6.25/6.28 4951 -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(401)]. 6.25/6.28 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(4951,a,4908,a)]. 6.25/6.28 4952 class_Orderings_Olinorder(tc_Int_Oint) # label(arity_Int__Oint__Orderings_Olinorder) # label(axiom). [assumption]. 6.25/6.28 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),f1(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(4952,a,4909,a)]. 6.25/6.28 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),f1(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(4952,a,4910,a)]. 6.25/6.28 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(4952,a,4911,a)]. 6.25/6.28 Derived: c_List_Olinorder__class_Osorted(tc_Int_Oint,c_List_Olist_ONil(tc_Int_Oint)). [resolve(4952,a,4912,a)]. 6.25/6.28 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(4952,a,4913,a)]. 6.25/6.28 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(4952,a,4914,a)]. 6.25/6.28 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(4952,a,4915,a)]. 6.25/6.31 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(4952,a,4916,a)]. 6.25/6.31 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)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),C)). [resolve(4952,a,4917,a)]. 6.25/6.31 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),B),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_Finite__Set_Ofold(tc_Int_Oint,tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint)),C),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),B),A)). [resolve(4952,a,4918,a)]. 6.25/6.31 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(4952,a,4919,a)]. 6.25/6.31 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | B != A | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)). [resolve(4952,a,4920,a)]. 6.25/6.31 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),f13(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(4952,a,4922,a)]. 6.25/6.31 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),f13(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(4952,a,4923,a)]. 6.25/6.31 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(4952,a,4924,a)]. 6.25/6.31 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | hAPP(c_List_Oset(tc_Int_Oint),f15(A,tc_Int_Oint)) = A. [resolve(4952,a,4925,a)]. 6.25/6.31 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | c_List_Olinorder__class_Osorted(tc_Int_Oint,f15(A,tc_Int_Oint)). [resolve(4952,a,4926,a)]. 6.25/6.31 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 = f15(A,tc_Int_Oint). [resolve(4952,a,4927,a)]. 6.25/6.31 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | hBOOL(hAPP(c_List_Odistinct(tc_Int_Oint),f15(A,tc_Int_Oint))). [resolve(4952,a,4928,a)]. 6.25/6.31 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(4952,a,4929,a)]. 6.25/6.31 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(4952,a,4930,a)]. 6.25/6.31 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(4952,a,4931,a)]. 6.25/6.31 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(4952,a,4932,a)]. 6.25/6.31 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(4952,a,4933,a)]. 6.25/6.31 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(4952,a,4934,a)]. 6.25/6.31 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(4952,a,4935,a)]. 6.25/6.31 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(4952,a,4936,a)]. 6.25/6.31 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(4952,a,4937,a)]. 6.25/6.31 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) = A | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),hAPP(c_Finite__Set_Ofold1(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint)),A))) | -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),C),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),C)). [resolve(4952,a,4938,a)]. 6.34/6.33 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) = A | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),hAPP(c_Finite__Set_Ofold1(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint)),A))) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),f34(B,A,tc_Int_Oint)),A)). [resolve(4952,a,4939,a)]. 6.34/6.33 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) = A | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),hAPP(c_Finite__Set_Ofold1(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint)),A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),f34(B,A,tc_Int_Oint))). [resolve(4952,a,4940,a)]. 6.34/6.33 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(4952,a,4941,a)]. 6.34/6.33 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),hAPP(A,f41(B,A,tc_Int_Oint))),hAPP(A,f42(B,A,tc_Int_Oint))) != hAPP(A,hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),f41(B,A,tc_Int_Oint)),f42(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(4952,a,4942,a)]. 6.34/6.33 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_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(4952,a,4943,a)]. 6.34/6.33 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(4952,a,4944,a)]. 6.34/6.33 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(4952,a,4945,a)]. 6.34/6.33 Derived: hBOOL(hAPP(hAPP(c_member(A),f47(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(4952,a,4946,a)]. 6.34/6.33 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(A,B)),hAPP(A,f47(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(4952,a,4947,a)]. 6.34/6.34 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(4952,a,4948,a)]. 6.34/6.34 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(4952,a,4949,a)]. 6.34/6.34 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(4952,a,4950,a)]. 6.34/6.34 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(4952,a,4951,a)]. 6.34/6.34 4953 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f57(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(421)]. 6.34/6.34 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),f57(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(4953,a,4908,a)]. 6.34/6.34 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),f57(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(4953,a,4952,a)]. 6.34/6.34 4954 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),f57(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(421)]. 6.34/6.34 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),f57(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(4954,a,4908,a)]. 6.34/6.34 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),f57(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(4954,a,4952,a)]. 6.34/6.34 4955 -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(422)]. 6.34/6.35 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(4955,a,4908,a)]. 6.34/6.35 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(4955,a,4952,a)]. 6.34/6.35 4956 -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(453)]. 6.34/6.35 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(4956,a,4908,a)]. 6.34/6.35 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(4956,a,4952,a)]. 6.34/6.35 4957 -class_Orderings_Olinorder(A) | -class_Complete__Lattice_Ocomplete__lattice(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),f64(E,D,C,B,A)),D)) # label(fact_less__SUP__iff) # label(axiom). [clausify(461)]. 6.34/6.35 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Code__Numeral_Ocode__numeral) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OSUPR(B,tc_Code__Numeral_Ocode__numeral),C),D))) | hBOOL(hAPP(hAPP(c_member(B),f64(D,C,B,A,tc_Code__Numeral_Ocode__numeral)),C)). [resolve(4957,a,4908,a)]. 6.34/6.35 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),f64(D,C,B,A,tc_Int_Oint)),C)). [resolve(4957,a,4952,a)]. 6.34/6.35 4958 -class_Orderings_Olinorder(A) | -class_Complete__Lattice_Ocomplete__lattice(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,f64(E,D,C,B,A)))) # label(fact_less__SUP__iff) # label(axiom). [clausify(461)]. 6.34/6.35 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Code__Numeral_Ocode__numeral) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),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,f64(D,C,B,A,tc_Code__Numeral_Ocode__numeral)))). [resolve(4958,a,4908,a)]. 6.34/6.35 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,f64(D,C,B,A,tc_Int_Oint)))). [resolve(4958,a,4952,a)]. 6.34/6.35 4959 -class_Orderings_Olinorder(A) | -class_Complete__Lattice_Ocomplete__lattice(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(461)]. 6.34/6.36 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Code__Numeral_Ocode__numeral) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),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(4959,a,4908,a)]. 6.34/6.36 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(4959,a,4952,a)]. 6.34/6.36 4960 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = B | hBOOL(hAPP(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(463)]. 6.34/6.36 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)) = 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(4960,a,4908,a)]. 6.34/6.36 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(4960,a,4952,a)]. 6.34/6.36 4961 -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(465)]. 6.34/6.36 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)) = 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(4961,a,4908,a)]. 6.34/6.36 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(4961,a,4952,a)]. 6.40/6.38 4962 -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(481)]. 6.40/6.38 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(4962,a,4908,a)]. 6.40/6.38 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(4962,a,4952,a)]. 6.40/6.38 4963 -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(511)]. 6.40/6.38 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(4963,a,4908,a)]. 6.40/6.38 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(4963,a,4952,a)]. 6.40/6.38 4964 -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(515)]. 6.40/6.38 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(4964,a,4908,a)]. 6.40/6.38 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(4964,a,4952,a)]. 6.40/6.38 4965 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),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(535)]. 6.40/6.38 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),B),C))). [resolve(4965,a,4908,a)]. 6.40/6.38 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),B),C))). [resolve(4965,a,4952,a)]. 6.40/6.38 4966 -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(581)]. 6.40/6.38 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(4966,a,4908,a)]. 6.40/6.39 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(4966,a,4952,a)]. 6.40/6.39 4967 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),B)) # label(fact_linorder__le__less__linear) # label(axiom). [clausify(586)]. 6.40/6.39 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),A)). [resolve(4967,a,4908,a)]. 6.40/6.39 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),A)). [resolve(4967,a,4952,a)]. 6.40/6.39 4968 -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(645)]. 6.40/6.39 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)) != 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(4968,a,4908,a)]. 6.40/6.39 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) != 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(4968,a,4952,a)]. 6.40/6.39 4969 -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(645)]. 6.40/6.39 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)) = 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(4969,a,4908,a)]. 6.40/6.40 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) = 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(4969,a,4952,a)]. 6.40/6.40 4970 -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(647)]. 6.40/6.40 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(4970,a,4908,a)]. 6.40/6.40 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(4970,a,4952,a)]. 6.40/6.40 4971 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),f77(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(662)]. 6.40/6.40 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),f77(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(4971,a,4908,a)]. 6.40/6.40 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),f77(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(4971,a,4952,a)]. 6.40/6.40 4972 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),f78(B,A)),f77(B,A))) | c_List_Olinorder__class_Osorted(A,B) # label(fact_sorted__equals__nth__mono) # label(axiom). [clausify(662)]. 6.40/6.40 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),f78(A,tc_Code__Numeral_Ocode__numeral)),f77(A,tc_Code__Numeral_Ocode__numeral))) | c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A). [resolve(4972,a,4908,a)]. 6.40/6.40 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),f78(A,tc_Int_Oint)),f77(A,tc_Int_Oint))) | c_List_Olinorder__class_Osorted(tc_Int_Oint,A). [resolve(4972,a,4952,a)]. 6.40/6.40 4973 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_List_Onth(A),B),f78(B,A))),hAPP(hAPP(c_List_Onth(A),B),f77(B,A)))) | c_List_Olinorder__class_Osorted(A,B) # label(fact_sorted__equals__nth__mono) # label(axiom). [clausify(662)]. 6.40/6.40 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),f78(A,tc_Code__Numeral_Ocode__numeral))),hAPP(hAPP(c_List_Onth(tc_Code__Numeral_Ocode__numeral),A),f77(A,tc_Code__Numeral_Ocode__numeral)))) | c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A). [resolve(4973,a,4908,a)]. 6.40/6.41 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_List_Onth(tc_Int_Oint),A),f78(A,tc_Int_Oint))),hAPP(hAPP(c_List_Onth(tc_Int_Oint),A),f77(A,tc_Int_Oint)))) | c_List_Olinorder__class_Osorted(tc_Int_Oint,A). [resolve(4973,a,4952,a)]. 6.40/6.41 4974 -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(662)]. 6.40/6.41 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_Code__Numeral_Ocode__numeral)),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),C),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),B),C)),hAPP(hAPP(c_List_Onth(tc_Code__Numeral_Ocode__numeral),B),A))) | -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,B). [resolve(4974,a,4908,a)]. 6.40/6.41 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_Int_Oint)),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),C),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_List_Onth(tc_Int_Oint),B),C)),hAPP(hAPP(c_List_Onth(tc_Int_Oint),B),A))) | -c_List_Olinorder__class_Osorted(tc_Int_Oint,B). [resolve(4974,a,4952,a)]. 6.40/6.41 4975 -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(699)]. 6.40/6.41 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | -hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),B),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)))) | c_Big__Operators_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(4975,a,4908,a)]. 6.40/6.41 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(4975,a,4952,a)]. 6.40/6.43 4976 -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(699)]. 6.40/6.43 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | -hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),B),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),A),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),B),c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)))) | c_Big__Operators_Olinorder__class_OMin(tc_Code__Numeral_Ocode__numeral,A) = B. [resolve(4976,a,4908,a)]. 6.40/6.43 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(4976,a,4952,a)]. 6.40/6.43 4977 -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(718)]. 6.40/6.43 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(4977,a,4908,a)]. 6.40/6.43 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(4977,a,4952,a)]. 6.40/6.43 4978 -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(738)]. 6.40/6.43 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(4978,a,4908,a)]. 6.40/6.43 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(4978,a,4952,a)]. 6.40/6.43 4979 -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(738)]. 6.40/6.43 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(4979,a,4908,a)]. 6.40/6.43 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(4979,a,4952,a)]. 6.40/6.43 4980 -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(774)]. 6.40/6.43 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_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(4980,a,4908,a)]. 6.48/6.46 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),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(4980,a,4952,a)]. 6.48/6.46 4981 -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(796)]. 6.48/6.46 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(4981,a,4908,a)]. 6.48/6.46 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(4981,a,4952,a)]. 6.48/6.46 4982 -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(813)]. 6.48/6.46 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(4982,a,4908,a)]. 6.48/6.46 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(4982,a,4952,a)]. 6.48/6.46 4983 -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(849)]. 6.48/6.46 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(4983,a,4908,a)]. 6.48/6.46 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(4983,a,4952,a)]. 6.48/6.46 4984 -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),f103(B,A)),f104(B,A))),hAPP(hAPP(c_Set_Oinsert(A),f103(B,A)),hAPP(hAPP(c_Set_Oinsert(A),f104(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(866)]. 6.48/6.47 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 | -hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),f103(A,tc_Code__Numeral_Ocode__numeral)),f104(A,tc_Code__Numeral_Ocode__numeral))),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),f103(A,tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),f104(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(4984,a,4908,a)]. 6.48/6.47 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) = A | -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),f103(A,tc_Int_Oint)),f104(A,tc_Int_Oint))),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),f103(A,tc_Int_Oint)),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),f104(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(4984,a,4952,a)]. 6.48/6.47 4985 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),C)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),D)) # label(fact_min__le__iff__disj) # label(axiom). [clausify(888)]. 6.48/6.47 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)),C)) | 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),B),C)). [resolve(4985,a,4908,a)]. 6.48/6.47 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)),C)) | 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),B),C)). [resolve(4985,a,4952,a)]. 6.48/6.47 4986 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),C)),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),D)) # label(fact_min__le__iff__disj) # label(axiom). [clausify(888)]. 6.48/6.47 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)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),C)). [resolve(4986,a,4908,a)]. 6.48/6.47 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)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),C)). [resolve(4986,a,4952,a)]. 6.48/6.47 4987 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),C)),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),D)) # label(fact_min__le__iff__disj) # label(axiom). [clausify(888)]. 6.48/6.47 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)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),B),C)). [resolve(4987,a,4908,a)]. 6.48/6.47 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)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),C)). [resolve(4987,a,4952,a)]. 6.50/6.48 4988 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_List_Odistinct(B),C)) | hBOOL(hAPP(c_List_Odistinct(B),c_List_Olinorder__class_Osort__key(B,A,D,C))) # label(fact_distinct__sort) # label(axiom). [clausify(907)]. 6.50/6.48 Derived: -hBOOL(hAPP(c_List_Odistinct(A),B)) | hBOOL(hAPP(c_List_Odistinct(A),c_List_Olinorder__class_Osort__key(A,tc_Code__Numeral_Ocode__numeral,C,B))). [resolve(4988,a,4908,a)]. 6.50/6.48 Derived: -hBOOL(hAPP(c_List_Odistinct(A),B)) | hBOOL(hAPP(c_List_Odistinct(A),c_List_Olinorder__class_Osort__key(A,tc_Int_Oint,C,B))). [resolve(4988,a,4952,a)]. 6.50/6.48 4989 -class_Orderings_Olinorder(A) | hBOOL(hAPP(c_List_Odistinct(B),C)) | -hBOOL(hAPP(c_List_Odistinct(B),c_List_Olinorder__class_Osort__key(B,A,D,C))) # label(fact_distinct__sort) # label(axiom). [clausify(907)]. 6.50/6.48 Derived: hBOOL(hAPP(c_List_Odistinct(A),B)) | -hBOOL(hAPP(c_List_Odistinct(A),c_List_Olinorder__class_Osort__key(A,tc_Code__Numeral_Ocode__numeral,C,B))). [resolve(4989,a,4908,a)]. 6.50/6.48 Derived: hBOOL(hAPP(c_List_Odistinct(A),B)) | -hBOOL(hAPP(c_List_Odistinct(A),c_List_Olinorder__class_Osort__key(A,tc_Int_Oint,C,B))). [resolve(4989,a,4952,a)]. 6.50/6.48 4990 -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(932)]. 6.50/6.48 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)) = A | -hBOOL(hAPP(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(4990,a,4908,a)]. 6.50/6.48 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) = A | -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),B)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) = B | c_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(4990,a,4952,a)]. 6.50/6.51 4991 -class_Orderings_Olinorder(A) | c_List_Olinorder__class_Osorted(A,c_List_Olist_ONil(A)) # label(fact_sorted_ONil) # label(axiom). [clausify(946)]. 6.50/6.51 4992 -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(952)]. 6.50/6.51 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(4992,a,4908,a)]. 6.50/6.51 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(4992,a,4952,a)]. 6.50/6.51 4993 -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(968)]. 6.50/6.51 Derived: c_List_Olist_ONil(A) != hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Code__Numeral_Ocode__numeral,B),C),D). [resolve(4993,a,4908,a)]. 6.50/6.51 Derived: c_List_Olist_ONil(A) != hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Int_Oint,B),C),D). [resolve(4993,a,4952,a)]. 6.50/6.51 4994 -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(972)]. 6.50/6.51 4995 -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(979)]. 6.50/6.51 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),C),B))). [resolve(4995,a,4908,a)]. 6.50/6.51 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(4995,a,4952,a)]. 6.50/6.51 4996 -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(979)]. 6.50/6.51 4997 -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),D),C))) # label(fact_less__max__iff__disj) # label(axiom). [clausify(979)]. 6.50/6.51 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),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),C),B))). [resolve(4997,a,4908,a)]. 6.50/6.51 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),C),B))). [resolve(4997,a,4952,a)]. 6.50/6.51 4998 -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(987)]. 6.55/6.53 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(4998,a,4908,a)]. 6.55/6.53 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(4998,a,4952,a)]. 6.55/6.53 4999 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | C = B | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),B)) # label(fact_not__less__iff__gr__or__eq) # label(axiom). [clausify(999)]. 6.55/6.53 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),B)) | B = A | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),B),A)). [resolve(4999,a,4908,a)]. 6.55/6.53 5000 -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(999)]. 6.55/6.53 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),B)) | B != A. [resolve(5000,a,4908,a)]. 6.55/6.53 5001 -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(999)]. 6.55/6.53 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(5001,a,4908,a)]. 6.55/6.53 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(5001,a,4952,a)]. 6.55/6.53 5002 -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(1008)]. 6.55/6.53 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(5002,a,4908,a)]. 6.55/6.53 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(5002,a,4952,a)]. 6.55/6.53 5003 -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_Ole__iff__inf) # label(axiom). [clausify(1017)]. 6.55/6.53 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),B)) | hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A),B) = A. [resolve(5003,a,4908,a)]. 6.55/6.53 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)) | hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),A),B) = A. [resolve(5003,a,4952,a)]. 6.55/6.53 5004 -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_Ole__iff__inf) # label(axiom). [clausify(1017)]. 6.55/6.53 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),B)) | hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A),B) != A. [resolve(5004,a,4908,a)]. 6.57/6.54 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)) | hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),A),B) != A. [resolve(5004,a,4952,a)]. 6.57/6.54 5005 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),B) = B # label(fact_min__max_Osup_Oidem) # label(axiom). [clausify(1023)]. 6.57/6.54 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A),A) = A. [resolve(5005,a,4908,a)]. 6.57/6.54 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),A) = A. [resolve(5005,a,4952,a)]. 6.57/6.54 5006 -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(1034)]. 6.57/6.54 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(5006,a,4908,a)]. 6.57/6.54 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(5006,a,4952,a)]. 6.57/6.54 5007 -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(1040)]. 6.57/6.54 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(5007,a,4908,a)]. 6.57/6.54 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(5007,a,4952,a)]. 6.57/6.54 5008 -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(1049)]. 6.57/6.54 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(5008,a,4908,a)]. 6.57/6.54 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(5008,a,4952,a)]. 6.57/6.56 5009 -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,B),C) != hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,D),E) | C = E # label(fact_atLeastLessThan__eq__iff) # label(axiom). [clausify(1051)]. 6.57/6.56 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,A),B) != hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Code__Numeral_Ocode__numeral,C),D) | B = D. [resolve(5009,a,4908,a)]. 6.57/6.56 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),C),D)) | hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,A),B) != hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,C),D) | B = D. [resolve(5009,a,4952,a)]. 6.57/6.56 5010 -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,B),C) != hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,D),E) | B = D # label(fact_atLeastLessThan__eq__iff) # label(axiom). [clausify(1051)]. 6.57/6.56 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,A),B) != hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Code__Numeral_Ocode__numeral,C),D) | A = C. [resolve(5010,a,4908,a)]. 6.57/6.56 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),C),D)) | hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,A),B) != hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,C),D) | A = C. [resolve(5010,a,4952,a)]. 6.57/6.56 5011 -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,B),C) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,D),E) | C != E | B != D # label(fact_atLeastLessThan__eq__iff) # label(axiom). [clausify(1051)]. 6.57/6.56 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,A),B) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Code__Numeral_Ocode__numeral,C),D) | B != D | A != C. [resolve(5011,a,4908,a)]. 6.57/6.56 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),C),D)) | hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,A),B) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Int_Oint,C),D) | B != D | A != C. [resolve(5011,a,4952,a)]. 6.57/6.56 5012 -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),B),C)) | C = B # label(fact_linorder__antisym__conv2) # label(axiom). [clausify(1052)]. 6.57/6.56 5013 -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),B),C)) | C != B # label(fact_linorder__antisym__conv2) # label(axiom). [clausify(1052)]. 6.57/6.56 5014 -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(1056)]. 6.57/6.57 5015 -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(1059)]. 6.57/6.57 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(5015,a,4908,a)]. 6.57/6.57 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(5015,a,4952,a)]. 6.57/6.57 5016 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_List_Odistinct(B),C)) | hBOOL(hAPP(hAPP(c_member(B),D),hAPP(c_List_Oset(B),C))) | hBOOL(hAPP(c_List_Odistinct(B),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(B,A,E),D),C))) # label(fact_distinct__insort) # label(axiom). [clausify(1090)]. 6.57/6.57 Derived: -hBOOL(hAPP(c_List_Odistinct(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),hAPP(c_List_Oset(A),B))) | hBOOL(hAPP(c_List_Odistinct(A),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Code__Numeral_Ocode__numeral,D),C),B))). [resolve(5016,a,4908,a)]. 6.57/6.57 Derived: -hBOOL(hAPP(c_List_Odistinct(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),hAPP(c_List_Oset(A),B))) | hBOOL(hAPP(c_List_Odistinct(A),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Int_Oint,D),C),B))). [resolve(5016,a,4952,a)]. 6.57/6.57 5017 -class_Orderings_Olinorder(A) | hBOOL(hAPP(c_List_Odistinct(B),C)) | -hBOOL(hAPP(c_List_Odistinct(B),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(B,A,D),E),C))) # label(fact_distinct__insort) # label(axiom). [clausify(1090)]. 6.57/6.60 Derived: hBOOL(hAPP(c_List_Odistinct(A),B)) | -hBOOL(hAPP(c_List_Odistinct(A),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Code__Numeral_Ocode__numeral,C),D),B))). [resolve(5017,a,4908,a)]. 6.57/6.60 Derived: hBOOL(hAPP(c_List_Odistinct(A),B)) | -hBOOL(hAPP(c_List_Odistinct(A),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Int_Oint,C),D),B))). [resolve(5017,a,4952,a)]. 6.57/6.60 5018 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_member(B),C),hAPP(c_List_Oset(B),D))) | -hBOOL(hAPP(c_List_Odistinct(B),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(B,A,E),C),D))) # label(fact_distinct__insort) # label(axiom). [clausify(1090)]. 6.57/6.60 Derived: -hBOOL(hAPP(hAPP(c_member(A),B),hAPP(c_List_Oset(A),C))) | -hBOOL(hAPP(c_List_Odistinct(A),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Code__Numeral_Ocode__numeral,D),B),C))). [resolve(5018,a,4908,a)]. 6.57/6.60 Derived: -hBOOL(hAPP(hAPP(c_member(A),B),hAPP(c_List_Oset(A),C))) | -hBOOL(hAPP(c_List_Odistinct(A),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Int_Oint,D),B),C))). [resolve(5018,a,4952,a)]. 6.57/6.60 5019 -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(1115)]. 6.57/6.60 5020 -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(1115)]. 6.57/6.60 5021 -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(1145)]. 6.57/6.60 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),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(5021,a,4908,a)]. 6.57/6.60 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(5021,a,4952,a)]. 6.57/6.60 5022 -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(1170)]. 6.57/6.60 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(5022,a,4908,a)]. 6.57/6.60 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(5022,a,4952,a)]. 6.57/6.60 5023 -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(1180)]. 6.57/6.60 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(5023,a,4908,a)]. 6.57/6.60 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(5023,a,4952,a)]. 6.57/6.60 5024 -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(1180)]. 6.57/6.60 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(5024,a,4908,a)]. 6.57/6.60 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(5024,a,4952,a)]. 6.65/6.62 5025 -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(1194)]. 6.65/6.62 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(5025,a,4908,a)]. 6.65/6.62 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(5025,a,4952,a)]. 6.65/6.62 5026 -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),f139(B,A)),f140(B,A))),hAPP(hAPP(c_Set_Oinsert(A),f139(B,A)),hAPP(hAPP(c_Set_Oinsert(A),f140(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(1209)]. 6.65/6.62 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),f139(A,tc_Code__Numeral_Ocode__numeral)),f140(A,tc_Code__Numeral_Ocode__numeral))),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),f139(A,tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),f140(A,tc_Code__Numeral_Ocode__numeral)),c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)))))) | hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),c_Big__Operators_Olinorder__class_OMax(tc_Code__Numeral_Ocode__numeral,A)),A)). [resolve(5026,a,4908,a)]. 6.65/6.62 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),f139(A,tc_Int_Oint)),f140(A,tc_Int_Oint))),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),f139(A,tc_Int_Oint)),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),f140(A,tc_Int_Oint)),c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)))))) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),c_Big__Operators_Olinorder__class_OMax(tc_Int_Oint,A)),A)). [resolve(5026,a,4952,a)]. 6.65/6.62 5027 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | C != B # label(fact_linorder__neq__iff) # label(axiom). [clausify(1220)]. 6.65/6.62 5028 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | B != C # label(fact_linorder__neq__iff) # label(axiom). [clausify(1220)]. 6.65/6.62 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),B)) | A != B. [resolve(5028,a,4908,a)]. 6.65/6.62 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | A != B. [resolve(5028,a,4952,a)]. 6.65/6.62 5029 -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_linorder__neq__iff) # label(axiom). [clausify(1220)]. 6.65/6.62 5030 -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(1227)]. 6.65/6.62 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(5030,a,4908,a)]. 6.65/6.64 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(5030,a,4952,a)]. 6.65/6.64 5031 -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(1238)]. 6.65/6.64 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(5031,a,4908,a)]. 6.65/6.64 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(5031,a,4952,a)]. 6.65/6.64 5032 -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(1251)]. 6.65/6.64 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(5032,a,4908,a)]. 6.65/6.64 Derived: c_Finite__Set_Ofun__left__comm__idem(tc_Int_Oint,tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint)). [resolve(5032,a,4952,a)]. 6.65/6.64 5033 -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(1267)]. 6.65/6.64 5034 -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(1282)]. 6.65/6.64 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(5034,a,4908,a)]. 6.65/6.66 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(5034,a,4952,a)]. 6.65/6.66 5035 -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),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),E)) | C = E # label(fact_atLeastLessThan__inj_I2_J) # label(axiom). [clausify(1290)]. 6.65/6.66 5036 -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(1295)]. 6.65/6.66 Derived: -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(c_List_Orev(tc_Code__Numeral_Ocode__numeral),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),C),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_Code__Numeral_Ocode__numeral)),A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Onth(tc_Code__Numeral_Ocode__numeral),A),C)),hAPP(hAPP(c_List_Onth(tc_Code__Numeral_Ocode__numeral),A),B))). [resolve(5036,a,4908,a)]. 6.65/6.66 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(5036,a,4952,a)]. 6.65/6.66 5037 -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(1315)]. 6.65/6.66 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(5037,a,4908,a)]. 6.65/6.66 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),A)). [resolve(5037,a,4952,a)]. 6.65/6.66 5038 -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(1320)]. 6.65/6.67 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(5038,a,4908,a)]. 6.65/6.67 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(5038,a,4952,a)]. 6.65/6.67 5039 -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(1330)]. 6.65/6.67 5040 -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(1351)]. 6.65/6.67 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(5040,a,4908,a)]. 6.65/6.67 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(5040,a,4952,a)]. 6.65/6.67 5041 -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(1352)]. 6.65/6.67 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(5041,a,4908,a)]. 6.65/6.67 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(5041,a,4952,a)]. 6.65/6.67 5042 -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(1353)]. 6.65/6.67 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(5042,a,4908,a)]. 6.71/6.70 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(5042,a,4952,a)]. 6.71/6.70 5043 -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(1353)]. 6.71/6.70 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(5043,a,4908,a)]. 6.71/6.70 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(5043,a,4952,a)]. 6.71/6.70 5044 -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(1371)]. 6.71/6.70 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(5044,a,4908,a)]. 6.71/6.70 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(5044,a,4952,a)]. 6.71/6.70 5045 -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(1421)]. 6.71/6.70 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(5045,a,4908,a)]. 6.71/6.70 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(5045,a,4952,a)]. 6.71/6.72 5046 -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(1453)]. 6.71/6.72 5047 -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(1464)]. 6.71/6.72 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(5047,a,4908,a)]. 6.71/6.72 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(5047,a,4952,a)]. 6.71/6.72 5048 -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(1472)]. 6.71/6.72 5049 -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(1477)]. 6.71/6.72 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(5049,a,4908,a)]. 6.71/6.72 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(5049,a,4952,a)]. 6.71/6.72 5050 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),C),D)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(A),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),C)),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),D)) # label(fact_min__max_Oinf__sup__distrib1) # label(axiom). [clausify(1478)]. 6.71/6.72 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),B),C)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A),B)),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A),C)). [resolve(5050,a,4908,a)]. 6.71/6.72 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),B),C)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),A),B)),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),A),C)). [resolve(5050,a,4952,a)]. 6.71/6.72 5051 -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(1502)]. 6.71/6.72 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(5051,a,4908,a)]. 6.71/6.72 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(5051,a,4952,a)]. 6.71/6.75 5052 -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(1532)]. 6.71/6.75 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),B),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)) = A | c_Big__Operators_Olinorder__class_OMin(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),B),A)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),B),c_Big__Operators_Olinorder__class_OMin(tc_Code__Numeral_Ocode__numeral,A)). [resolve(5052,a,4908,a)]. 6.71/6.75 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) = A | c_Big__Operators_Olinorder__class_OMin(tc_Int_Oint,hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),B),A)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),B),c_Big__Operators_Olinorder__class_OMin(tc_Int_Oint,A)). [resolve(5052,a,4952,a)]. 6.71/6.75 5053 -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(1546)]. 6.71/6.75 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(5053,a,4908,a)]. 6.71/6.75 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(5053,a,4952,a)]. 6.71/6.75 5054 -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(1556)]. 6.71/6.75 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(5054,a,4908,a)]. 6.71/6.75 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(5054,a,4952,a)]. 6.71/6.75 5055 -class_Complete__Lattice_Ocomplete__lattice(A) | -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_member(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),c_Complete__Lattice_OSup__class_OSup(A,C))) # label(fact_less__Sup__iff) # label(axiom). [clausify(1584)]. 6.71/6.75 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Code__Numeral_Ocode__numeral) | -hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),C),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),C),c_Complete__Lattice_OSup__class_OSup(tc_Code__Numeral_Ocode__numeral,B))). [resolve(5055,b,4908,a)]. 6.71/6.75 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),C),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),C),c_Complete__Lattice_OSup__class_OSup(tc_Int_Oint,B))). [resolve(5055,b,4952,a)]. 6.71/6.75 5056 -class_Complete__Lattice_Ocomplete__lattice(A) | -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_member(A),f178(B,C,A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),c_Complete__Lattice_OSup__class_OSup(A,B))) # label(fact_less__Sup__iff) # label(axiom). [clausify(1584)]. 6.79/6.79 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Code__Numeral_Ocode__numeral) | hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),f178(A,B,tc_Code__Numeral_Ocode__numeral)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),B),c_Complete__Lattice_OSup__class_OSup(tc_Code__Numeral_Ocode__numeral,A))). [resolve(5056,b,4908,a)]. 6.79/6.79 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),f178(A,B,tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),c_Complete__Lattice_OSup__class_OSup(tc_Int_Oint,A))). [resolve(5056,b,4952,a)]. 6.79/6.79 5057 -class_Complete__Lattice_Ocomplete__lattice(A) | -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),f178(C,B,A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Complete__Lattice_OSup__class_OSup(A,C))) # label(fact_less__Sup__iff) # label(axiom). [clausify(1584)]. 6.79/6.79 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),f178(B,A,tc_Code__Numeral_Ocode__numeral))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),c_Complete__Lattice_OSup__class_OSup(tc_Code__Numeral_Ocode__numeral,B))). [resolve(5057,b,4908,a)]. 6.79/6.79 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),f178(B,A,tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Complete__Lattice_OSup__class_OSup(tc_Int_Oint,B))). [resolve(5057,b,4952,a)]. 6.79/6.79 5058 -class_Lattices_Osemilattice__inf(A) | -class_Orderings_Olinorder(A) | c_Orderings_Oord__class_Omin(A) = c_Lattices_Osemilattice__inf__class_Oinf(A) # label(fact_inf__min) # label(axiom). [clausify(1585)]. 6.79/6.79 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(5058,b,4908,a)]. 6.79/6.79 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(5058,b,4952,a)]. 6.79/6.79 5059 -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(1621)]. 6.79/6.79 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(5059,a,4908,a)]. 6.79/6.79 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(5059,a,4952,a)]. 6.79/6.79 5060 -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),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),E)) | B = D # label(fact_atLeastLessThan__inj_I1_J) # label(axiom). [clausify(1624)]. 6.79/6.79 5061 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),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(1668)]. 6.82/6.80 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(5061,a,4908,a)]. 6.82/6.80 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),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(5061,a,4952,a)]. 6.82/6.80 5062 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omin(A),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),C)),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),D),C)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(A),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),D)),C) # label(fact_min__max_Osup__inf__distrib2) # label(axiom). [clausify(1678)]. 6.82/6.80 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A),B)),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),C),B)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A),C)),B). [resolve(5062,a,4908,a)]. 6.82/6.80 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),B)),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),C),B)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),A),C)),B). [resolve(5062,a,4952,a)]. 6.82/6.80 5063 -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(1688)]. 6.82/6.80 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(5063,a,4908,a)]. 6.82/6.80 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(5063,a,4952,a)]. 6.82/6.80 5064 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = B | hBOOL(hAPP(hAPP(c_member(A),f187(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(1688)]. 6.82/6.82 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),f187(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(5064,a,4908,a)]. 6.82/6.82 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),f187(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(5064,a,4952,a)]. 6.82/6.82 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__eq(A),f187(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(1688)]. 6.82/6.82 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),f187(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(5065,a,4908,a)]. 6.82/6.82 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),f187(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(5065,a,4952,a)]. 6.82/6.82 5066 -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(1728)]. 6.82/6.82 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),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(5066,a,4908,a)]. 6.82/6.82 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(5066,a,4952,a)]. 6.82/6.82 5067 -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(1743)]. 6.87/6.84 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(5067,a,4908,a)]. 6.87/6.84 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(5067,a,4952,a)]. 6.87/6.84 5068 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),B)) # label(fact_linorder__not__less) # label(axiom). [clausify(1747)]. 6.87/6.84 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),B),A)). [resolve(5068,a,4908,a)]. 6.87/6.84 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),A)). [resolve(5068,a,4952,a)]. 6.87/6.84 5069 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),B)) # label(fact_linorder__not__less) # label(axiom). [clausify(1747)]. 6.87/6.84 5070 -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(1782)]. 6.87/6.84 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(5070,a,4908,a)]. 6.87/6.84 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(5070,a,4952,a)]. 6.87/6.84 5071 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omin(A),hAPP(B,f197(C,B,A))),hAPP(B,f198(C,B,A))) != hAPP(B,hAPP(hAPP(c_Orderings_Oord__class_Omin(A),f197(C,B,A)),f198(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(1787)]. 6.87/6.84 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),hAPP(A,f197(B,A,tc_Code__Numeral_Ocode__numeral))),hAPP(A,f198(B,A,tc_Code__Numeral_Ocode__numeral))) != hAPP(A,hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),f197(B,A,tc_Code__Numeral_Ocode__numeral)),f198(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(5071,a,4908,a)]. 6.89/6.87 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),hAPP(A,f197(B,A,tc_Int_Oint))),hAPP(A,f198(B,A,tc_Int_Oint))) != hAPP(A,hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),f197(B,A,tc_Int_Oint)),f198(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(5071,a,4952,a)]. 6.89/6.87 5072 -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(1798)]. 6.89/6.87 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(5072,a,4908,a)]. 6.89/6.87 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(5072,a,4952,a)]. 6.89/6.87 5073 -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(1829)]. 6.89/6.87 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(5073,a,4908,a)]. 6.89/6.87 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(5073,a,4952,a)]. 6.89/6.87 5074 -class_Orderings_Olinorder(A) | B != C | hAPP(c_SetInterval_Oord__class_OatLeast(A),B) = hAPP(c_SetInterval_Oord__class_OatLeast(A),C) # label(fact_atLeast__eq__iff) # label(axiom). [clausify(1856)]. 6.89/6.87 Derived: A != B | hAPP(c_SetInterval_Oord__class_OatLeast(tc_Code__Numeral_Ocode__numeral),A) = hAPP(c_SetInterval_Oord__class_OatLeast(tc_Code__Numeral_Ocode__numeral),B). [resolve(5074,a,4908,a)]. 6.89/6.87 Derived: A != B | hAPP(c_SetInterval_Oord__class_OatLeast(tc_Int_Oint),A) = hAPP(c_SetInterval_Oord__class_OatLeast(tc_Int_Oint),B). [resolve(5074,a,4952,a)]. 6.89/6.87 5075 -class_Orderings_Olinorder(A) | B = C | hAPP(c_SetInterval_Oord__class_OatLeast(A),B) != hAPP(c_SetInterval_Oord__class_OatLeast(A),C) # label(fact_atLeast__eq__iff) # label(axiom). [clausify(1856)]. 6.89/6.87 Derived: A = B | hAPP(c_SetInterval_Oord__class_OatLeast(tc_Code__Numeral_Ocode__numeral),A) != hAPP(c_SetInterval_Oord__class_OatLeast(tc_Code__Numeral_Ocode__numeral),B). [resolve(5075,a,4908,a)]. 6.89/6.87 Derived: A = B | hAPP(c_SetInterval_Oord__class_OatLeast(tc_Int_Oint),A) != hAPP(c_SetInterval_Oord__class_OatLeast(tc_Int_Oint),B). [resolve(5075,a,4952,a)]. 6.89/6.90 5076 -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(1870)]. 6.89/6.90 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(5076,a,4908,a)]. 6.89/6.90 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(5076,a,4952,a)]. 6.89/6.90 5077 -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(1885)]. 6.89/6.90 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(5077,a,4908,a)]. 6.89/6.90 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(5077,a,4952,a)]. 6.89/6.90 5078 -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(1894)]. 6.89/6.90 5079 -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(1896)]. 6.89/6.90 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(5079,a,4908,a)]. 6.89/6.90 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(5079,a,4952,a)]. 6.89/6.90 5080 -class_Orderings_Olinorder(A) | hAPP(c_SetInterval_Oord__class_OatMost(A),B) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(A,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OgreaterThan(A),B)) # label(fact_Compl__greaterThan) # label(axiom). [clausify(1910)]. 6.89/6.90 Derived: hAPP(c_SetInterval_Oord__class_OatMost(tc_Code__Numeral_Ocode__numeral),A) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Code__Numeral_Ocode__numeral),A)). [resolve(5080,a,4908,a)]. 6.89/6.90 Derived: hAPP(c_SetInterval_Oord__class_OatMost(tc_Int_Oint),A) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Int_Oint),A)). [resolve(5080,a,4952,a)]. 6.89/6.90 5081 -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(1911)]. 6.89/6.90 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(5081,a,4908,a)]. 6.89/6.90 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(5081,a,4952,a)]. 6.89/6.90 5082 -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(1929)]. 6.96/6.92 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(5082,a,4908,a)]. 6.96/6.92 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(5082,a,4952,a)]. 6.96/6.92 5083 -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(1933)]. 6.96/6.92 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(5083,a,4908,a)]. 6.96/6.92 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(5083,a,4952,a)]. 6.96/6.92 5084 -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(1958)]. 6.96/6.92 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(5084,a,4908,a)]. 6.96/6.92 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(5084,a,4952,a)]. 6.96/6.94 5085 -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(1984)]. 6.96/6.94 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(5085,a,4908,a)]. 6.96/6.94 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(5085,a,4952,a)]. 6.96/6.94 5086 -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(1984)]. 6.96/6.94 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(5086,a,4908,a)]. 6.96/6.94 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(5086,a,4952,a)]. 6.96/6.94 5087 -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(2002)]. 6.96/6.94 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(5087,a,4908,a)]. 6.96/6.94 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(5087,a,4952,a)]. 6.96/6.94 5088 -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(2002)]. 6.96/6.94 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(5088,a,4908,a)]. 6.96/6.94 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(5088,a,4952,a)]. 7.00/6.98 5089 -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(2011)]. 7.00/6.98 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(5089,a,4908,a)]. 7.00/6.98 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(5089,a,4952,a)]. 7.00/6.98 5090 -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(2013)]. 7.00/6.98 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)) = 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(5090,a,4908,a)]. 7.00/6.98 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) = 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(5090,a,4952,a)]. 7.00/6.98 5091 -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(2060)]. 7.00/6.98 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(5091,a,4908,a)]. 7.00/6.98 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(5091,a,4952,a)]. 7.00/6.98 5092 -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(2103)]. 7.00/6.98 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(5092,a,4908,a)]. 7.00/6.98 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(5092,a,4952,a)]. 7.00/6.99 5093 -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(2117)]. 7.00/6.99 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(5093,a,4908,a)]. 7.00/6.99 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(5093,a,4952,a)]. 7.00/6.99 5094 -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),f247(B,A)),f248(B,A))),hAPP(hAPP(c_Set_Oinsert(A),f247(B,A)),hAPP(hAPP(c_Set_Oinsert(A),f248(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(2128)]. 7.00/6.99 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_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),f247(A,tc_Code__Numeral_Ocode__numeral)),f248(A,tc_Code__Numeral_Ocode__numeral))),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),f247(A,tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),f248(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(5094,a,4908,a)]. 7.00/6.99 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_Orderings_Oord__class_Omax(tc_Int_Oint),f247(A,tc_Int_Oint)),f248(A,tc_Int_Oint))),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),f247(A,tc_Int_Oint)),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),f248(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(5094,a,4952,a)]. 7.00/6.99 5095 -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(2133)]. 7.00/6.99 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)) = 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(5095,a,4908,a)]. 7.00/7.02 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(5095,a,4952,a)]. 7.00/7.02 5096 -class_Orderings_Olinorder(A) | -c_List_Olinorder__class_Osorted(A,B) | hBOOL(hAPP(hAPP(c_member(A),f257(B,C,A)),hAPP(c_List_Oset(A),B))) | c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Olist_OCons(A),C),B)) # label(fact_sorted__Cons) # label(axiom). [clausify(2172)]. 7.00/7.02 Derived: -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A) | hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),f257(A,B,tc_Code__Numeral_Ocode__numeral)),hAPP(c_List_Oset(tc_Code__Numeral_Ocode__numeral),A))) | c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Olist_OCons(tc_Code__Numeral_Ocode__numeral),B),A)). [resolve(5096,a,4908,a)]. 7.00/7.02 Derived: -c_List_Olinorder__class_Osorted(tc_Int_Oint,A) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),f257(A,B,tc_Int_Oint)),hAPP(c_List_Oset(tc_Int_Oint),A))) | c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Olist_OCons(tc_Int_Oint),B),A)). [resolve(5096,a,4952,a)]. 7.00/7.02 5097 -class_Orderings_Olinorder(A) | -c_List_Olinorder__class_Osorted(A,B) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),f257(B,C,A))) | c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Olist_OCons(A),C),B)) # label(fact_sorted__Cons) # label(axiom). [clausify(2172)]. 7.00/7.02 Derived: -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),B),f257(A,B,tc_Code__Numeral_Ocode__numeral))) | c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Olist_OCons(tc_Code__Numeral_Ocode__numeral),B),A)). [resolve(5097,a,4908,a)]. 7.00/7.02 Derived: -c_List_Olinorder__class_Osorted(tc_Int_Oint,A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),f257(A,B,tc_Int_Oint))) | c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Olist_OCons(tc_Int_Oint),B),A)). [resolve(5097,a,4952,a)]. 7.00/7.02 5098 -class_Orderings_Olinorder(A) | c_List_Olinorder__class_Osorted(A,B) | -c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Olist_OCons(A),C),B)) # label(fact_sorted__Cons) # label(axiom). [clausify(2172)]. 7.00/7.02 Derived: c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A) | -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Olist_OCons(tc_Code__Numeral_Ocode__numeral),B),A)). [resolve(5098,a,4908,a)]. 7.00/7.02 Derived: c_List_Olinorder__class_Osorted(tc_Int_Oint,A) | -c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Olist_OCons(tc_Int_Oint),B),A)). [resolve(5098,a,4952,a)]. 7.00/7.02 5099 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_member(A),B),hAPP(c_List_Oset(A),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),D),B)) | -c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Olist_OCons(A),D),C)) # label(fact_sorted__Cons) # label(axiom). [clausify(2172)]. 7.00/7.02 Derived: -hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),A),hAPP(c_List_Oset(tc_Code__Numeral_Ocode__numeral),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),C),A)) | -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Olist_OCons(tc_Code__Numeral_Ocode__numeral),C),B)). [resolve(5099,a,4908,a)]. 7.00/7.02 Derived: -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),A),hAPP(c_List_Oset(tc_Int_Oint),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),C),A)) | -c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Olist_OCons(tc_Int_Oint),C),B)). [resolve(5099,a,4952,a)]. 7.00/7.02 5100 -class_Int_Onumber(A) | -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(c_Int_Onumber__class_Onumber__of(A),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Int_Onumber__class_Onumber__of(A),C)),hAPP(c_Int_Onumber__class_Onumber__of(A),B))) # label(fact_le__number__of__eq__not__less) # label(axiom). [clausify(2177)]. 7.12/7.10 5101 -class_Int_Onumber(A) | -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(c_Int_Onumber__class_Onumber__of(A),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Int_Onumber__class_Onumber__of(A),C)),hAPP(c_Int_Onumber__class_Onumber__of(A),B))) # label(fact_le__number__of__eq__not__less) # label(axiom). [clausify(2177)]. 7.12/7.10 5102 -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(2181)]. 7.12/7.10 5103 -class_Orderings_Olinorder(A) | hAPP(c_SetInterval_Oord__class_OatLeast(A),B) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(A,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(A),B)) # label(fact_Compl__lessThan) # label(axiom). [clausify(2184)]. 7.12/7.10 Derived: hAPP(c_SetInterval_Oord__class_OatLeast(tc_Code__Numeral_Ocode__numeral),A) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Code__Numeral_Ocode__numeral),A)). [resolve(5103,a,4908,a)]. 7.12/7.10 Derived: hAPP(c_SetInterval_Oord__class_OatLeast(tc_Int_Oint),A) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Int_Oint),A)). [resolve(5103,a,4952,a)]. 7.12/7.10 5104 -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(2307)]. 7.12/7.10 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(5104,a,4908,a)]. 7.12/7.10 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(5104,a,4952,a)]. 7.12/7.10 5105 -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(2307)]. 7.12/7.10 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(5105,a,4908,a)]. 7.12/7.10 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(5105,a,4952,a)]. 7.12/7.10 5106 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omin(A),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),C)),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),D)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),C),D)) # label(fact_min__max_Osup__inf__distrib1) # label(axiom). [clausify(2356)]. 7.12/7.10 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A),B)),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A),C)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),B),C)). [resolve(5106,a,4908,a)]. 7.12/7.12 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),B)),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),C)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),B),C)). [resolve(5106,a,4952,a)]. 7.12/7.12 5107 -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),D) != hAPP(c_List_Oset(B),E) | D = E # label(fact_map__sorted__distinct__set__unique) # label(axiom). [clausify(2361)]. 7.12/7.12 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),C) != hAPP(c_List_Oset(A),D) | C = D. [resolve(5107,a,4908,a)]. 7.12/7.12 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),C) != hAPP(c_List_Oset(A),D) | C = D. [resolve(5107,a,4952,a)]. 7.12/7.12 5108 -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(2368)]. 7.12/7.12 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(5108,a,4908,a)]. 7.12/7.12 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(5108,a,4952,a)]. 7.12/7.12 5109 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),C) != C | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) # label(fact_min__max_Ole__iff__sup) # label(axiom). [clausify(2395)]. 7.12/7.13 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A),B) != B | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),B)). [resolve(5109,a,4908,a)]. 7.12/7.13 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),B) != B | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)). [resolve(5109,a,4952,a)]. 7.12/7.13 5110 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),C) = C | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) # label(fact_min__max_Ole__iff__sup) # label(axiom). [clausify(2395)]. 7.12/7.13 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A),B) = B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),B)). [resolve(5110,a,4908,a)]. 7.12/7.13 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),B) = B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)). [resolve(5110,a,4952,a)]. 7.12/7.13 5111 -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(2406)]. 7.12/7.13 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_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(5111,a,4908,a)]. 7.12/7.13 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(5111,a,4952,a)]. 7.12/7.13 5112 -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(2406)]. 7.12/7.13 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_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(5112,a,4908,a)]. 7.20/7.16 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) = 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(5112,a,4952,a)]. 7.20/7.16 5113 -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),B) != hAPP(c_List_Oset(A),C) | B = C # label(fact_sorted__distinct__set__unique) # label(axiom). [clausify(2408)]. 7.20/7.16 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),A) != hAPP(c_List_Oset(tc_Code__Numeral_Ocode__numeral),B) | A = B. [resolve(5113,a,4908,a)]. 7.20/7.16 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),A) != hAPP(c_List_Oset(tc_Int_Oint),B) | A = B. [resolve(5113,a,4952,a)]. 7.20/7.16 5114 -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(2442)]. 7.20/7.16 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(5114,a,4908,a)]. 7.20/7.16 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),B),C))). [resolve(5114,a,4952,a)]. 7.20/7.16 5115 -class_Orderings_Olinorder(A) | B != C | hAPP(c_SetInterval_Oord__class_OlessThan(A),B) = hAPP(c_SetInterval_Oord__class_OlessThan(A),C) # label(fact_lessThan__eq__iff) # label(axiom). [clausify(2452)]. 7.20/7.16 Derived: A != B | hAPP(c_SetInterval_Oord__class_OlessThan(tc_Code__Numeral_Ocode__numeral),A) = hAPP(c_SetInterval_Oord__class_OlessThan(tc_Code__Numeral_Ocode__numeral),B). [resolve(5115,a,4908,a)]. 7.20/7.16 Derived: A != B | hAPP(c_SetInterval_Oord__class_OlessThan(tc_Int_Oint),A) = hAPP(c_SetInterval_Oord__class_OlessThan(tc_Int_Oint),B). [resolve(5115,a,4952,a)]. 7.20/7.16 5116 -class_Orderings_Olinorder(A) | B = C | hAPP(c_SetInterval_Oord__class_OlessThan(A),B) != hAPP(c_SetInterval_Oord__class_OlessThan(A),C) # label(fact_lessThan__eq__iff) # label(axiom). [clausify(2452)]. 7.20/7.16 Derived: A = B | hAPP(c_SetInterval_Oord__class_OlessThan(tc_Code__Numeral_Ocode__numeral),A) != hAPP(c_SetInterval_Oord__class_OlessThan(tc_Code__Numeral_Ocode__numeral),B). [resolve(5116,a,4908,a)]. 7.20/7.18 Derived: A = B | hAPP(c_SetInterval_Oord__class_OlessThan(tc_Int_Oint),A) != hAPP(c_SetInterval_Oord__class_OlessThan(tc_Int_Oint),B). [resolve(5116,a,4952,a)]. 7.20/7.18 5117 -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(2453)]. 7.20/7.18 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(5117,a,4908,a)]. 7.20/7.18 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(5117,a,4952,a)]. 7.20/7.18 5118 -class_Orderings_Olinorder(A) | -class_Complete__Lattice_Ocomplete__lattice(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(2470)]. 7.20/7.18 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(5118,a,4908,a)]. 7.20/7.18 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),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(5118,a,4952,a)]. 7.20/7.18 5119 -class_Orderings_Olinorder(A) | -class_Complete__Lattice_Ocomplete__lattice(A) | hBOOL(hAPP(hAPP(c_member(A),f294(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(2470)]. 7.20/7.18 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Code__Numeral_Ocode__numeral) | hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),f294(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(5119,a,4908,a)]. 7.20/7.18 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),f294(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(5119,a,4952,a)]. 7.20/7.18 5120 -class_Orderings_Olinorder(A) | -class_Complete__Lattice_Ocomplete__lattice(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),f294(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(2470)]. 7.20/7.18 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Code__Numeral_Ocode__numeral) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),f294(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(5120,a,4908,a)]. 7.20/7.18 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),f294(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(5120,a,4952,a)]. 7.20/7.18 5121 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),B) = B # label(fact_min__max_Oinf_Oidem) # label(axiom). [clausify(2479)]. 7.20/7.18 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A),A) = A. [resolve(5121,a,4908,a)]. 7.20/7.20 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),A),A) = A. [resolve(5121,a,4952,a)]. 7.20/7.20 5122 -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(2485)]. 7.20/7.20 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)) = A | 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(5122,a,4908,a)]. 7.20/7.20 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) = A | 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(5122,a,4952,a)]. 7.20/7.20 5123 -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(2489)]. 7.20/7.20 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(5123,a,4908,a)]. 7.20/7.20 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(5123,a,4952,a)]. 7.20/7.20 5124 -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(2502)]. 7.20/7.20 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(5124,a,4908,a)]. 7.20/7.20 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(5124,a,4952,a)]. 7.20/7.20 5125 -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(2516)]. 7.20/7.20 5126 -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(2524)]. 7.26/7.25 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_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(5126,a,4908,a)]. 7.26/7.25 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_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(5126,a,4952,a)]. 7.26/7.25 5127 -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(2557)]. 7.26/7.25 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(5127,a,4908,a)]. 7.26/7.25 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(5127,a,4952,a)]. 7.26/7.25 5128 -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(2565)]. 7.26/7.25 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(5128,a,4908,a)]. 7.26/7.25 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(5128,a,4952,a)]. 7.26/7.25 5129 -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(2621)]. 7.26/7.25 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(5129,a,4908,a)]. 7.30/7.30 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(5129,a,4952,a)]. 7.30/7.30 5130 -class_Orderings_Olinorder(A) | B != C | hAPP(c_SetInterval_Oord__class_OgreaterThan(A),B) = hAPP(c_SetInterval_Oord__class_OgreaterThan(A),C) # label(fact_greaterThan__eq__iff) # label(axiom). [clausify(2635)]. 7.30/7.30 Derived: A != B | hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Code__Numeral_Ocode__numeral),A) = hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Code__Numeral_Ocode__numeral),B). [resolve(5130,a,4908,a)]. 7.30/7.30 Derived: A != B | hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Int_Oint),A) = hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Int_Oint),B). [resolve(5130,a,4952,a)]. 7.30/7.30 5131 -class_Orderings_Olinorder(A) | B = C | hAPP(c_SetInterval_Oord__class_OgreaterThan(A),B) != hAPP(c_SetInterval_Oord__class_OgreaterThan(A),C) # label(fact_greaterThan__eq__iff) # label(axiom). [clausify(2635)]. 7.30/7.30 Derived: A = B | hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Code__Numeral_Ocode__numeral),A) != hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Code__Numeral_Ocode__numeral),B). [resolve(5131,a,4908,a)]. 7.30/7.30 Derived: A = B | hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Int_Oint),A) != hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Int_Oint),B). [resolve(5131,a,4952,a)]. 7.30/7.30 5132 -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(2656)]. 7.30/7.30 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(5132,a,4908,a)]. 7.30/7.30 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(5132,a,4952,a)]. 7.30/7.30 5133 -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(2694)]. 7.30/7.30 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)) = A | c_Big__Operators_Olinorder__class_OMax(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),B),A)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),B),c_Big__Operators_Olinorder__class_OMax(tc_Code__Numeral_Ocode__numeral,A)). [resolve(5133,a,4908,a)]. 7.30/7.30 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_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(5133,a,4952,a)]. 7.30/7.30 5134 -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(2697)]. 7.30/7.30 5135 -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(2706)]. 7.30/7.30 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(5135,a,4908,a)]. 7.30/7.34 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(5135,a,4952,a)]. 7.30/7.34 5136 -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(2749)]. 7.30/7.34 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(5136,a,4908,a)]. 7.30/7.34 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(5136,a,4952,a)]. 7.30/7.34 5137 -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(2762)]. 7.30/7.34 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(5137,a,4908,a)]. 7.30/7.34 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(5137,a,4952,a)]. 7.30/7.34 5138 -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(2784)]. 7.30/7.34 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | c_Big__Operators_Olattice_OInf__fin(tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),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(5138,a,4908,a)]. 7.40/7.36 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(5138,a,4952,a)]. 7.40/7.36 5139 -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(2799)]. 7.40/7.36 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(5139,a,4908,a)]. 7.40/7.36 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(5139,a,4952,a)]. 7.40/7.36 5140 -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(2817)]. 7.40/7.36 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(5140,a,4908,a)]. 7.40/7.36 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(5140,a,4952,a)]. 7.40/7.36 5141 -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(2817)]. 7.40/7.36 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(5141,a,4908,a)]. 7.40/7.36 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(5141,a,4952,a)]. 7.40/7.36 5142 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),C),D))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) # label(fact_min__max_Ole__inf__iff) # label(axiom). [clausify(2817)]. 7.40/7.36 5143 -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(2819)]. 7.43/7.43 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(5143,a,4908,a)]. 7.43/7.43 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(5143,a,4952,a)]. 7.43/7.43 5144 -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__neqE) # label(axiom). [clausify(2823)]. 7.43/7.43 5145 -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__cases) # label(axiom). [clausify(2854)]. 7.43/7.43 5146 -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(2892)]. 7.43/7.43 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | -hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),B),A)) | c_Big__Operators_Olinorder__class_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(5146,a,4908,a)]. 7.43/7.43 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(5146,a,4952,a)]. 7.43/7.43 5147 -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(2903)]. 7.43/7.43 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(5147,a,4908,a)]. 7.43/7.43 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(5147,a,4952,a)]. 7.43/7.43 5148 -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(2950)]. 7.43/7.43 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(5148,a,4908,a)]. 7.50/7.45 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(5148,a,4952,a)]. 7.50/7.45 5149 -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(2954)]. 7.50/7.45 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)) = 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(5149,a,4908,a)]. 7.50/7.45 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(5149,a,4952,a)]. 7.50/7.45 5150 -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(2975)]. 7.50/7.45 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(5150,a,4908,a)]. 7.50/7.45 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(5150,a,4952,a)]. 7.50/7.45 5151 -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(2977)]. 7.50/7.45 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(5151,a,4908,a)]. 7.50/7.48 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(5151,a,4952,a)]. 7.50/7.48 5152 -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(2977)]. 7.50/7.48 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(5152,a,4908,a)]. 7.50/7.48 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(5152,a,4952,a)]. 7.50/7.48 5153 -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(2986)]. 7.50/7.48 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(5153,a,4908,a)]. 7.50/7.48 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(5153,a,4952,a)]. 7.50/7.48 5154 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_List_Oremove1(A,C,c_List_Olinorder__class_Osorted__list__of__set(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))))) # label(fact_sorted__list__of__set__remove) # label(axiom). [clausify(2987)]. 7.50/7.48 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | c_List_Oremove1(tc_Code__Numeral_Ocode__numeral,B,c_List_Olinorder__class_Osorted__list__of__set(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))))). [resolve(5154,a,4908,a)]. 7.50/7.48 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | c_List_Oremove1(tc_Int_Oint,B,c_List_Olinorder__class_Osorted__list__of__set(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))))). [resolve(5154,a,4952,a)]. 7.50/7.48 5155 -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(3019)]. 7.55/7.50 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(5155,a,4908,a)]. 7.55/7.50 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(5155,a,4952,a)]. 7.55/7.50 5156 -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(3038)]. 7.55/7.50 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(5156,a,4908,a)]. 7.55/7.50 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(5156,a,4952,a)]. 7.55/7.50 5157 -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),f341(B,A)),f342(B,A))),hAPP(hAPP(c_Set_Oinsert(A),f341(B,A)),hAPP(hAPP(c_Set_Oinsert(A),f342(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(3042)]. 7.55/7.50 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)) = A | -hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),f341(A,tc_Code__Numeral_Ocode__numeral)),f342(A,tc_Code__Numeral_Ocode__numeral))),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),f341(A,tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),f342(A,tc_Code__Numeral_Ocode__numeral)),c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)))))) | hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),c_Big__Operators_Olinorder__class_OMin(tc_Code__Numeral_Ocode__numeral,A)),A)). [resolve(5157,a,4908,a)]. 7.55/7.50 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(hAPP(c_member(tc_Int_Oint),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),f341(A,tc_Int_Oint)),f342(A,tc_Int_Oint))),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),f341(A,tc_Int_Oint)),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),f342(A,tc_Int_Oint)),c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)))))) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),c_Big__Operators_Olinorder__class_OMin(tc_Int_Oint,A)),A)). [resolve(5157,a,4952,a)]. 7.55/7.50 5158 -class_Orderings_Olinorder(A) | -c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Oappend(A),B),C)) | c_List_Olinorder__class_Osorted(A,B) # label(fact_sorted__append) # label(axiom). [clausify(3071)]. 7.55/7.51 Derived: -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Oappend(tc_Code__Numeral_Ocode__numeral),A),B)) | c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A). [resolve(5158,a,4908,a)]. 7.55/7.51 Derived: -c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Oappend(tc_Int_Oint),A),B)) | c_List_Olinorder__class_Osorted(tc_Int_Oint,A). [resolve(5158,a,4952,a)]. 7.55/7.51 5159 -class_Orderings_Olinorder(A) | -c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Oappend(A),B),C)) | -hBOOL(hAPP(hAPP(c_member(A),D),hAPP(c_List_Oset(A),B))) | -hBOOL(hAPP(hAPP(c_member(A),E),hAPP(c_List_Oset(A),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),D),E)) # label(fact_sorted__append) # label(axiom). [clausify(3071)]. 7.55/7.51 Derived: -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Oappend(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),A))) | -hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),D),hAPP(c_List_Oset(tc_Code__Numeral_Ocode__numeral),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),C),D)). [resolve(5159,a,4908,a)]. 7.55/7.51 Derived: -c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Oappend(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),C),hAPP(c_List_Oset(tc_Int_Oint),A))) | -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),D),hAPP(c_List_Oset(tc_Int_Oint),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),C),D)). [resolve(5159,a,4952,a)]. 7.55/7.51 5160 -class_Orderings_Olinorder(A) | -c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Oappend(A),B),C)) | c_List_Olinorder__class_Osorted(A,C) # label(fact_sorted__append) # label(axiom). [clausify(3071)]. 7.55/7.51 Derived: -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Oappend(tc_Code__Numeral_Ocode__numeral),A),B)) | c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,B). [resolve(5160,a,4908,a)]. 7.55/7.51 Derived: -c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Oappend(tc_Int_Oint),A),B)) | c_List_Olinorder__class_Osorted(tc_Int_Oint,B). [resolve(5160,a,4952,a)]. 7.55/7.51 5161 -class_Orderings_Olinorder(A) | c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Oappend(A),B),C)) | -c_List_Olinorder__class_Osorted(A,B) | hBOOL(hAPP(hAPP(c_member(A),f346(C,B,A)),hAPP(c_List_Oset(A),B))) | -c_List_Olinorder__class_Osorted(A,C) # label(fact_sorted__append) # label(axiom). [clausify(3071)]. 7.55/7.51 Derived: c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Oappend(tc_Code__Numeral_Ocode__numeral),A),B)) | -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A) | hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),f346(B,A,tc_Code__Numeral_Ocode__numeral)),hAPP(c_List_Oset(tc_Code__Numeral_Ocode__numeral),A))) | -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,B). [resolve(5161,a,4908,a)]. 7.55/7.51 Derived: c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Oappend(tc_Int_Oint),A),B)) | -c_List_Olinorder__class_Osorted(tc_Int_Oint,A) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),f346(B,A,tc_Int_Oint)),hAPP(c_List_Oset(tc_Int_Oint),A))) | -c_List_Olinorder__class_Osorted(tc_Int_Oint,B). [resolve(5161,a,4952,a)]. 7.55/7.51 5162 -class_Orderings_Olinorder(A) | c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Oappend(A),B),C)) | -c_List_Olinorder__class_Osorted(A,B) | hBOOL(hAPP(hAPP(c_member(A),f347(C,B,A)),hAPP(c_List_Oset(A),C))) | -c_List_Olinorder__class_Osorted(A,C) # label(fact_sorted__append) # label(axiom). [clausify(3071)]. 7.55/7.51 Derived: c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Oappend(tc_Code__Numeral_Ocode__numeral),A),B)) | -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A) | hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),f347(B,A,tc_Code__Numeral_Ocode__numeral)),hAPP(c_List_Oset(tc_Code__Numeral_Ocode__numeral),B))) | -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,B). [resolve(5162,a,4908,a)]. 7.55/7.54 Derived: c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Oappend(tc_Int_Oint),A),B)) | -c_List_Olinorder__class_Osorted(tc_Int_Oint,A) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),f347(B,A,tc_Int_Oint)),hAPP(c_List_Oset(tc_Int_Oint),B))) | -c_List_Olinorder__class_Osorted(tc_Int_Oint,B). [resolve(5162,a,4952,a)]. 7.55/7.54 5163 -class_Orderings_Olinorder(A) | c_List_Olinorder__class_Osorted(A,hAPP(hAPP(c_List_Oappend(A),B),C)) | -c_List_Olinorder__class_Osorted(A,B) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),f346(C,B,A)),f347(C,B,A))) | -c_List_Olinorder__class_Osorted(A,C) # label(fact_sorted__append) # label(axiom). [clausify(3071)]. 7.55/7.54 Derived: c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_List_Oappend(tc_Code__Numeral_Ocode__numeral),A),B)) | -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),f346(B,A,tc_Code__Numeral_Ocode__numeral)),f347(B,A,tc_Code__Numeral_Ocode__numeral))) | -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,B). [resolve(5163,a,4908,a)]. 7.55/7.54 Derived: c_List_Olinorder__class_Osorted(tc_Int_Oint,hAPP(hAPP(c_List_Oappend(tc_Int_Oint),A),B)) | -c_List_Olinorder__class_Osorted(tc_Int_Oint,A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),f346(B,A,tc_Int_Oint)),f347(B,A,tc_Int_Oint))) | -c_List_Olinorder__class_Osorted(tc_Int_Oint,B). [resolve(5163,a,4952,a)]. 7.55/7.54 5164 -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(3081)]. 7.55/7.54 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(5164,a,4908,a)]. 7.55/7.54 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(5164,a,4952,a)]. 7.55/7.54 5165 -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(3084)]. 7.55/7.54 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(5165,a,4908,a)]. 7.55/7.54 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(5165,a,4952,a)]. 7.55/7.54 5166 -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(3094)]. 7.55/7.54 5167 -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(3113)]. 7.59/7.60 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(5167,a,4908,a)]. 7.59/7.60 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(5167,a,4952,a)]. 7.59/7.60 5168 -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(3176)]. 7.59/7.60 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(5168,a,4908,a)]. 7.59/7.60 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(5168,a,4952,a)]. 7.59/7.60 5169 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),C),D))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),D)) # label(fact_min__less__iff__conj) # label(axiom). [clausify(3182)]. 7.59/7.60 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),B),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),C)). [resolve(5169,a,4908,a)]. 7.59/7.60 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),B),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),C)). [resolve(5169,a,4952,a)]. 7.59/7.60 5170 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),C),D))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) # label(fact_min__less__iff__conj) # label(axiom). [clausify(3182)]. 7.59/7.60 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),B),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),B)). [resolve(5170,a,4908,a)]. 7.59/7.60 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),B),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)). [resolve(5170,a,4952,a)]. 7.59/7.60 5171 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),C),D))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) # label(fact_min__less__iff__conj) # label(axiom). [clausify(3182)]. 7.59/7.60 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),B),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),B)). [resolve(5171,a,4908,a)]. 7.59/7.60 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),B),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)). [resolve(5171,a,4952,a)]. 7.59/7.63 5172 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = B | hBOOL(hAPP(hAPP(c_member(A),c_Big__Operators_Olinorder__class_OMax(A,B)),B)) # label(fact_Max__in) # label(axiom). [clausify(3187)]. 7.59/7.63 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)) = A | hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),c_Big__Operators_Olinorder__class_OMax(tc_Code__Numeral_Ocode__numeral,A)),A)). [resolve(5172,a,4908,a)]. 7.59/7.63 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(5172,a,4952,a)]. 7.59/7.63 5173 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),B) = B # label(fact_Max_Oidem) # label(axiom). [clausify(3198)]. 7.59/7.63 5174 -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(3206)]. 7.59/7.63 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(5174,a,4908,a)]. 7.59/7.63 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(5174,a,4952,a)]. 7.59/7.63 5175 -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(3209)]. 7.59/7.63 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(5175,a,4908,a)]. 7.59/7.63 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(5175,a,4952,a)]. 7.59/7.63 5176 -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(3232)]. 7.59/7.63 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(5176,a,4908,a)]. 7.59/7.63 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(5176,a,4952,a)]. 7.73/7.69 5177 -class_Orderings_Olinorder(A) | c_List_Olinorder__class_Osort__key(A,A,c_COMBI(A),c_List_Oremdups(A,B)) = c_List_Olinorder__class_Osorted__list__of__set(A,hAPP(c_List_Oset(A),B)) # label(fact_sorted__list__of__set__sort__remdups) # label(axiom). [clausify(3267)]. 7.73/7.69 Derived: 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)) = c_List_Olinorder__class_Osorted__list__of__set(tc_Code__Numeral_Ocode__numeral,hAPP(c_List_Oset(tc_Code__Numeral_Ocode__numeral),A)). [resolve(5177,a,4908,a)]. 7.73/7.69 Derived: c_List_Olinorder__class_Osort__key(tc_Int_Oint,tc_Int_Oint,c_COMBI(tc_Int_Oint),c_List_Oremdups(tc_Int_Oint,A)) = c_List_Olinorder__class_Osorted__list__of__set(tc_Int_Oint,hAPP(c_List_Oset(tc_Int_Oint),A)). [resolve(5177,a,4952,a)]. 7.73/7.69 5178 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),f376(B,A)),f377(B,A))) | c_List_Olinorder__class_Osorted(A,B) # label(fact_sorted__nth__monoI) # label(axiom). [clausify(3278)]. 7.73/7.69 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),f376(A,tc_Code__Numeral_Ocode__numeral)),f377(A,tc_Code__Numeral_Ocode__numeral))) | c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A). [resolve(5178,a,4908,a)]. 7.73/7.69 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),f376(A,tc_Int_Oint)),f377(A,tc_Int_Oint))) | c_List_Olinorder__class_Osorted(tc_Int_Oint,A). [resolve(5178,a,4952,a)]. 7.73/7.69 5179 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),f377(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(3278)]. 7.73/7.69 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),f377(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(5179,a,4908,a)]. 7.73/7.69 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),f377(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(5179,a,4952,a)]. 7.73/7.69 5180 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_List_Onth(A),B),f376(B,A))),hAPP(hAPP(c_List_Onth(A),B),f377(B,A)))) | c_List_Olinorder__class_Osorted(A,B) # label(fact_sorted__nth__monoI) # label(axiom). [clausify(3278)]. 7.73/7.69 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),f376(A,tc_Code__Numeral_Ocode__numeral))),hAPP(hAPP(c_List_Onth(tc_Code__Numeral_Ocode__numeral),A),f377(A,tc_Code__Numeral_Ocode__numeral)))) | c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A). [resolve(5180,a,4908,a)]. 7.73/7.69 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_List_Onth(tc_Int_Oint),A),f376(A,tc_Int_Oint))),hAPP(hAPP(c_List_Onth(tc_Int_Oint),A),f377(A,tc_Int_Oint)))) | c_List_Olinorder__class_Osorted(tc_Int_Oint,A). [resolve(5180,a,4952,a)]. 7.73/7.69 5181 -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(3279)]. 7.73/7.69 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(5181,a,4908,a)]. 7.73/7.69 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(5181,a,4952,a)]. 7.73/7.69 5182 -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(3311)]. 7.73/7.71 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(5182,a,4908,a)]. 7.73/7.71 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(5182,a,4952,a)]. 7.73/7.71 5183 -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(3319)]. 7.73/7.71 5184 -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(3319)]. 7.73/7.71 5185 -class_Orderings_Olinorder(A) | -c_List_Olinorder__class_Osorted(A,B) | hAPP(hAPP(c_List_Olist_OCons(A),f382(B,A)),f381(B,A)) = B | c_List_Olist_ONil(A) = B # label(fact_sorted_Osimps) # label(axiom). [clausify(3335)]. 7.73/7.71 Derived: -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A) | hAPP(hAPP(c_List_Olist_OCons(tc_Code__Numeral_Ocode__numeral),f382(A,tc_Code__Numeral_Ocode__numeral)),f381(A,tc_Code__Numeral_Ocode__numeral)) = A | c_List_Olist_ONil(tc_Code__Numeral_Ocode__numeral) = A. [resolve(5185,a,4908,a)]. 7.73/7.71 Derived: -c_List_Olinorder__class_Osorted(tc_Int_Oint,A) | hAPP(hAPP(c_List_Olist_OCons(tc_Int_Oint),f382(A,tc_Int_Oint)),f381(A,tc_Int_Oint)) = A | c_List_Olist_ONil(tc_Int_Oint) = A. [resolve(5185,a,4952,a)]. 7.73/7.71 5186 -class_Orderings_Olinorder(A) | -c_List_Olinorder__class_Osorted(A,B) | c_List_Olinorder__class_Osorted(A,f381(B,A)) | c_List_Olist_ONil(A) = B # label(fact_sorted_Osimps) # label(axiom). [clausify(3335)]. 7.73/7.71 Derived: -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A) | c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,f381(A,tc_Code__Numeral_Ocode__numeral)) | c_List_Olist_ONil(tc_Code__Numeral_Ocode__numeral) = A. [resolve(5186,a,4908,a)]. 7.73/7.71 Derived: -c_List_Olinorder__class_Osorted(tc_Int_Oint,A) | c_List_Olinorder__class_Osorted(tc_Int_Oint,f381(A,tc_Int_Oint)) | c_List_Olist_ONil(tc_Int_Oint) = A. [resolve(5186,a,4952,a)]. 7.73/7.71 5187 -class_Orderings_Olinorder(A) | -c_List_Olinorder__class_Osorted(A,B) | -hBOOL(hAPP(hAPP(c_member(A),C),hAPP(c_List_Oset(A),f381(B,A)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),f382(B,A)),C)) | c_List_Olist_ONil(A) = B # label(fact_sorted_Osimps) # label(axiom). [clausify(3335)]. 7.73/7.71 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),f381(A,tc_Code__Numeral_Ocode__numeral)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),f382(A,tc_Code__Numeral_Ocode__numeral)),B)) | c_List_Olist_ONil(tc_Code__Numeral_Ocode__numeral) = A. [resolve(5187,a,4908,a)]. 7.73/7.71 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),f381(A,tc_Int_Oint)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),f382(A,tc_Int_Oint)),B)) | c_List_Olist_ONil(tc_Int_Oint) = A. [resolve(5187,a,4952,a)]. 7.73/7.71 5188 -class_Orderings_Olinorder(A) | c_List_Olinorder__class_Osorted(A,B) | hAPP(hAPP(c_List_Olist_OCons(A),C),D) != B | -c_List_Olinorder__class_Osorted(A,D) | hBOOL(hAPP(hAPP(c_member(A),f383(B,A,D,C)),hAPP(c_List_Oset(A),D))) # label(fact_sorted_Osimps) # label(axiom). [clausify(3335)]. 7.73/7.71 Derived: c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A) | hAPP(hAPP(c_List_Olist_OCons(tc_Code__Numeral_Ocode__numeral),B),C) != A | -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,C) | hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),f383(A,tc_Code__Numeral_Ocode__numeral,C,B)),hAPP(c_List_Oset(tc_Code__Numeral_Ocode__numeral),C))). [resolve(5188,a,4908,a)]. 7.73/7.73 Derived: c_List_Olinorder__class_Osorted(tc_Int_Oint,A) | hAPP(hAPP(c_List_Olist_OCons(tc_Int_Oint),B),C) != A | -c_List_Olinorder__class_Osorted(tc_Int_Oint,C) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),f383(A,tc_Int_Oint,C,B)),hAPP(c_List_Oset(tc_Int_Oint),C))). [resolve(5188,a,4952,a)]. 7.73/7.73 5189 -class_Orderings_Olinorder(A) | c_List_Olinorder__class_Osorted(A,B) | hAPP(hAPP(c_List_Olist_OCons(A),C),D) != B | -c_List_Olinorder__class_Osorted(A,D) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),f383(B,A,D,C))) # label(fact_sorted_Osimps) # label(axiom). [clausify(3335)]. 7.73/7.73 Derived: c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A) | hAPP(hAPP(c_List_Olist_OCons(tc_Code__Numeral_Ocode__numeral),B),C) != A | -c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,C) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),B),f383(A,tc_Code__Numeral_Ocode__numeral,C,B))). [resolve(5189,a,4908,a)]. 7.73/7.73 Derived: c_List_Olinorder__class_Osorted(tc_Int_Oint,A) | hAPP(hAPP(c_List_Olist_OCons(tc_Int_Oint),B),C) != A | -c_List_Olinorder__class_Osorted(tc_Int_Oint,C) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),f383(A,tc_Int_Oint,C,B))). [resolve(5189,a,4952,a)]. 7.73/7.73 5190 -class_Orderings_Olinorder(A) | c_List_Olinorder__class_Osorted(A,B) | c_List_Olist_ONil(A) != B # label(fact_sorted_Osimps) # label(axiom). [clausify(3335)]. 7.73/7.73 Derived: c_List_Olinorder__class_Osorted(tc_Code__Numeral_Ocode__numeral,A) | c_List_Olist_ONil(tc_Code__Numeral_Ocode__numeral) != A. [resolve(5190,a,4908,a)]. 7.73/7.73 Derived: c_List_Olinorder__class_Osorted(tc_Int_Oint,A) | c_List_Olist_ONil(tc_Int_Oint) != A. [resolve(5190,a,4952,a)]. 7.73/7.73 5191 -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(3348)]. 7.73/7.73 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(5191,a,4908,a)]. 7.73/7.73 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(5191,a,4952,a)]. 7.73/7.73 5192 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),C)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),D)) # label(fact_min__less__iff__disj) # label(axiom). [clausify(3355)]. 7.73/7.73 Derived: -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),B)),C)) | 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),A),C)). [resolve(5192,a,4908,a)]. 7.73/7.73 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),A),B)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),C)). [resolve(5192,a,4952,a)]. 7.80/7.77 5193 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),C)),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),D)) # label(fact_min__less__iff__disj) # label(axiom). [clausify(3355)]. 7.80/7.77 Derived: 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),B)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),B),C)). [resolve(5193,a,4908,a)]. 7.80/7.77 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),A),B)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),C)). [resolve(5193,a,4952,a)]. 7.80/7.77 5194 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),C)),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),D)) # label(fact_min__less__iff__disj) # label(axiom). [clausify(3355)]. 7.80/7.77 5195 -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(3369)]. 7.80/7.77 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(5195,a,4908,a)]. 7.80/7.77 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(5195,a,4952,a)]. 7.80/7.77 5196 -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(3393)]. 7.80/7.77 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)) | 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(5196,a,4908,a)]. 7.80/7.77 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) = A | c_Big__Operators_Olattice_OSup__fin(tc_Int_Oint,c_Orderings_Oord__class_Omax(tc_Int_Oint),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),B),A)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),B),c_Big__Operators_Olattice_OSup__fin(tc_Int_Oint,c_Orderings_Oord__class_Omax(tc_Int_Oint),A)). [resolve(5196,a,4952,a)]. 7.80/7.77 5197 -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(3414)]. 7.80/7.77 5198 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),C),D))) # label(fact_le__max__iff__disj) # label(axiom). [clausify(3414)]. 7.80/7.80 5199 -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(3414)]. 7.80/7.80 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(5199,a,4908,a)]. 7.80/7.80 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),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(5199,a,4952,a)]. 7.80/7.80 5200 -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(3433)]. 7.80/7.80 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(5200,a,4908,a)]. 7.80/7.80 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(5200,a,4952,a)]. 7.80/7.80 5201 -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(3444)]. 7.80/7.80 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(5201,a,4908,a)]. 7.80/7.80 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(5201,a,4952,a)]. 7.80/7.80 5202 -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(3457)]. 7.85/7.82 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(5202,a,4908,a)]. 7.85/7.82 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(5202,a,4952,a)]. 7.85/7.82 5203 -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(3471)]. 7.85/7.82 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(5203,a,4908,a)]. 7.85/7.82 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(5203,a,4952,a)]. 7.85/7.82 5204 -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(3477)]. 7.85/7.82 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(5204,a,4908,a)]. 7.85/7.82 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(5204,a,4952,a)]. 7.85/7.82 5205 -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(3485)]. 7.85/7.82 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(5205,a,4908,a)]. 7.85/7.82 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(5205,a,4952,a)]. 7.85/7.82 5206 -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(3485)]. 7.85/7.82 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(5206,a,4908,a)]. 7.85/7.84 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(5206,a,4952,a)]. 7.85/7.84 5207 -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(3500)]. 7.85/7.84 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(5207,a,4908,a)]. 7.85/7.84 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(5207,a,4952,a)]. 7.85/7.84 5208 -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(3506)]. 7.85/7.84 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(5208,a,4908,a)]. 7.85/7.84 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(5208,a,4952,a)]. 7.85/7.84 5209 -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(3522)]. 7.85/7.86 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(5209,a,4908,a)]. 7.85/7.86 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(5209,a,4952,a)]. 7.85/7.86 5210 -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(3524)]. 7.85/7.86 5211 -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(3532)]. 7.85/7.86 5212 -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(3536)]. 7.85/7.86 5213 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omax(A),hAPP(B,f403(C,B,A))),hAPP(B,f404(C,B,A))) != hAPP(B,hAPP(hAPP(c_Orderings_Oord__class_Omax(A),f403(C,B,A)),f404(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(3539)]. 7.85/7.86 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),hAPP(A,f403(B,A,tc_Code__Numeral_Ocode__numeral))),hAPP(A,f404(B,A,tc_Code__Numeral_Ocode__numeral))) != hAPP(A,hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),f403(B,A,tc_Code__Numeral_Ocode__numeral)),f404(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(5213,a,4908,a)]. 7.85/7.86 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),hAPP(A,f403(B,A,tc_Int_Oint))),hAPP(A,f404(B,A,tc_Int_Oint))) != hAPP(A,hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),f403(B,A,tc_Int_Oint)),f404(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(5213,a,4952,a)]. 7.85/7.86 5214 -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(3547)]. 7.91/7.92 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(5214,a,4908,a)]. 7.91/7.92 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(5214,a,4952,a)]. 7.91/7.92 5215 -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(3592)]. 7.91/7.92 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(5215,a,4908,a)]. 7.91/7.92 Derived: c_List_Olinorder__class_Osort__key(A,tc_Int_Oint,B,c_List_Olist_ONil(A)) = c_List_Olist_ONil(A). [resolve(5215,a,4952,a)]. 7.91/7.92 5216 -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(3606)]. 7.91/7.92 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),B),A)) | c_Big__Operators_Olinorder__class_OMax(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),B),A)) = hAPP(hAPP(c_Finite__Set_Ofold(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral)),B),A). [resolve(5216,a,4908,a)]. 7.91/7.92 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),B),A)) | c_Big__Operators_Olinorder__class_OMax(tc_Int_Oint,hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),B),A)) = hAPP(hAPP(c_Finite__Set_Ofold(tc_Int_Oint,tc_Int_Oint,c_Orderings_Oord__class_Omax(tc_Int_Oint)),B),A). [resolve(5216,a,4952,a)]. 7.91/7.92 5217 -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(3624)]. 7.91/7.92 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(5217,a,4908,a)]. 7.91/7.92 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(5217,a,4952,a)]. 7.91/7.92 5218 -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(3641)]. 7.91/7.92 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(5218,a,4908,a)]. 7.91/7.92 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(5218,a,4952,a)]. 7.91/7.92 5219 -class_Orderings_Olinorder(A) | hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(A,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeast(A),B)) = hAPP(c_SetInterval_Oord__class_OlessThan(A),B) # label(fact_Compl__atLeast) # label(axiom). [clausify(3647)]. 7.91/7.93 Derived: hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeast(tc_Code__Numeral_Ocode__numeral),A)) = hAPP(c_SetInterval_Oord__class_OlessThan(tc_Code__Numeral_Ocode__numeral),A). [resolve(5219,a,4908,a)]. 7.91/7.93 Derived: hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeast(tc_Int_Oint),A)) = hAPP(c_SetInterval_Oord__class_OlessThan(tc_Int_Oint),A). [resolve(5219,a,4952,a)]. 7.91/7.93 5220 -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(3652)]. 7.91/7.93 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(5220,a,4908,a)]. 7.91/7.93 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(5220,a,4952,a)]. 7.91/7.93 5221 -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(3652)]. 7.91/7.93 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(5221,a,4908,a)]. 7.91/7.93 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(5221,a,4952,a)]. 7.91/7.94 5222 -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(3666)]. 7.91/7.94 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(5222,a,4908,a)]. 7.91/7.94 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(5222,a,4952,a)]. 7.91/7.94 5223 -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(3674)]. 7.91/7.94 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(5223,a,4908,a)]. 8.01/7.96 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(5223,a,4952,a)]. 8.01/7.96 5224 class_Orderings_Olinorder(tc_Nat_Onat) # label(arity_Nat__Onat__Orderings_Olinorder) # label(axiom). [assumption]. 8.01/7.96 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),f1(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(5224,a,4909,a)]. 8.01/7.96 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),f1(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(5224,a,4910,a)]. 8.01/7.96 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),C)) | 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(5224,a,4911,a)]. 8.01/7.96 Derived: c_List_Olinorder__class_Osorted(tc_Nat_Onat,c_List_Olist_ONil(tc_Nat_Onat)). [resolve(5224,a,4912,a)]. 8.01/7.96 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(5224,a,4913,a)]. 8.01/7.96 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(5224,a,4914,a)]. 8.01/7.96 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(5224,a,4915,a)]. 8.01/7.96 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(5224,a,4916,a)]. 8.01/7.96 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)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),C)). [resolve(5224,a,4917,a)]. 8.01/7.96 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),B),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_Finite__Set_Ofold(tc_Nat_Onat,tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat)),C),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),B),A)). [resolve(5224,a,4918,a)]. 8.01/7.96 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(5224,a,4919,a)]. 8.01/7.96 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | B != A | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)). [resolve(5224,a,4920,a)]. 8.01/7.96 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | B = A | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)). [resolve(5224,a,4921,a)]. 8.01/7.96 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),f13(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(5224,a,4922,a)]. 8.01/7.96 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),f13(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(5224,a,4923,a)]. 8.01/7.96 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(5224,a,4924,a)]. 8.01/7.96 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | hAPP(c_List_Oset(tc_Nat_Onat),f15(A,tc_Nat_Onat)) = A. [resolve(5224,a,4925,a)]. 8.01/7.96 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_List_Olinorder__class_Osorted(tc_Nat_Onat,f15(A,tc_Nat_Onat)). [resolve(5224,a,4926,a)]. 8.01/7.96 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 = f15(A,tc_Nat_Onat). [resolve(5224,a,4927,a)]. 8.01/7.96 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | hBOOL(hAPP(c_List_Odistinct(tc_Nat_Onat),f15(A,tc_Nat_Onat))). [resolve(5224,a,4928,a)]. 8.01/7.96 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(5224,a,4929,a)]. 8.01/7.96 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(5224,a,4930,a)]. 8.01/7.96 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(5224,a,4931,a)]. 8.01/7.97 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(5224,a,4932,a)]. 8.01/7.97 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = A | -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),B)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = B | c_Big__Operators_Olattice_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(5224,a,4933,a)]. 8.01/7.97 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(5224,a,4934,a)]. 8.01/7.97 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),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(5224,a,4935,a)]. 8.01/7.97 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(5224,a,4936,a)]. 8.01/7.97 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),B),A) = A. [resolve(5224,a,4937,a)]. 8.01/7.97 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = A | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),hAPP(c_Finite__Set_Ofold1(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat)),A))) | -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),C),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),C)). [resolve(5224,a,4938,a)]. 8.01/7.97 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = A | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),hAPP(c_Finite__Set_Ofold1(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat)),A))) | hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),f34(B,A,tc_Nat_Onat)),A)). [resolve(5224,a,4939,a)]. 8.01/7.97 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = A | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),hAPP(c_Finite__Set_Ofold1(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat)),A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),f34(B,A,tc_Nat_Onat))). [resolve(5224,a,4940,a)]. 8.01/7.97 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(5224,a,4941,a)]. 8.01/7.97 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),hAPP(A,f41(B,A,tc_Nat_Onat))),hAPP(A,f42(B,A,tc_Nat_Onat))) != hAPP(A,hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),f41(B,A,tc_Nat_Onat)),f42(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(5224,a,4942,a)]. 8.01/7.97 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = A | -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),B)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = B | c_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_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(5224,a,4943,a)]. 8.01/7.97 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)) = 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(5224,a,4944,a)]. 8.01/7.98 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(5224,a,4945,a)]. 8.01/7.98 Derived: hBOOL(hAPP(hAPP(c_member(A),f47(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(5224,a,4946,a)]. 8.01/7.98 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(A,B)),hAPP(A,f47(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(5224,a,4947,a)]. 8.01/7.98 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(5224,a,4948,a)]. 8.01/7.98 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(5224,a,4949,a)]. 8.01/7.98 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(5224,a,4950,a)]. 8.01/7.98 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(5224,a,4951,a)]. 8.01/7.98 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),f57(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(5224,a,4953,a)]. 8.01/7.98 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),f57(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(5224,a,4954,a)]. 8.01/7.98 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(5224,a,4955,a)]. 8.01/7.98 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(5224,a,4956,a)]. 8.01/7.98 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),f64(D,C,B,A,tc_Nat_Onat)),C)). [resolve(5224,a,4957,a)]. 8.01/7.98 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,f64(D,C,B,A,tc_Nat_Onat)))). [resolve(5224,a,4958,a)]. 8.01/7.98 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(5224,a,4959,a)]. 8.01/7.98 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = A | hBOOL(hAPP(hAPP(c_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(5224,a,4960,a)]. 8.01/7.98 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = A | -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),B)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = B | c_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(5224,a,4961,a)]. 8.01/7.98 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(5224,a,4962,a)]. 8.01/7.98 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(5224,a,4963,a)]. 8.01/7.98 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(5224,a,4964,a)]. 8.01/7.98 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),B),C))). [resolve(5224,a,4965,a)]. 8.01/7.98 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(5224,a,4966,a)]. 8.01/7.98 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),A)). [resolve(5224,a,4967,a)]. 8.01/7.98 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) != 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(5224,a,4968,a)]. 8.01/7.98 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = 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(5224,a,4969,a)]. 8.01/7.98 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(5224,a,4970,a)]. 8.01/7.99 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),f77(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(5224,a,4971,a)]. 8.01/7.99 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),f78(A,tc_Nat_Onat)),f77(A,tc_Nat_Onat))) | c_List_Olinorder__class_Osorted(tc_Nat_Onat,A). [resolve(5224,a,4972,a)]. 8.01/7.99 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_List_Onth(tc_Nat_Onat),A),f78(A,tc_Nat_Onat))),hAPP(hAPP(c_List_Onth(tc_Nat_Onat),A),f77(A,tc_Nat_Onat)))) | c_List_Olinorder__class_Osorted(tc_Nat_Onat,A). [resolve(5224,a,4973,a)]. 8.01/7.99 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),hAPP(c_Nat_Osize__class_Osize(tc_List_Olist(tc_Nat_Onat)),B))) | -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),hAPP(hAPP(c_List_Onth(tc_Nat_Onat),B),C)),hAPP(hAPP(c_List_Onth(tc_Nat_Onat),B),A))) | -c_List_Olinorder__class_Osorted(tc_Nat_Onat,B). [resolve(5224,a,4974,a)]. 8.01/7.99 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(5224,a,4975,a)]. 8.01/7.99 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(5224,a,4976,a)]. 8.01/7.99 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(5224,a,4977,a)]. 8.01/7.99 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(5224,a,4978,a)]. 8.01/7.99 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(5224,a,4979,a)]. 8.01/8.00 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(5224,a,4980,a)]. 8.01/8.00 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(5224,a,4981,a)]. 8.01/8.00 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(5224,a,4982,a)]. 8.01/8.00 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(5224,a,4983,a)]. 8.01/8.00 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),f103(A,tc_Nat_Onat)),f104(A,tc_Nat_Onat))),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),f103(A,tc_Nat_Onat)),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),f104(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(5224,a,4984,a)]. 8.01/8.00 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)),C)) | 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),B),C)). [resolve(5224,a,4985,a)]. 8.01/8.00 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)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),C)). [resolve(5224,a,4986,a)]. 8.01/8.00 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)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),C)). [resolve(5224,a,4987,a)]. 8.01/8.00 Derived: -hBOOL(hAPP(c_List_Odistinct(A),B)) | hBOOL(hAPP(c_List_Odistinct(A),c_List_Olinorder__class_Osort__key(A,tc_Nat_Onat,C,B))). [resolve(5224,a,4988,a)]. 8.01/8.00 Derived: hBOOL(hAPP(c_List_Odistinct(A),B)) | -hBOOL(hAPP(c_List_Odistinct(A),c_List_Olinorder__class_Osort__key(A,tc_Nat_Onat,C,B))). [resolve(5224,a,4989,a)]. 8.01/8.00 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(5224,a,4990,a)]. 8.01/8.00 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(5224,a,4992,a)]. 8.01/8.00 Derived: c_List_Olist_ONil(A) != hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Nat_Onat,B),C),D). [resolve(5224,a,4993,a)]. 8.01/8.00 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(5224,a,4995,a)]. 8.01/8.00 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),C),B))). [resolve(5224,a,4997,a)]. 8.01/8.00 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(5224,a,4998,a)]. 8.01/8.00 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(5224,a,5001,a)]. 8.01/8.00 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(5224,a,5002,a)]. 8.07/8.01 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),A),B) = A. [resolve(5224,a,5003,a)]. 8.07/8.01 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),A),B) != A. [resolve(5224,a,5004,a)]. 8.07/8.01 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),A),A) = A. [resolve(5224,a,5005,a)]. 8.07/8.01 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(5224,a,5006,a)]. 8.07/8.01 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(5224,a,5007,a)]. 8.07/8.01 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(5224,a,5008,a)]. 8.07/8.01 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,A),B) != hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,C),D) | B = D. [resolve(5224,a,5009,a)]. 8.07/8.01 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,A),B) != hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,C),D) | A = C. [resolve(5224,a,5010,a)]. 8.07/8.01 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,A),B) = hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,C),D) | B != D | A != C. [resolve(5224,a,5011,a)]. 8.07/8.01 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(5224,a,5015,a)]. 8.07/8.01 Derived: -hBOOL(hAPP(c_List_Odistinct(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),hAPP(c_List_Oset(A),B))) | hBOOL(hAPP(c_List_Odistinct(A),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Nat_Onat,D),C),B))). [resolve(5224,a,5016,a)]. 8.07/8.01 Derived: hBOOL(hAPP(c_List_Odistinct(A),B)) | -hBOOL(hAPP(c_List_Odistinct(A),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Nat_Onat,C),D),B))). [resolve(5224,a,5017,a)]. 8.07/8.01 Derived: -hBOOL(hAPP(hAPP(c_member(A),B),hAPP(c_List_Oset(A),C))) | -hBOOL(hAPP(c_List_Odistinct(A),hAPP(hAPP(c_List_Olinorder__class_Oinsort__key(A,tc_Nat_Onat,D),B),C))). [resolve(5224,a,5018,a)]. 8.07/8.01 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(5224,a,5021,a)]. 8.07/8.01 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(5224,a,5022,a)]. 8.07/8.01 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(5224,a,5023,a)]. 8.07/8.01 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(5224,a,5024,a)]. 8.07/8.01 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(5224,a,5025,a)]. 8.07/8.01 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),f139(A,tc_Nat_Onat)),f140(A,tc_Nat_Onat))),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),f139(A,tc_Nat_Onat)),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),f140(A,tc_Nat_Onat)),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))))) | hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),c_Big__Operators_Olinorder__class_OMax(tc_Nat_Onat,A)),A)). [resolve(5224,a,5026,a)]. 8.07/8.01 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(5224,a,5030,a)]. 8.07/8.01 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(5224,a,5031,a)]. 8.07/8.01 Derived: c_Finite__Set_Ofun__left__comm__idem(tc_Nat_Onat,tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat)). [resolve(5224,a,5032,a)]. 8.07/8.01 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(5224,a,5034,a)]. 8.07/8.02 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(5224,a,5036,a)]. 8.07/8.02 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(5224,a,5038,a)]. 8.07/8.02 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(5224,a,5040,a)]. 8.07/8.02 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(5224,a,5041,a)]. 8.07/8.02 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(5224,a,5042,a)]. 8.07/8.02 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(5224,a,5043,a)]. 8.07/8.02 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(5224,a,5044,a)]. 8.07/8.02 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(5224,a,5045,a)]. 8.07/8.02 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(5224,a,5047,a)]. 8.07/8.02 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(5224,a,5049,a)]. 8.07/8.02 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),B),C)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),A),B)),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),A),C)). [resolve(5224,a,5050,a)]. 8.07/8.02 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(5224,a,5051,a)]. 8.07/8.03 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = A | c_Big__Operators_Olinorder__class_OMin(tc_Nat_Onat,hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),B),A)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),B),c_Big__Operators_Olinorder__class_OMin(tc_Nat_Onat,A)). [resolve(5224,a,5052,a)]. 8.07/8.03 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(5224,a,5053,a)]. 8.07/8.03 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(5224,a,5054,a)]. 8.07/8.03 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Nat_Onat) | -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),C),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),C),c_Complete__Lattice_OSup__class_OSup(tc_Nat_Onat,B))). [resolve(5224,a,5055,b)]. 8.07/8.03 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Nat_Onat) | hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),f178(A,B,tc_Nat_Onat)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),c_Complete__Lattice_OSup__class_OSup(tc_Nat_Onat,A))). [resolve(5224,a,5056,b)]. 8.07/8.03 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Nat_Onat) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),f178(B,A,tc_Nat_Onat))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),c_Complete__Lattice_OSup__class_OSup(tc_Nat_Onat,B))). [resolve(5224,a,5057,b)]. 8.07/8.03 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(5224,a,5058,b)]. 8.07/8.03 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(5224,a,5059,a)]. 8.07/8.03 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),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(5224,a,5061,a)]. 8.07/8.03 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),A),B)),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),C),B)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),A),C)),B). [resolve(5224,a,5062,a)]. 8.07/8.03 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(5224,a,5063,a)]. 8.07/8.03 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),f187(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(5224,a,5064,a)]. 8.07/8.03 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),f187(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(5224,a,5065,a)]. 8.07/8.04 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(5224,a,5066,a)]. 8.07/8.04 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(5224,a,5067,a)]. 8.07/8.04 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),A)). [resolve(5224,a,5068,a)]. 8.07/8.04 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(5224,a,5070,a)]. 8.07/8.04 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),hAPP(A,f197(B,A,tc_Nat_Onat))),hAPP(A,f198(B,A,tc_Nat_Onat))) != hAPP(A,hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),f197(B,A,tc_Nat_Onat)),f198(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(5224,a,5071,a)]. 8.07/8.04 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),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(5224,a,5072,a)]. 8.07/8.04 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),A),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),C),B))). [resolve(5224,a,5073,a)]. 8.07/8.04 Derived: A != B | hAPP(c_SetInterval_Oord__class_OatLeast(tc_Nat_Onat),A) = hAPP(c_SetInterval_Oord__class_OatLeast(tc_Nat_Onat),B). [resolve(5224,a,5074,a)]. 8.07/8.04 Derived: A = B | hAPP(c_SetInterval_Oord__class_OatLeast(tc_Nat_Onat),A) != hAPP(c_SetInterval_Oord__class_OatLeast(tc_Nat_Onat),B). [resolve(5224,a,5075,a)]. 8.07/8.04 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(5224,a,5076,a)]. 8.07/8.04 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(5224,a,5077,a)]. 8.07/8.04 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(5224,a,5079,a)]. 8.07/8.04 Derived: hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),A) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Nat_Onat),A)). [resolve(5224,a,5080,a)]. 8.07/8.04 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(5224,a,5081,a)]. 8.07/8.04 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(5224,a,5082,a)]. 8.07/8.04 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(5224,a,5083,a)]. 8.07/8.04 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(5224,a,5084,a)]. 8.07/8.04 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(5224,a,5085,a)]. 8.07/8.04 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(5224,a,5086,a)]. 8.07/8.04 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(5224,a,5087,a)]. 8.07/8.04 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(5224,a,5088,a)]. 8.07/8.04 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(5224,a,5089,a)]. 8.07/8.04 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = 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(5224,a,5090,a)]. 8.07/8.04 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(5224,a,5091,a)]. 8.07/8.04 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(5224,a,5092,a)]. 8.07/8.04 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(5224,a,5093,a)]. 8.11/8.05 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),f247(A,tc_Nat_Onat)),f248(A,tc_Nat_Onat))),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),f247(A,tc_Nat_Onat)),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),f248(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(5224,a,5094,a)]. 8.11/8.05 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(5224,a,5095,a)]. 8.11/8.05 Derived: -c_List_Olinorder__class_Osorted(tc_Nat_Onat,A) | hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),f257(A,B,tc_Nat_Onat)),hAPP(c_List_Oset(tc_Nat_Onat),A))) | c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Olist_OCons(tc_Nat_Onat),B),A)). [resolve(5224,a,5096,a)]. 8.11/8.05 Derived: -c_List_Olinorder__class_Osorted(tc_Nat_Onat,A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),f257(A,B,tc_Nat_Onat))) | c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Olist_OCons(tc_Nat_Onat),B),A)). [resolve(5224,a,5097,a)]. 8.11/8.05 Derived: c_List_Olinorder__class_Osorted(tc_Nat_Onat,A) | -c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Olist_OCons(tc_Nat_Onat),B),A)). [resolve(5224,a,5098,a)]. 8.11/8.05 Derived: -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),A),hAPP(c_List_Oset(tc_Nat_Onat),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),C),A)) | -c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Olist_OCons(tc_Nat_Onat),C),B)). [resolve(5224,a,5099,a)]. 8.11/8.05 Derived: hAPP(c_SetInterval_Oord__class_OatLeast(tc_Nat_Onat),A) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),A)). [resolve(5224,a,5103,a)]. 8.11/8.05 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(5224,a,5104,a)]. 8.11/8.05 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(5224,a,5105,a)]. 8.11/8.05 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),A),B)),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),A),C)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),A),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),B),C)). [resolve(5224,a,5106,a)]. 8.11/8.05 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),C) != hAPP(c_List_Oset(A),D) | C = D. [resolve(5224,a,5107,a)]. 8.11/8.05 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(5224,a,5108,a)]. 8.11/8.06 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),A),B) != B | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)). [resolve(5224,a,5109,a)]. 8.11/8.06 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),A),B) = B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)). [resolve(5224,a,5110,a)]. 8.11/8.06 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_Olattice_OInf__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat),A) = B. [resolve(5224,a,5111,a)]. 8.11/8.06 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_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(5224,a,5112,a)]. 8.11/8.06 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),A) != hAPP(c_List_Oset(tc_Nat_Onat),B) | A = B. [resolve(5224,a,5113,a)]. 8.11/8.06 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(5224,a,5114,a)]. 8.11/8.06 Derived: A != B | hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),A) = hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),B). [resolve(5224,a,5115,a)]. 8.11/8.06 Derived: A = B | hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),A) != hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),B). [resolve(5224,a,5116,a)]. 8.11/8.06 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(5224,a,5117,a)]. 8.11/8.06 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(5224,a,5118,a)]. 8.11/8.06 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Nat_Onat) | hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),f294(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(5224,a,5119,a)]. 8.11/8.06 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Nat_Onat) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),f294(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(5224,a,5120,a)]. 8.11/8.06 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),A),A) = A. [resolve(5224,a,5121,a)]. 8.11/8.06 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = A | 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(5224,a,5122,a)]. 8.11/8.07 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(5224,a,5123,a)]. 8.11/8.07 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(5224,a,5124,a)]. 8.11/8.07 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(5224,a,5126,a)]. 8.11/8.07 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(5224,a,5127,a)]. 8.11/8.07 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(5224,a,5128,a)]. 8.11/8.07 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(5224,a,5129,a)]. 8.11/8.07 Derived: A != B | hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Nat_Onat),A) = hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Nat_Onat),B). [resolve(5224,a,5130,a)]. 8.11/8.07 Derived: A = B | hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Nat_Onat),A) != hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Nat_Onat),B). [resolve(5224,a,5131,a)]. 8.11/8.07 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(5224,a,5132,a)]. 8.11/8.07 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(5224,a,5133,a)]. 8.11/8.07 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(5224,a,5135,a)]. 8.11/8.07 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(5224,a,5136,a)]. 8.11/8.07 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(5224,a,5137,a)]. 8.11/8.07 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(5224,a,5138,a)]. 8.11/8.07 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(5224,a,5139,a)]. 8.11/8.07 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(5224,a,5140,a)]. 8.11/8.07 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(5224,a,5141,a)]. 8.11/8.07 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(5224,a,5143,a)]. 8.11/8.07 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(5224,a,5146,a)]. 8.11/8.07 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(5224,a,5147,a)]. 8.11/8.07 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(5224,a,5148,a)]. 8.11/8.07 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(5224,a,5149,a)]. 8.11/8.07 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(5224,a,5150,a)]. 8.11/8.07 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(5224,a,5151,a)]. 8.11/8.07 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(5224,a,5152,a)]. 8.11/8.08 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(5224,a,5153,a)]. 8.11/8.08 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_List_Oremove1(tc_Nat_Onat,B,c_List_Olinorder__class_Osorted__list__of__set(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))))). [resolve(5224,a,5154,a)]. 8.11/8.08 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(5224,a,5155,a)]. 8.11/8.08 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(5224,a,5156,a)]. 8.11/8.08 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),f341(A,tc_Nat_Onat)),f342(A,tc_Nat_Onat))),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),f341(A,tc_Nat_Onat)),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),f342(A,tc_Nat_Onat)),c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)))))) | hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),c_Big__Operators_Olinorder__class_OMin(tc_Nat_Onat,A)),A)). [resolve(5224,a,5157,a)]. 8.11/8.08 Derived: -c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Oappend(tc_Nat_Onat),A),B)) | c_List_Olinorder__class_Osorted(tc_Nat_Onat,A). [resolve(5224,a,5158,a)]. 8.11/8.08 Derived: -c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Oappend(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),C),hAPP(c_List_Oset(tc_Nat_Onat),A))) | -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),D),hAPP(c_List_Oset(tc_Nat_Onat),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),C),D)). [resolve(5224,a,5159,a)]. 8.11/8.08 Derived: -c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Oappend(tc_Nat_Onat),A),B)) | c_List_Olinorder__class_Osorted(tc_Nat_Onat,B). [resolve(5224,a,5160,a)]. 8.11/8.08 Derived: c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Oappend(tc_Nat_Onat),A),B)) | -c_List_Olinorder__class_Osorted(tc_Nat_Onat,A) | hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),f346(B,A,tc_Nat_Onat)),hAPP(c_List_Oset(tc_Nat_Onat),A))) | -c_List_Olinorder__class_Osorted(tc_Nat_Onat,B). [resolve(5224,a,5161,a)]. 8.11/8.08 Derived: c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Oappend(tc_Nat_Onat),A),B)) | -c_List_Olinorder__class_Osorted(tc_Nat_Onat,A) | hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),f347(B,A,tc_Nat_Onat)),hAPP(c_List_Oset(tc_Nat_Onat),B))) | -c_List_Olinorder__class_Osorted(tc_Nat_Onat,B). [resolve(5224,a,5162,a)]. 8.11/8.08 Derived: c_List_Olinorder__class_Osorted(tc_Nat_Onat,hAPP(hAPP(c_List_Oappend(tc_Nat_Onat),A),B)) | -c_List_Olinorder__class_Osorted(tc_Nat_Onat,A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),f346(B,A,tc_Nat_Onat)),f347(B,A,tc_Nat_Onat))) | -c_List_Olinorder__class_Osorted(tc_Nat_Onat,B). [resolve(5224,a,5163,a)]. 8.11/8.08 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(5224,a,5164,a)]. 8.11/8.08 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(5224,a,5165,a)]. 8.11/8.09 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(5224,a,5167,a)]. 8.11/8.09 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(5224,a,5168,a)]. 8.11/8.09 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),B),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),C)). [resolve(5224,a,5169,a)]. 8.11/8.09 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),B),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)). [resolve(5224,a,5170,a)]. 8.11/8.09 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),B),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)). [resolve(5224,a,5171,a)]. 8.11/8.09 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(5224,a,5172,a)]. 8.11/8.09 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(5224,a,5174,a)]. 8.11/8.09 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(5224,a,5175,a)]. 8.11/8.09 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(5224,a,5176,a)]. 8.11/8.09 Derived: c_List_Olinorder__class_Osort__key(tc_Nat_Onat,tc_Nat_Onat,c_COMBI(tc_Nat_Onat),c_List_Oremdups(tc_Nat_Onat,A)) = c_List_Olinorder__class_Osorted__list__of__set(tc_Nat_Onat,hAPP(c_List_Oset(tc_Nat_Onat),A)). [resolve(5224,a,5177,a)]. 8.11/8.09 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),f376(A,tc_Nat_Onat)),f377(A,tc_Nat_Onat))) | c_List_Olinorder__class_Osorted(tc_Nat_Onat,A). [resolve(5224,a,5178,a)]. 8.11/8.09 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),f377(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(5224,a,5179,a)]. 8.11/8.09 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(hAPP(c_List_Onth(tc_Nat_Onat),A),f376(A,tc_Nat_Onat))),hAPP(hAPP(c_List_Onth(tc_Nat_Onat),A),f377(A,tc_Nat_Onat)))) | c_List_Olinorder__class_Osorted(tc_Nat_Onat,A). [resolve(5224,a,5180,a)]. 8.11/8.09 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(5224,a,5181,a)]. 8.11/8.09 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(5224,a,5182,a)]. 8.11/8.09 Derived: -c_List_Olinorder__class_Osorted(tc_Nat_Onat,A) | hAPP(hAPP(c_List_Olist_OCons(tc_Nat_Onat),f382(A,tc_Nat_Onat)),f381(A,tc_Nat_Onat)) = A | c_List_Olist_ONil(tc_Nat_Onat) = A. [resolve(5224,a,5185,a)]. 8.11/8.10 Derived: -c_List_Olinorder__class_Osorted(tc_Nat_Onat,A) | c_List_Olinorder__class_Osorted(tc_Nat_Onat,f381(A,tc_Nat_Onat)) | c_List_Olist_ONil(tc_Nat_Onat) = A. [resolve(5224,a,5186,a)]. 8.11/8.10 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),f381(A,tc_Nat_Onat)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),f382(A,tc_Nat_Onat)),B)) | c_List_Olist_ONil(tc_Nat_Onat) = A. [resolve(5224,a,5187,a)]. 8.11/8.10 Derived: c_List_Olinorder__class_Osorted(tc_Nat_Onat,A) | hAPP(hAPP(c_List_Olist_OCons(tc_Nat_Onat),B),C) != A | -c_List_Olinorder__class_Osorted(tc_Nat_Onat,C) | hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),f383(A,tc_Nat_Onat,C,B)),hAPP(c_List_Oset(tc_Nat_Onat),C))). [resolve(5224,a,5188,a)]. 8.11/8.10 Derived: c_List_Olinorder__class_Osorted(tc_Nat_Onat,A) | hAPP(hAPP(c_List_Olist_OCons(tc_Nat_Onat),B),C) != A | -c_List_Olinorder__class_Osorted(tc_Nat_Onat,C) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),f383(A,tc_Nat_Onat,C,B))). [resolve(5224,a,5189,a)]. 8.11/8.10 Derived: c_List_Olinorder__class_Osorted(tc_Nat_Onat,A) | c_List_Olist_ONil(tc_Nat_Onat) != A. [resolve(5224,a,5190,a)]. 8.11/8.10 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(5224,a,5191,a)]. 8.11/8.10 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),A),B)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),C)). [resolve(5224,a,5192,a)]. 8.11/8.10 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),A),B)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),C)). [resolve(5224,a,5193,a)]. 8.11/8.10 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(5224,a,5195,a)]. 8.11/8.10 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = A | c_Big__Operators_Olattice_OSup__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omax(tc_Nat_Onat),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),B),A)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),B),c_Big__Operators_Olattice_OSup__fin(tc_Nat_Onat,c_Orderings_Oord__class_Omax(tc_Nat_Onat),A)). [resolve(5224,a,5196,a)]. 8.11/8.10 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(5224,a,5199,a)]. 8.11/8.10 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(5224,a,5200,a)]. 8.11/8.10 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(5224,a,5201,a)]. 8.11/8.10 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(5224,a,5202,a)]. 8.11/8.10 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(5224,a,5203,a)]. 8.11/8.10 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(5224,a,5204,a)]. 8.11/8.10 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(5224,a,5205,a)]. 8.11/8.10 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(5224,a,5206,a)]. 8.11/8.10 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(5224,a,5207,a)]. 8.11/8.10 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(5224,a,5208,a)]. 8.11/8.10 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(5224,a,5209,a)]. 8.11/8.10 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),hAPP(A,f403(B,A,tc_Nat_Onat))),hAPP(A,f404(B,A,tc_Nat_Onat))) != hAPP(A,hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),f403(B,A,tc_Nat_Onat)),f404(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(5224,a,5213,a)]. 8.11/8.10 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(5224,a,5214,a)]. 8.11/8.10 Derived: c_List_Olinorder__class_Osort__key(A,tc_Nat_Onat,B,c_List_Olist_ONil(A)) = c_List_Olist_ONil(A). [resolve(5224,a,5215,a)]. 8.11/8.10 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),B),A)) | c_Big__Operators_Olinorder__class_OMax(tc_Nat_Onat,hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),B),A)) = hAPP(hAPP(c_Finite__Set_Ofold(tc_Nat_Onat,tc_Nat_Onat,c_Orderings_Oord__class_Omax(tc_Nat_Onat)),B),A). [resolve(5224,a,5216,a)]. 8.11/8.10 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(5224,a,5217,a)]. 8.11/8.13 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(5224,a,5218,a)]. 8.11/8.13 Derived: hAPP(c_Groups_Ouminus__class_Ouminus(tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(c_SetInterval_Oord__class_OatLeast(tc_Nat_Onat),A)) = hAPP(c_SetInterval_Oord__class_OlessThan(tc_Nat_Onat),A). [resolve(5224,a,5219,a)]. 8.11/8.13 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(5224,a,5220,a)]. 8.11/8.13 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(5224,a,5221,a)]. 8.11/8.13 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(5224,a,5222,a)]. 8.11/8.13 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(5224,a,5223,a)]. 8.11/8.13 5225 -class_Orderings_Olinorder(A) | hBOOL(hAPP(hAPP(c_member(A),f423(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(3724)]. 8.11/8.13 Derived: hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),f423(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(5225,a,4908,a)]. 8.20/8.15 Derived: hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),f423(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(5225,a,4952,a)]. 8.20/8.15 Derived: hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),f423(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(5225,a,5224,a)]. 8.20/8.15 5226 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),f423(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(3724)]. 8.20/8.15 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),f423(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(5226,a,4908,a)]. 8.20/8.15 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),f423(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(5226,a,4952,a)]. 8.20/8.15 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),f423(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(5226,a,5224,a)]. 8.20/8.15 5227 -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(3736)]. 8.20/8.15 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(5227,a,4908,a)]. 8.20/8.15 Derived: c_Finite__Set_Ofun__left__comm__idem(tc_Int_Oint,tc_Int_Oint,c_Orderings_Oord__class_Omax(tc_Int_Oint)). [resolve(5227,a,4952,a)]. 8.20/8.15 Derived: c_Finite__Set_Ofun__left__comm__idem(tc_Nat_Onat,tc_Nat_Onat,c_Orderings_Oord__class_Omax(tc_Nat_Onat)). [resolve(5227,a,5224,a)]. 8.20/8.15 5228 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f428(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(3751)]. 8.20/8.15 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),f428(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(5228,a,4908,a)]. 8.20/8.15 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),f428(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(5228,a,4952,a)]. 8.20/8.15 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),f428(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(5228,a,5224,a)]. 8.20/8.15 5229 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),f428(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(3751)]. 8.20/8.18 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),f428(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(5229,a,4908,a)]. 8.20/8.18 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),f428(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(5229,a,4952,a)]. 8.20/8.18 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),f428(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(5229,a,5224,a)]. 8.20/8.18 5230 -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(3754)]. 8.20/8.18 5231 -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(3758)]. 8.20/8.18 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(5231,a,4908,a)]. 8.20/8.18 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) = A | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),c_Big__Operators_Olinorder__class_OMin(tc_Int_Oint,A)),A)). [resolve(5231,a,4952,a)]. 8.20/8.18 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(5231,a,5224,a)]. 8.20/8.18 5232 -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(3788)]. 8.20/8.18 5233 -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(3788)]. 8.20/8.18 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(5233,a,4908,a)]. 8.20/8.18 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(5233,a,4952,a)]. 8.20/8.18 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(5233,a,5224,a)]. 8.20/8.18 5234 -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)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) # label(fact_sorted__many__eq) # label(axiom). [clausify(3788)]. 8.25/8.20 5235 -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(3814)]. 8.25/8.20 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(5235,a,4908,a)]. 8.25/8.20 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(5235,a,4952,a)]. 8.25/8.20 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(5235,a,5224,a)]. 8.25/8.20 5236 -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(3819)]. 8.25/8.20 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(5236,a,4908,a)]. 8.25/8.20 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)) | hAPP(c_SetInterval_Oord__class_OatLeast(tc_Int_Oint),A) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(tc_Int_Oint,tc_HOL_Obool)),c_SetInterval_Oord__class_OatLeastAtMost(tc_Int_Oint,A,B)),hAPP(c_SetInterval_Oord__class_OgreaterThan(tc_Int_Oint),B)). [resolve(5236,a,4952,a)]. 8.25/8.20 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(5236,a,5224,a)]. 8.25/8.20 5237 -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(3820)]. 8.25/8.20 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(5237,a,4908,a)]. 8.25/8.20 Derived: hAPP(c_SetInterval_Oord__class_OatMost(tc_Int_Oint),A) != hAPP(c_SetInterval_Oord__class_OatMost(tc_Int_Oint),B) | A = B. [resolve(5237,a,4952,a)]. 8.25/8.24 Derived: hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),A) != hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),B) | A = B. [resolve(5237,a,5224,a)]. 8.25/8.24 5238 -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(3820)]. 8.25/8.24 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(5238,a,4908,a)]. 8.25/8.24 Derived: hAPP(c_SetInterval_Oord__class_OatMost(tc_Int_Oint),A) = hAPP(c_SetInterval_Oord__class_OatMost(tc_Int_Oint),B) | A != B. [resolve(5238,a,4952,a)]. 8.25/8.24 Derived: hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),A) = hAPP(c_SetInterval_Oord__class_OatMost(tc_Nat_Onat),B) | A != B. [resolve(5238,a,5224,a)]. 8.25/8.24 5239 -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(3823)]. 8.25/8.24 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(5239,a,4908,a)]. 8.25/8.24 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(5239,a,4952,a)]. 8.25/8.24 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(5239,a,5224,a)]. 8.25/8.24 5240 -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(3854)]. 8.25/8.24 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(5240,a,4908,a)]. 8.25/8.24 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(5240,a,4952,a)]. 8.25/8.24 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(5240,a,5224,a)]. 8.25/8.24 5241 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omax(A),hAPP(B,f442(C,B,A))),hAPP(B,f443(C,B,A))) != hAPP(B,hAPP(hAPP(c_Orderings_Oord__class_Omax(A),f442(C,B,A)),f443(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(3871)]. 8.25/8.24 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),hAPP(A,f442(B,A,tc_Code__Numeral_Ocode__numeral))),hAPP(A,f443(B,A,tc_Code__Numeral_Ocode__numeral))) != hAPP(A,hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),f442(B,A,tc_Code__Numeral_Ocode__numeral)),f443(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(5241,a,4908,a)]. 8.37/8.31 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),hAPP(A,f442(B,A,tc_Int_Oint))),hAPP(A,f443(B,A,tc_Int_Oint))) != hAPP(A,hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),f442(B,A,tc_Int_Oint)),f443(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(5241,a,4952,a)]. 8.37/8.31 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),hAPP(A,f442(B,A,tc_Nat_Onat))),hAPP(A,f443(B,A,tc_Nat_Onat))) != hAPP(A,hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),f442(B,A,tc_Nat_Onat)),f443(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(5241,a,5224,a)]. 8.37/8.31 5242 -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(3886)]. 8.37/8.31 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(5242,a,4908,a)]. 8.37/8.31 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(5242,a,4952,a)]. 8.37/8.31 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(5242,a,5224,a)]. 8.37/8.31 5243 -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(3903)]. 8.37/8.31 5244 -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(3903)]. 8.37/8.31 5245 -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(3931)]. 8.37/8.31 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)) = A | 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(5245,a,4908,a)]. 8.37/8.31 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(5245,a,4952,a)]. 8.37/8.31 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_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(5245,a,5224,a)]. 8.41/8.37 5246 -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(3978)]. 8.41/8.37 5247 -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(3981)]. 8.41/8.37 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(5247,a,4908,a)]. 8.41/8.37 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(5247,a,4952,a)]. 8.41/8.37 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(5247,a,5224,a)]. 8.41/8.37 5248 -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(3985)]. 8.41/8.37 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(5248,a,4908,a)]. 8.41/8.37 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(5248,a,4952,a)]. 8.41/8.37 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(5248,a,5224,a)]. 8.41/8.37 5249 -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(4000)]. 8.41/8.37 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(5249,a,4908,a)]. 8.41/8.41 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(5249,a,4952,a)]. 8.41/8.41 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(5249,a,5224,a)]. 8.41/8.41 5250 -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(4000)]. 8.41/8.41 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(5250,a,4908,a)]. 8.41/8.41 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(5250,a,4952,a)]. 8.41/8.41 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(5250,a,5224,a)]. 8.41/8.41 5251 -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)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),D)) # label(fact_max__less__iff__conj) # label(axiom). [clausify(4000)]. 8.41/8.41 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)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),C)). [resolve(5251,a,4908,a)]. 8.41/8.41 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)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),C)). [resolve(5251,a,4952,a)]. 8.41/8.41 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)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),C)). [resolve(5251,a,5224,a)]. 8.41/8.41 5252 -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(4061)]. 8.41/8.41 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(5252,a,4908,a)]. 8.41/8.41 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(5252,a,4952,a)]. 8.50/8.44 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(5252,a,5224,a)]. 8.50/8.44 5253 -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(4063)]. 8.50/8.44 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)) = A | -hBOOL(hAPP(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(5253,a,4908,a)]. 8.50/8.44 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(5253,a,4952,a)]. 8.50/8.44 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = A | -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),B)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = B | c_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(5253,a,5224,a)]. 8.50/8.44 5254 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = B | -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(4089)]. 8.50/8.44 5255 -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(4101)]. 8.50/8.47 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(5255,a,4908,a)]. 8.50/8.47 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(5255,a,4952,a)]. 8.50/8.47 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(5255,a,5224,a)]. 8.50/8.47 5256 -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(4121)]. 8.50/8.47 5257 -class_Orderings_Olinorder(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = B | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Finite__Set_Ofold1(A,c_Orderings_Oord__class_Omin(A)),B)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),f475(C,B,A)),C)) # label(fact_fold1__strict__below__iff) # label(axiom). [clausify(4133)]. 8.50/8.47 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 | -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)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),f475(B,A,tc_Code__Numeral_Ocode__numeral)),B)). [resolve(5257,a,4908,a)]. 8.50/8.47 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) = 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)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),f475(B,A,tc_Int_Oint)),B)). [resolve(5257,a,4952,a)]. 8.50/8.47 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = A | -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)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),f475(B,A,tc_Nat_Onat)),B)). [resolve(5257,a,5224,a)]. 8.50/8.47 5258 -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),hAPP(c_Finite__Set_Ofold1(A,c_Orderings_Oord__class_Omin(A)),B)),C)) | hBOOL(hAPP(hAPP(c_member(A),f475(C,B,A)),B)) # label(fact_fold1__strict__below__iff) # label(axiom). [clausify(4133)]. 8.50/8.47 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 | -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)) | hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),f475(B,A,tc_Code__Numeral_Ocode__numeral)),A)). [resolve(5258,a,4908,a)]. 8.50/8.47 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) = 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)) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),f475(B,A,tc_Int_Oint)),A)). [resolve(5258,a,4952,a)]. 8.50/8.50 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = 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)) | hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),f475(B,A,tc_Nat_Onat)),A)). [resolve(5258,a,5224,a)]. 8.50/8.50 5259 -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),hAPP(c_Finite__Set_Ofold1(A,c_Orderings_Oord__class_Omin(A)),B)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),C)) | -hBOOL(hAPP(hAPP(c_member(A),D),B)) # label(fact_fold1__strict__below__iff) # label(axiom). [clausify(4133)]. 8.50/8.50 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)) = A | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(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)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),C),B)) | -hBOOL(hAPP(hAPP(c_member(tc_Code__Numeral_Ocode__numeral),C),A)). [resolve(5259,a,4908,a)]. 8.50/8.50 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(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)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),C),B)) | -hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),C),A)). [resolve(5259,a,4952,a)]. 8.50/8.50 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = 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)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),C),B)) | -hBOOL(hAPP(hAPP(c_member(tc_Nat_Onat),C),A)). [resolve(5259,a,5224,a)]. 8.50/8.50 5260 -class_Orderings_Olinorder(A) | -class_Lattices_Osemilattice__sup(A) | c_Lattices_Osemilattice__sup__class_Osup(A) = c_Orderings_Oord__class_Omax(A) # label(fact_sup__max) # label(axiom). [clausify(4138)]. 8.50/8.50 Derived: -class_Lattices_Osemilattice__sup(tc_Code__Numeral_Ocode__numeral) | c_Lattices_Osemilattice__sup__class_Osup(tc_Code__Numeral_Ocode__numeral) = c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral). [resolve(5260,a,4908,a)]. 8.50/8.50 Derived: -class_Lattices_Osemilattice__sup(tc_Int_Oint) | c_Lattices_Osemilattice__sup__class_Osup(tc_Int_Oint) = c_Orderings_Oord__class_Omax(tc_Int_Oint). [resolve(5260,a,4952,a)]. 8.50/8.50 Derived: -class_Lattices_Osemilattice__sup(tc_Nat_Onat) | c_Lattices_Osemilattice__sup__class_Osup(tc_Nat_Onat) = c_Orderings_Oord__class_Omax(tc_Nat_Onat). [resolve(5260,a,5224,a)]. 8.50/8.50 5261 -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(4168)]. 8.57/8.52 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(5261,a,4908,a)]. 8.57/8.52 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(5261,a,4952,a)]. 8.57/8.52 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(5261,a,5224,a)]. 8.57/8.52 5262 -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(4179)]. 8.57/8.52 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(5262,a,4908,a)]. 8.57/8.52 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(5262,a,4952,a)]. 8.57/8.52 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(5262,a,5224,a)]. 8.57/8.52 5263 -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(4197)]. 8.64/8.57 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(5263,a,4908,a)]. 8.64/8.57 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(5263,a,4952,a)]. 8.64/8.57 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(5263,a,5224,a)]. 8.64/8.57 5264 -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(4248)]. 8.64/8.57 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(5264,a,4908,a)]. 8.64/8.57 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(5264,a,4952,a)]. 8.70/8.66 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(5264,a,5224,a)]. 8.70/8.66 5265 -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(4312)]. 8.70/8.66 5266 -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(4320)]. 8.70/8.66 5267 -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(4320)]. 8.70/8.66 5268 -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(4358)]. 8.70/8.66 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(5268,a,4908,a)]. 8.70/8.66 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(5268,a,4952,a)]. 8.70/8.66 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(5268,a,5224,a)]. 8.70/8.66 5269 -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(4364)]. 8.70/8.66 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(5269,a,4908,a)]. 8.70/8.66 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(5269,a,4952,a)]. 8.70/8.71 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(5269,a,5224,a)]. 8.70/8.71 5270 -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(4393)]. 8.70/8.71 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(5270,a,4908,a)]. 8.70/8.71 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(5270,a,4952,a)]. 8.70/8.71 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,5224,a)]. 8.70/8.71 5271 -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),f511(C,B,A)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),hAPP(c_Finite__Set_Ofold1(A,c_Orderings_Oord__class_Omin(A)),B))) # label(fact_strict__below__fold1__iff) # label(axiom). [clausify(4430)]. 8.70/8.71 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),f511(B,A,tc_Code__Numeral_Ocode__numeral)),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))). [resolve(5271,a,4908,a)]. 8.70/8.71 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),f511(B,A,tc_Int_Oint)),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))). [resolve(5271,a,4952,a)]. 8.70/8.71 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),f511(B,A,tc_Nat_Onat)),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))). [resolve(5271,a,5224,a)]. 8.70/8.71 5272 -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),f511(C,B,A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),hAPP(c_Finite__Set_Ofold1(A,c_Orderings_Oord__class_Omin(A)),B))) # label(fact_strict__below__fold1__iff) # label(axiom). [clausify(4430)]. 8.70/8.71 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),f511(B,A,tc_Code__Numeral_Ocode__numeral))) | 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))). [resolve(5272,a,4908,a)]. 8.70/8.71 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),f511(B,A,tc_Int_Oint))) | 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))). [resolve(5272,a,4952,a)]. 8.70/8.71 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),f511(B,A,tc_Nat_Onat))) | 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))). [resolve(5272,a,5224,a)]. 8.70/8.71 5273 -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),D),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),hAPP(c_Finite__Set_Ofold1(A,c_Orderings_Oord__class_Omin(A)),B))) # label(fact_strict__below__fold1__iff) # label(axiom). [clausify(4430)]. 8.70/8.71 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),C),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(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(5273,a,4908,a)]. 8.70/8.71 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),C),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),C),hAPP(c_Finite__Set_Ofold1(tc_Int_Oint,c_Orderings_Oord__class_Omin(tc_Int_Oint)),A))). [resolve(5273,a,4952,a)]. 8.70/8.71 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),C),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),C),hAPP(c_Finite__Set_Ofold1(tc_Nat_Onat,c_Orderings_Oord__class_Omin(tc_Nat_Onat)),A))). [resolve(5273,a,5224,a)]. 8.70/8.71 5274 -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(4434)]. 8.70/8.71 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(5274,a,4908,a)]. 8.70/8.71 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(5274,a,4952,a)]. 8.80/8.75 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(5274,a,5224,a)]. 8.80/8.75 5275 -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(4448)]. 8.80/8.75 5276 -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(4459)]. 8.80/8.75 5277 -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(4477)]. 8.80/8.75 5278 -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(4478)]. 8.80/8.75 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(5278,a,4908,a)]. 8.80/8.75 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(5278,a,4952,a)]. 8.80/8.75 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(5278,a,5224,a)]. 8.80/8.75 5279 -class_Orderings_Olinorder(A) | -class_Complete__Lattice_Ocomplete__lattice(A) | -hBOOL(hAPP(hAPP(c_member(B),C),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(E,C)),F)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(B,A),D),E)),F)) # label(fact_INF__less__iff) # label(axiom). [clausify(4485)]. 8.80/8.75 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Code__Numeral_Ocode__numeral) | -hBOOL(hAPP(hAPP(c_member(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(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(A,tc_Code__Numeral_Ocode__numeral),C),D)),E)). [resolve(5279,a,4908,a)]. 8.80/8.75 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_member(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(D,B)),E)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(A,tc_Int_Oint),C),D)),E)). [resolve(5279,a,4952,a)]. 8.80/8.75 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Nat_Onat) | -hBOOL(hAPP(hAPP(c_member(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(D,B)),E)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Complete__Lattice_Ocomplete__lattice__class_OINFI(A,tc_Nat_Onat),C),D)),E)). [resolve(5279,a,5224,a)]. 8.80/8.75 5280 -class_Orderings_Olinorder(A) | -class_Complete__Lattice_Ocomplete__lattice(A) | hBOOL(hAPP(hAPP(c_member(B),f516(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(4485)]. 8.80/8.75 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Code__Numeral_Ocode__numeral) | hBOOL(hAPP(hAPP(c_member(A),f516(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(5280,a,4908,a)]. 8.80/8.77 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_member(A),f516(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(5280,a,4952,a)]. 8.80/8.77 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Nat_Onat) | hBOOL(hAPP(hAPP(c_member(A),f516(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(5280,a,5224,a)]. 8.80/8.77 5281 -class_Orderings_Olinorder(A) | -class_Complete__Lattice_Ocomplete__lattice(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(B,f516(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(4485)]. 8.80/8.77 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,f516(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(5281,a,4908,a)]. 8.80/8.77 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(A,f516(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(5281,a,4952,a)]. 8.80/8.77 Derived: -class_Complete__Lattice_Ocomplete__lattice(tc_Nat_Onat) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(A,f516(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(5281,a,5224,a)]. 8.80/8.77 5282 -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(4506)]. 8.80/8.77 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(5282,a,4908,a)]. 8.80/8.77 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(5282,a,4952,a)]. 8.80/8.77 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(5282,a,5224,a)]. 8.80/8.77 5283 -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(4506)]. 8.86/8.83 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(5283,a,4908,a)]. 8.86/8.83 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(5283,a,4952,a)]. 8.86/8.83 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(5283,a,5224,a)]. 8.86/8.83 5284 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omin(A),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),C)),D) = hAPP(hAPP(c_Orderings_Oord__class_Omax(A),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),D)),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),C),D)) # label(fact_min__max_Oinf__sup__distrib2) # label(axiom). [clausify(4508)]. 8.86/8.83 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A),B)),C) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A),C)),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),B),C)). [resolve(5284,a,4908,a)]. 8.86/8.83 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),B)),C) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),A),C)),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),B),C)). [resolve(5284,a,4952,a)]. 8.86/8.83 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),A),B)),C) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),A),C)),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),B),C)). [resolve(5284,a,5224,a)]. 8.86/8.83 5285 -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(4572)]. 8.86/8.83 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(5285,a,4908,a)]. 8.86/8.83 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(5285,a,4952,a)]. 8.94/8.90 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = 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(5285,a,5224,a)]. 8.94/8.90 5286 -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(4574)]. 8.94/8.90 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(5286,a,4908,a)]. 8.94/8.90 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(5286,a,4952,a)]. 8.94/8.90 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(5286,a,5224,a)]. 8.94/8.90 5287 -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(4609)]. 8.94/8.90 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(5287,a,4908,a)]. 8.94/8.90 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(5287,a,4952,a)]. 8.94/8.90 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(5287,a,5224,a)]. 8.94/8.90 5288 -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(4628)]. 8.94/8.90 5289 -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(4661)]. 8.98/8.92 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(5289,a,4908,a)]. 8.98/8.92 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(5289,a,4952,a)]. 8.98/8.92 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(5289,a,5224,a)]. 8.98/8.92 5290 -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(4674)]. 8.98/8.92 5291 -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(4682)]. 8.98/8.92 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(5291,a,4908,a)]. 8.98/8.92 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(5291,a,4952,a)]. 8.98/8.92 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(5291,a,5224,a)]. 8.98/8.92 5292 -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(4682)]. 8.98/8.92 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(5292,a,4908,a)]. 9.00/8.96 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(5292,a,4952,a)]. 9.00/8.96 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(5292,a,5224,a)]. 9.00/8.96 5293 -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(4722)]. 9.00/8.96 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(5293,a,4908,a)]. 9.00/8.96 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(5293,a,4952,a)]. 9.00/8.96 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(5293,a,5224,a)]. 9.00/8.96 5294 -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(4738)]. 9.00/8.96 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(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(4739)]. 9.00/8.96 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(5295,a,4908,a)]. 9.00/8.96 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(5295,a,4952,a)]. 9.00/8.96 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(5295,a,5224,a)]. 9.00/8.96 5296 -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_Oequations_I2_J) # label(axiom). [clausify(4740)]. 9.00/8.96 5297 -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_Oequations_I2_J) # label(axiom). [clausify(4740)]. 9.00/8.96 5298 -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),f554(C,B,A)),hAPP(c_List_Oset(A),C))) # label(fact_sorted_Oequations_I2_J) # label(axiom). [clausify(4740)]. 9.00/8.96 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),f554(B,A,tc_Code__Numeral_Ocode__numeral)),hAPP(c_List_Oset(tc_Code__Numeral_Ocode__numeral),B))). [resolve(5298,a,4908,a)]. 9.00/8.96 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),f554(B,A,tc_Int_Oint)),hAPP(c_List_Oset(tc_Int_Oint),B))). [resolve(5298,a,4952,a)]. 9.00/8.96 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),f554(B,A,tc_Nat_Onat)),hAPP(c_List_Oset(tc_Nat_Onat),B))). [resolve(5298,a,5224,a)]. 9.00/8.96 5299 -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),f554(C,B,A))) # label(fact_sorted_Oequations_I2_J) # label(axiom). [clausify(4740)]. 9.00/8.96 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),f554(B,A,tc_Code__Numeral_Ocode__numeral))). [resolve(5299,a,4908,a)]. 9.00/8.96 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),f554(B,A,tc_Int_Oint))). [resolve(5299,a,4952,a)]. 9.00/8.96 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),f554(B,A,tc_Nat_Onat))). [resolve(5299,a,5224,a)]. 9.00/8.98 5300 -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(4747)]. 9.00/8.98 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(5300,a,4908,a)]. 9.00/8.98 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(5300,a,4952,a)]. 9.00/8.98 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(5300,a,5224,a)]. 9.00/8.98 5301 -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(4756)]. 9.00/8.98 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(5301,a,4908,a)]. 9.00/8.98 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(5301,a,4952,a)]. 9.00/8.98 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(5301,a,5224,a)]. 9.00/8.98 5302 -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(4756)]. 9.00/8.98 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(5302,a,4908,a)]. 9.00/8.98 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(5302,a,4952,a)]. 9.00/8.98 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(5302,a,5224,a)]. 9.00/8.98 5303 -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(4759)]. 9.00/8.99 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)) | 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(5303,a,4908,a)]. 9.00/8.99 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(5303,a,4952,a)]. 9.00/8.99 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(5303,a,5224,a)]. 9.00/8.99 5304 -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(4759)]. 9.00/8.99 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)) | 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(5304,a,4908,a)]. 9.00/8.99 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(5304,a,4952,a)]. 9.00/9.02 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(5304,a,5224,a)]. 9.00/9.02 5305 -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(4767)]. 9.00/9.02 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(5305,a,4908,a)]. 9.00/9.02 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(5305,a,4952,a)]. 9.00/9.02 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(5305,a,5224,a)]. 9.00/9.02 5306 -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(4771)]. 9.00/9.02 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(5306,a,4908,a)]. 9.00/9.02 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(5306,a,4952,a)]. 9.00/9.02 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(5306,a,5224,a)]. 9.00/9.02 5307 -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(4798)]. 10.01/9.99 5308 -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(4830)]. 10.01/9.99 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_OatMost(tc_Code__Numeral_Ocode__numeral),A)),c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Code__Numeral_Ocode__numeral,A,B)). [resolve(5308,a,4908,a)]. 10.01/9.99 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_OatMost(tc_Int_Oint),A)),c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Int_Oint,A,B)). [resolve(5308,a,4952,a)]. 10.01/9.99 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_OatMost(tc_Nat_Onat),A)),c_SetInterval_Oord__class_OgreaterThanAtMost(tc_Nat_Onat,A,B)). [resolve(5308,a,5224,a)]. 10.01/9.99 5309 -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(4864)]. 10.01/9.99 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(5309,a,4908,a)]. 10.01/9.99 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(5309,a,4952,a)]. 10.01/9.99 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(5309,a,5224,a)]. 10.01/9.99 5310 -class_Orderings_Olinorder(A) | hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),B) = B # label(fact_Min_Oidem) # label(axiom). [clausify(4878)]. 10.01/9.99 5311 -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(4891)]. 10.01/9.99 5312 -class_Power_Opower(A) | -class_Rings_Ono__zero__divisors(A) | -class_Rings_Ozero__neq__one(A) | -class_Rings_Omult__zero(A) | c_Groups_Ozero__class_Ozero(A) != hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != C # label(fact_power__eq__0__iff) # label(axiom). [clausify(264)]. 10.01/9.99 5313 class_Rings_Ono__zero__divisors(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Ono__zero__divisors) # label(axiom). [assumption]. 10.01/9.99 Derived: -class_Power_Opower(tc_Nat_Onat) | -class_Rings_Ozero__neq__one(tc_Nat_Onat) | -class_Rings_Omult__zero(tc_Nat_Onat) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != B. [resolve(5312,b,5313,a)]. 10.01/9.99 5314 -class_Power_Opower(A) | -class_Rings_Ono__zero__divisors(A) | -class_Rings_Ozero__neq__one(A) | -class_Rings_Omult__zero(A) | c_Groups_Ozero__class_Ozero(A) != hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C) | c_Groups_Ozero__class_Ozero(A) = B # label(fact_power__eq__0__iff) # label(axiom). [clausify(264)]. 10.01/9.99 Derived: -class_Power_Opower(tc_Nat_Onat) | -class_Rings_Ozero__neq__one(tc_Nat_Onat) | -class_Rings_Omult__zero(tc_Nat_Onat) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A. [resolve(5314,b,5313,a)]. 10.09/10.05 5315 -class_Power_Opower(A) | -class_Rings_Ono__zero__divisors(A) | -class_Rings_Ozero__neq__one(A) | -class_Rings_Omult__zero(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = C | c_Groups_Ozero__class_Ozero(A) != B # label(fact_power__eq__0__iff) # label(axiom). [clausify(264)]. 10.09/10.05 Derived: -class_Power_Opower(tc_Nat_Onat) | -class_Rings_Ozero__neq__one(tc_Nat_Onat) | -class_Rings_Omult__zero(tc_Nat_Onat) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = B | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != A. [resolve(5315,b,5313,a)]. 10.09/10.05 5316 class_Rings_Ono__zero__divisors(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Rings_Ono__zero__divisors) # label(axiom). [assumption]. 10.09/10.05 Derived: -class_Power_Opower(tc_Code__Numeral_Ocode__numeral) | -class_Rings_Ozero__neq__one(tc_Code__Numeral_Ocode__numeral) | -class_Rings_Omult__zero(tc_Code__Numeral_Ocode__numeral) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),B) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != B. [resolve(5316,a,5312,b)]. 10.09/10.05 Derived: -class_Power_Opower(tc_Code__Numeral_Ocode__numeral) | -class_Rings_Ozero__neq__one(tc_Code__Numeral_Ocode__numeral) | -class_Rings_Omult__zero(tc_Code__Numeral_Ocode__numeral) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),B) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = A. [resolve(5316,a,5314,b)]. 10.09/10.05 Derived: -class_Power_Opower(tc_Code__Numeral_Ocode__numeral) | -class_Rings_Ozero__neq__one(tc_Code__Numeral_Ocode__numeral) | -class_Rings_Omult__zero(tc_Code__Numeral_Ocode__numeral) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),B) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = B | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != A. [resolve(5316,a,5315,b)]. 10.09/10.05 5317 -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(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),D),C) | c_Groups_Ozero__class_Ozero(A) = hAPP(D,f119(D,C,B,A)) # label(fact_setprod__zero__iff) # label(axiom). [clausify(953)]. 10.09/10.05 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(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),C),B) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(C,f119(C,B,A,tc_Nat_Onat)). [resolve(5317,a,5313,a)]. 10.09/10.05 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(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),C),B) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = hAPP(C,f119(C,B,A,tc_Code__Numeral_Ocode__numeral)). [resolve(5317,a,5316,a)]. 10.09/10.05 5318 -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(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),D),C) | hBOOL(hAPP(hAPP(c_member(B),f119(D,C,B,A)),C)) # label(fact_setprod__zero__iff) # label(axiom). [clausify(953)]. 10.09/10.05 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(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),C),B) | hBOOL(hAPP(hAPP(c_member(A),f119(C,B,A,tc_Nat_Onat)),B)). [resolve(5318,a,5313,a)]. 10.86/10.84 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(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),C),B) | hBOOL(hAPP(hAPP(c_member(A),f119(C,B,A,tc_Code__Numeral_Ocode__numeral)),B)). [resolve(5318,a,5316,a)]. 10.86/10.84 5319 -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(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),D),C) | c_Groups_Ozero__class_Ozero(A) != hAPP(D,E) | -hBOOL(hAPP(hAPP(c_member(B),E),C)) # label(fact_setprod__zero__iff) # label(axiom). [clausify(953)]. 10.86/10.84 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(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),C),B) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(C,D) | -hBOOL(hAPP(hAPP(c_member(A),D),B)). [resolve(5319,a,5313,a)]. 10.86/10.84 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(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),C),B) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != hAPP(C,D) | -hBOOL(hAPP(hAPP(c_member(A),D),B)). [resolve(5319,a,5316,a)]. 10.86/10.84 5320 -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(3329)]. 10.86/10.84 Derived: 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_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B). [resolve(5320,a,5313,a)]. 10.86/10.84 Derived: c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = A | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = B | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),B). [resolve(5320,a,5316,a)]. 10.86/10.84 5321 -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(3389)]. 10.86/10.84 5322 class_Rings_Ono__zero__divisors(tc_Int_Oint) # label(arity_Int__Oint__Rings_Ono__zero__divisors) # label(axiom). [assumption]. 10.86/10.84 Derived: -class_Power_Opower(tc_Int_Oint) | -class_Rings_Ozero__neq__one(tc_Int_Oint) | -class_Rings_Omult__zero(tc_Int_Oint) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != B. [resolve(5322,a,5312,b)]. 10.86/10.84 Derived: -class_Power_Opower(tc_Int_Oint) | -class_Rings_Ozero__neq__one(tc_Int_Oint) | -class_Rings_Omult__zero(tc_Int_Oint) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A. [resolve(5322,a,5314,b)]. 10.86/10.84 Derived: -class_Power_Opower(tc_Int_Oint) | -class_Rings_Ozero__neq__one(tc_Int_Oint) | -class_Rings_Omult__zero(tc_Int_Oint) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = B | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A. [resolve(5322,a,5315,b)]. 10.86/10.84 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(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),C),B) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(C,f119(C,B,A,tc_Int_Oint)). [resolve(5322,a,5317,a)]. 12.51/12.48 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(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),C),B) | hBOOL(hAPP(hAPP(c_member(A),f119(C,B,A,tc_Int_Oint)),B)). [resolve(5322,a,5318,a)]. 12.51/12.48 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(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),C),B) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(C,D) | -hBOOL(hAPP(hAPP(c_member(A),D),B)). [resolve(5322,a,5319,a)]. 12.51/12.48 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = B | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B). [resolve(5322,a,5320,a)]. 12.51/12.48 5323 class_Rings_Olinordered__semidom(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Olinordered__semidom) # label(axiom). [assumption]. 12.51/12.48 5324 -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(20)]. 12.51/12.48 5325 -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(274)]. 12.51/12.48 5326 -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(301)]. 12.51/12.48 5327 -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(446)]. 12.51/12.48 5328 -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(446)]. 12.51/12.48 5329 -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(559)]. 12.51/12.48 5330 -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(794)]. 12.51/12.48 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(5323,a,5324,a)]. 12.51/12.48 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(5323,a,5325,a)]. 12.51/12.50 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(5323,a,5327,a)]. 12.51/12.50 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(5323,a,5328,a)]. 12.51/12.50 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(5323,a,5329,a)]. 12.51/12.50 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(5323,a,5330,a)]. 12.51/12.50 5331 class_Rings_Olinordered__semidom(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Rings_Olinordered__semidom) # label(axiom). [assumption]. 12.51/12.50 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(5331,a,5324,a)]. 12.51/12.50 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(5331,a,5325,a)]. 12.51/12.50 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(5331,a,5326,a)]. 12.51/12.50 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(5331,a,5327,a)]. 12.51/12.50 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(5331,a,5328,a)]. 12.58/12.56 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(5331,a,5329,a)]. 12.58/12.56 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(5331,a,5330,a)]. 12.58/12.56 5332 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),B)) | hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C) != hAPP(hAPP(c_Power_Opower__class_Opower(A),B),D) | C = D # label(fact_power__inject__exp) # label(axiom). [clausify(997)]. 12.58/12.56 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),A)) | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),C) | B = C. [resolve(5332,a,5323,a)]. 12.58/12.56 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),A)) | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),B) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),C) | B = C. [resolve(5332,a,5331,a)]. 12.58/12.56 5333 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),B)) | hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C) = hAPP(hAPP(c_Power_Opower__class_Opower(A),B),D) | C != D # label(fact_power__inject__exp) # label(axiom). [clausify(997)]. 12.58/12.56 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),A)) | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),C) | B != C. [resolve(5333,a,5323,a)]. 12.58/12.56 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),A)) | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),B) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),C) | B != C. [resolve(5333,a,5331,a)]. 12.58/12.56 5334 -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(1013)]. 12.58/12.56 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(5334,a,5331,a)]. 12.58/12.56 5335 -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(1224)]. 12.58/12.56 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(5335,a,5323,a)]. 12.58/12.56 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(5335,a,5331,a)]. 12.79/12.72 5336 -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(1224)]. 12.79/12.72 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(5336,a,5323,a)]. 12.79/12.72 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(5336,a,5331,a)]. 12.79/12.72 5337 -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(1595)]. 12.79/12.72 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(5337,a,5323,a)]. 12.79/12.72 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(5337,a,5331,a)]. 12.79/12.72 5338 -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(1902)]. 12.79/12.72 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),A)),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(5338,a,5323,a)]. 12.79/12.72 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)),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(5338,a,5331,a)]. 12.79/12.72 5339 -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(1902)]. 12.79/12.72 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),A)),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(5339,a,5323,a)]. 12.80/12.77 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(5339,a,5331,a)]. 12.80/12.77 5340 -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(1973)]. 12.80/12.77 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(5340,a,5323,a)]. 12.80/12.77 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(5340,a,5331,a)]. 12.80/12.77 5341 -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)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),C)) | B = C # label(fact_power2__eq__imp__eq) # label(axiom). [clausify(2024)]. 12.80/12.77 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)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B)) | A = B. [resolve(5341,a,5323,a)]. 12.80/12.77 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)),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)) | A = B. [resolve(5341,a,5331,a)]. 12.80/12.77 5342 -class_Rings_Olinordered__semidom(A) | hBOOL(hAPP(hAPP(c_member(B),f235(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(2054)]. 12.80/12.77 Derived: hBOOL(hAPP(hAPP(c_member(A),f235(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(5342,a,5323,a)]. 12.80/12.77 Derived: hBOOL(hAPP(hAPP(c_member(A),f235(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(5342,a,5331,a)]. 12.80/12.77 5343 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(B,f235(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(2054)]. 12.89/12.87 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(A,f235(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(5343,a,5323,a)]. 12.89/12.87 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,f235(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(5343,a,5331,a)]. 12.89/12.87 5344 -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(2073)]. 12.89/12.87 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(5344,a,5323,a)]. 12.89/12.87 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(5344,a,5331,a)]. 12.89/12.87 5345 -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(2135)]. 12.89/12.87 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(5345,a,5331,a)]. 12.89/12.87 5346 -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(2305)]. 12.89/12.87 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(5346,a,5331,a)]. 12.89/12.87 5347 -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(2322)]. 12.96/12.89 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(5347,a,5323,a)]. 12.96/12.89 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(5347,a,5331,a)]. 12.96/12.89 5348 class_Rings_Olinordered__semidom(tc_Int_Oint) # label(arity_Int__Oint__Rings_Olinordered__semidom) # label(axiom). [assumption]. 12.96/12.89 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(5348,a,5324,a)]. 12.96/12.89 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(5348,a,5325,a)]. 12.96/12.89 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(5348,a,5326,a)]. 12.96/12.89 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(5348,a,5327,a)]. 12.96/12.89 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(5348,a,5328,a)]. 12.96/12.89 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(5348,a,5329,a)]. 12.96/12.89 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(5348,a,5330,a)]. 12.96/12.89 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),A)) | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),C) | B = C. [resolve(5348,a,5332,a)]. 12.96/12.89 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),A)) | hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),C) | B != C. [resolve(5348,a,5333,a)]. 12.96/12.90 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(5348,a,5334,a)]. 12.96/12.90 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(5348,a,5335,a)]. 12.96/12.90 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(5348,a,5336,a)]. 12.96/12.90 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(5348,a,5337,a)]. 12.96/12.90 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(5348,a,5340,a)]. 12.96/12.90 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)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)) | A = B. [resolve(5348,a,5341,a)]. 12.96/12.90 Derived: hBOOL(hAPP(hAPP(c_member(A),f235(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(5348,a,5342,a)]. 12.96/12.90 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(A,f235(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(5348,a,5343,a)]. 12.96/12.90 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(5348,a,5344,a)]. 12.96/12.90 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(5348,a,5345,a)]. 12.96/12.90 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(5348,a,5346,a)]. 12.96/12.90 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(5348,a,5347,a)]. 12.96/12.94 5349 -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(2329)]. 12.96/12.94 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(5349,a,5331,a)]. 12.96/12.94 5350 -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),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),C))) # label(fact_of__nat__le__iff) # label(axiom). [clausify(2337)]. 12.96/12.94 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(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),A)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),B))). [resolve(5350,a,5323,a)]. 12.96/12.94 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),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(5350,a,5331,a)]. 12.96/12.94 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),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(5350,a,5348,a)]. 12.96/12.94 5351 -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),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),B)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),C))) # label(fact_of__nat__le__iff) # label(axiom). [clausify(2337)]. 12.96/12.94 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(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),A)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),B))). [resolve(5351,a,5323,a)]. 12.96/12.94 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),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(5351,a,5331,a)]. 12.96/12.94 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),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(5351,a,5348,a)]. 12.96/12.94 5352 -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(2414)]. 12.96/12.94 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(5352,a,5323,a)]. 12.96/12.94 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(5352,a,5331,a)]. 13.07/13.01 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(5352,a,5348,a)]. 13.07/13.01 5353 -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(2584)]. 13.07/13.01 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(5353,a,5323,a)]. 13.07/13.01 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(5353,a,5331,a)]. 13.07/13.01 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(5353,a,5348,a)]. 13.07/13.01 5354 -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(2584)]. 13.07/13.01 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(5354,a,5323,a)]. 13.07/13.01 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(5354,a,5331,a)]. 13.07/13.01 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(5354,a,5348,a)]. 13.12/13.06 5355 -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(2615)]. 13.12/13.06 5356 -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(2644)]. 13.12/13.06 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(5356,a,5323,a)]. 13.12/13.06 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(5356,a,5331,a)]. 13.12/13.06 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(5356,a,5348,a)]. 13.12/13.06 5357 -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(2685)]. 13.12/13.06 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),c_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(5357,a,5323,a)]. 13.12/13.06 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(5357,a,5331,a)]. 13.12/13.06 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(5357,a,5348,a)]. 13.42/13.36 5358 -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(2876)]. 13.42/13.36 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(5358,a,5323,a)]. 13.42/13.36 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(5358,a,5331,a)]. 13.42/13.36 5359 -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(3005)]. 13.42/13.36 5360 -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(3137)]. 13.42/13.36 5361 -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(3203)]. 13.42/13.36 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),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(5361,a,5323,a)]. 13.42/13.36 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(5361,a,5331,a)]. 13.42/13.36 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_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(5361,a,5348,a)]. 13.42/13.36 5362 -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(3294)]. 13.56/13.52 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),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(5362,a,5323,a)]. 13.56/13.52 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(5362,a,5331,a)]. 13.56/13.52 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(5362,a,5348,a)]. 13.56/13.52 5363 -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(3523)]. 13.56/13.52 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(5363,a,5323,a)]. 13.56/13.52 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),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(5363,a,5331,a)]. 13.56/13.52 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(5363,a,5348,a)]. 13.56/13.52 5364 -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(3602)]. 13.56/13.52 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(5364,a,5323,a)]. 13.72/13.63 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(5364,a,5331,a)]. 13.72/13.63 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(5364,a,5348,a)]. 13.72/13.63 5365 -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(3722)]. 13.72/13.63 5366 -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(3776)]. 13.72/13.63 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(5366,a,5323,a)]. 13.72/13.63 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),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(5366,a,5331,a)]. 13.72/13.63 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(5366,a,5348,a)]. 13.72/13.63 5367 -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(3786)]. 13.72/13.63 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(5367,a,5323,a)]. 13.72/13.63 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(5367,a,5331,a)]. 13.97/13.92 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(5367,a,5348,a)]. 13.97/13.92 5368 -class_Rings_Olinordered__semidom(A) | hBOOL(hAPP(hAPP(c_member(B),f437(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(3801)]. 13.97/13.92 Derived: hBOOL(hAPP(hAPP(c_member(A),f437(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(5368,a,5331,a)]. 13.97/13.92 Derived: hBOOL(hAPP(hAPP(c_member(A),f437(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(5368,a,5348,a)]. 13.97/13.92 5369 -class_Rings_Olinordered__semidom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),hAPP(B,f437(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(3801)]. 13.97/13.92 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),hAPP(A,f437(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(5369,a,5331,a)]. 13.97/13.92 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(A,f437(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(5369,a,5348,a)]. 13.97/13.92 5370 -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(4198)]. 13.97/13.92 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(5370,a,5323,a)]. 13.97/13.92 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(5370,a,5331,a)]. 14.15/14.11 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(5370,a,5348,a)]. 14.15/14.11 5371 -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(4298)]. 14.15/14.11 5372 -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(4323)]. 14.15/14.11 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Nat_Onat),A))). [resolve(5372,a,5323,a)]. 14.15/14.11 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),A))). [resolve(5372,a,5348,a)]. 14.15/14.11 5373 -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(4353)]. 14.15/14.11 5374 -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(4441)]. 14.15/14.11 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(5374,a,5323,a)]. 14.15/14.11 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(5374,a,5331,a)]. 14.15/14.11 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(5374,a,5348,a)]. 14.15/14.11 5375 -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(4456)]. 14.15/14.11 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)) | 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(5375,a,5323,a)]. 14.28/14.20 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(5375,a,5331,a)]. 14.28/14.20 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(5375,a,5348,a)]. 14.28/14.20 5376 -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(4548)]. 14.28/14.20 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(5376,a,5323,a)]. 14.28/14.20 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(5376,a,5331,a)]. 14.28/14.20 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(5376,a,5348,a)]. 14.28/14.20 5377 -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(4585)]. 14.28/14.20 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(5377,a,5323,a)]. 14.28/14.20 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(5377,a,5331,a)]. 15.68/15.61 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(5377,a,5348,a)]. 15.68/15.61 5378 -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(4865)]. 15.68/15.61 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(5378,a,5323,a)]. 15.68/15.61 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(5378,a,5331,a)]. 15.68/15.61 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(5378,a,5348,a)]. 15.68/15.61 5379 class_Groups_Oab__semigroup__mult(tc_Int_Oint) # label(arity_Int__Oint__Groups_Oab__semigroup__mult) # label(axiom). [assumption]. 15.68/15.61 5380 -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(33)]. 15.68/15.61 5381 -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(975)]. 15.68/15.61 5382 -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(1044)]. 15.68/15.61 5383 -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(1321)]. 15.82/15.74 5384 -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(1837)]. 15.82/15.74 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(5379,a,5380,a)]. 15.82/15.74 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(5379,a,5381,a)]. 15.82/15.74 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(5379,a,5383,a)]. 15.82/15.74 Derived: c_Finite__Set_Ofun__left__comm(tc_Int_Oint,tc_Int_Oint,c_Groups_Otimes__class_Otimes(tc_Int_Oint)). [resolve(5379,a,5384,a)]. 15.82/15.74 5385 -class_Groups_Oab__semigroup__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hBOOL(hAPP(hAPP(c_member(B),f269(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(2308)]. 15.82/15.74 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f269(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(5385,a,5379,a)]. 15.82/15.74 5386 -class_Groups_Oab__semigroup__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hAPP(D,f269(E,D,F,C,B,A)) != hAPP(F,f269(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(2308)]. 15.82/15.74 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(C,f269(D,C,E,B,A,tc_Int_Oint)) != hAPP(E,f269(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(5386,a,5379,a)]. 15.82/15.74 5387 class_Groups_Oab__semigroup__mult(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Oab__semigroup__mult) # label(axiom). [assumption]. 15.82/15.74 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(5387,a,5380,a)]. 16.46/16.37 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(5387,a,5381,a)]. 16.46/16.37 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(5387,a,5383,a)]. 16.46/16.37 Derived: c_Finite__Set_Ofun__left__comm(tc_Nat_Onat,tc_Nat_Onat,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)). [resolve(5387,a,5384,a)]. 16.46/16.37 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f269(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(5387,a,5385,a)]. 16.46/16.37 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(C,f269(D,C,E,B,A,tc_Nat_Onat)) != hAPP(E,f269(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(5387,a,5386,a)]. 16.46/16.37 5388 -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(3947)]. 16.46/16.37 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(5388,a,5379,a)]. 16.46/16.37 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(5388,a,5387,a)]. 16.46/16.37 5389 class_Groups_Oab__semigroup__mult(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Groups_Oab__semigroup__mult) # label(axiom). [assumption]. 16.46/16.37 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(5389,a,5380,a)]. 16.46/16.38 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(5389,a,5381,a)]. 16.46/16.38 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(5389,a,5382,a)]. 16.46/16.38 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) | 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(5389,a,5383,a)]. 16.46/16.38 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(5389,a,5384,a)]. 16.46/16.38 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f269(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(5389,a,5385,a)]. 16.46/16.38 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(C,f269(D,C,E,B,A,tc_Code__Numeral_Ocode__numeral)) != hAPP(E,f269(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(5389,a,5386,a)]. 16.46/16.38 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)) | 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(5389,a,5388,a)]. 16.67/16.65 5390 -class_Groups_Oab__semigroup__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -c_Fun_Oinj__on(B,D,E,C) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,D,c_Groups_Otimes__class_Otimes(A)),F),V6),hAPP(c_Set_Oimage(B,D,E),C)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,B,c_Groups_Otimes__class_Otimes(A)),hAPP(c_Fun_Ocomp(D,A,B,F),E)),V6),C) # label(fact_fold__image__reindex) # label(axiom). [clausify(4338)]. 16.67/16.65 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -c_Fun_Oinj__on(A,C,D,B) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,C,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),E),F),hAPP(c_Set_Oimage(A,C,D),B)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,A,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),hAPP(c_Fun_Ocomp(C,tc_Int_Oint,A,E),D)),F),B). [resolve(5390,a,5379,a)]. 16.67/16.65 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -c_Fun_Oinj__on(A,C,D,B) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,C,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),E),F),hAPP(c_Set_Oimage(A,C,D),B)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,A,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),hAPP(c_Fun_Ocomp(C,tc_Nat_Onat,A,E),D)),F),B). [resolve(5390,a,5387,a)]. 16.67/16.65 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -c_Fun_Oinj__on(A,C,D,B) | hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Code__Numeral_Ocode__numeral,C,c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral)),E),F),hAPP(c_Set_Oimage(A,C,D),B)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Code__Numeral_Ocode__numeral,A,c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral)),hAPP(c_Fun_Ocomp(C,tc_Code__Numeral_Ocode__numeral,A,E),D)),F),B). [resolve(5390,a,5389,a)]. 16.67/16.65 5391 -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(4676)]. 16.67/16.65 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(5391,a,5379,a)]. 16.67/16.65 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(5391,a,5387,a)]. 16.67/16.65 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(5391,a,5389,a)]. 16.67/16.65 5392 -class_Groups_Oab__semigroup__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hBOOL(hAPP(hAPP(c_member(B),D),C)) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(E,D)),hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,B,c_Groups_Otimes__class_Otimes(A)),E),F),C)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(A,B,c_Groups_Otimes__class_Otimes(A)),E),F),hAPP(hAPP(c_Set_Oinsert(B),D),C)) # label(fact_fold__image__insert) # label(axiom). [clausify(4677)]. 16.77/16.68 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(D,C)),hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,A,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),D),E),B)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Int_Oint,A,c_Groups_Otimes__class_Otimes(tc_Int_Oint)),D),E),hAPP(hAPP(c_Set_Oinsert(A),C),B)). [resolve(5392,a,5379,a)]. 16.77/16.68 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(D,C)),hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,A,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),D),E),B)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,A,c_Groups_Otimes__class_Otimes(tc_Nat_Onat)),D),E),hAPP(hAPP(c_Set_Oinsert(A),C),B)). [resolve(5392,a,5387,a)]. 16.77/16.68 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(D,C)),hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Code__Numeral_Ocode__numeral,A,c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral)),D),E),B)) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Code__Numeral_Ocode__numeral,A,c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral)),D),E),hAPP(hAPP(c_Set_Oinsert(A),C),B)). [resolve(5392,a,5389,a)]. 16.77/16.68 5393 -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),f552(B,A)),f553(B,A))),hAPP(hAPP(c_Set_Oinsert(A),f552(B,A)),hAPP(hAPP(c_Set_Oinsert(A),f553(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(4734)]. 16.77/16.68 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),f552(A,tc_Int_Oint)),f553(A,tc_Int_Oint))),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),f552(A,tc_Int_Oint)),hAPP(hAPP(c_Set_Oinsert(tc_Int_Oint),f553(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(5393,a,5379,a)]. 16.77/16.68 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),f552(A,tc_Nat_Onat)),f553(A,tc_Nat_Onat))),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),f552(A,tc_Nat_Onat)),hAPP(hAPP(c_Set_Oinsert(tc_Nat_Onat),f553(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(5393,a,5387,a)]. 16.77/16.68 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),f552(A,tc_Code__Numeral_Ocode__numeral)),f553(A,tc_Code__Numeral_Ocode__numeral))),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),f552(A,tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_Set_Oinsert(tc_Code__Numeral_Ocode__numeral),f553(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(5393,a,5389,a)]. 17.93/17.84 5394 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),D)) # label(fact_add__le__cancel__left) # label(axiom). [clausify(243)]. 17.93/17.84 5395 class_Groups_Oordered__ab__semigroup__add__imp__le(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Oordered__ab__semigroup__add__imp__le) # label(axiom). [assumption]. 17.93/17.84 5396 class_Groups_Oordered__ab__semigroup__add__imp__le(tc_Int_Oint) # label(arity_Int__Oint__Groups_Oordered__ab__semigroup__add__imp__le) # label(axiom). [assumption]. 17.93/17.84 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),C)). [resolve(5394,a,5396,a)]. 17.93/17.84 5397 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),D)) # label(fact_add__le__cancel__left) # label(axiom). [clausify(243)]. 17.93/17.84 5398 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),C)),D) = hAPP(hAPP(c_Orderings_Oord__class_Omax(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),D)) # label(fact_max__add__distrib__left) # label(axiom). [clausify(385)]. 17.93/17.84 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),A),B)),C) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),C)). [resolve(5398,a,5395,a)]. 17.93/17.84 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),B)),C) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),C)). [resolve(5398,a,5396,a)]. 17.93/17.84 5399 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),D)) # label(fact_add__less__cancel__right) # label(axiom). [clausify(672)]. 17.93/17.84 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),A),C)). [resolve(5399,a,5395,a)]. 17.93/17.84 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),C),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),C)). [resolve(5399,a,5396,a)]. 17.93/17.84 5400 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),D)) # label(fact_add__less__cancel__right) # label(axiom). [clausify(672)]. 17.93/17.84 5401 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),D)) # label(fact_add__less__imp__less__right) # label(axiom). [clausify(923)]. 18.34/18.26 5402 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | hAPP(hAPP(c_Orderings_Oord__class_Omin(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),D)),C) # label(fact_min__add__distrib__left) # label(axiom). [clausify(955)]. 18.34/18.26 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Nat_Onat),A),C)),B). [resolve(5402,a,5395,a)]. 18.34/18.26 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),C),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),A),C)),B). [resolve(5402,a,5396,a)]. 18.34/18.26 5403 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),D))) # label(fact_add__le__cancel__right) # label(axiom). [clausify(1714)]. 18.34/18.26 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),C))). [resolve(5403,a,5396,a)]. 18.34/18.26 5404 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),D))) # label(fact_add__le__cancel__right) # label(axiom). [clausify(1714)]. 18.34/18.26 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),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),C))). [resolve(5404,a,5395,a)]. 18.34/18.26 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),C))). [resolve(5404,a,5396,a)]. 18.34/18.26 5405 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),D)) # label(fact_add__le__imp__le__left) # label(axiom). [clausify(1818)]. 18.34/18.26 5406 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),C))) # label(fact_add__less__cancel__left) # label(axiom). [clausify(2721)]. 18.34/18.26 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),C),A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),C),B))). [resolve(5406,a,5396,a)]. 18.34/18.26 5407 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),C))) # label(fact_add__less__cancel__left) # label(axiom). [clausify(2721)]. 18.34/18.26 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),C),A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),C),B))). [resolve(5407,a,5396,a)]. 18.48/18.41 5408 class_Groups_Oordered__ab__semigroup__add__imp__le(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Groups_Oordered__ab__semigroup__add__imp__le) # label(axiom). [assumption]. 18.48/18.41 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),B),C)). [resolve(5408,a,5394,a)]. 18.48/18.41 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),B),C)). [resolve(5408,a,5397,a)]. 18.48/18.41 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),A),B)),C) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),B),C)). [resolve(5408,a,5398,a)]. 18.48/18.41 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),C),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),C)). [resolve(5408,a,5399,a)]. 18.48/18.41 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),C),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),C)). [resolve(5408,a,5400,a)]. 18.48/18.41 Derived: hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),C),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Code__Numeral_Ocode__numeral),A),C)),B). [resolve(5408,a,5402,a)]. 18.48/18.41 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),B),C))). [resolve(5408,a,5403,a)]. 18.48/18.41 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),B),C))). [resolve(5408,a,5404,a)]. 18.48/18.41 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),C),A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),C),B))). [resolve(5408,a,5406,a)]. 18.48/18.41 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),C),A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),C),B))). [resolve(5408,a,5407,a)]. 21.74/21.68 5409 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),D))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),D)) # label(fact_add__less__imp__less__left) # label(axiom). [clausify(3610)]. 21.74/21.68 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Nat_Onat),B),C)). [resolve(5409,a,5395,a)]. 21.74/21.68 5410 -class_Groups_Oordered__ab__semigroup__add__imp__le(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),D)) # label(fact_add__le__imp__le__right) # label(axiom). [clausify(3711)]. 21.74/21.68 5411 class_Groups_Ogroup__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Ogroup__add) # label(axiom). [assumption]. 21.74/21.68 5412 -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(47)]. 21.74/21.68 5413 -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(374)]. 21.74/21.68 5414 -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(495)]. 21.74/21.68 5415 -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(919)]. 21.74/21.68 5416 -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(919)]. 21.74/21.68 5417 -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(1334)]. 21.74/21.68 5418 -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(1334)]. 21.74/21.68 5419 -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(1569)]. 21.74/21.68 5420 -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(1733)]. 21.74/21.68 5421 -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(1733)]. 21.74/21.68 5422 -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(1830)]. 21.74/21.68 5423 -class_Groups_Ogroup__add(A) | c_Groups_Ozero__class_Ozero(A) != hAPP(c_Groups_Ouminus__class_Ouminus(A),B) | c_Groups_Ozero__class_Ozero(A) = B # label(fact_neg__equal__0__iff__equal) # label(axiom). [clausify(1862)]. 21.74/21.68 5424 -class_Groups_Ogroup__add(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(c_Groups_Ouminus__class_Ouminus(A),B) | c_Groups_Ozero__class_Ozero(A) != B # label(fact_neg__equal__0__iff__equal) # label(axiom). [clausify(1862)]. 21.74/21.68 5425 -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(1864)]. 21.74/21.68 5426 -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(2089)]. 21.74/21.68 5427 -class_Groups_Ogroup__add(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),hAPP(c_Groups_Ouminus__class_Ouminus(A),C)) # label(fact_diff__minus__eq__add) # label(axiom). [clausify(2366)]. 21.74/21.68 5428 -class_Groups_Ogroup__add(A) | c_Groups_Ozero__class_Ozero(A) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C) | B = C # label(fact_right__minus__eq) # label(axiom). [clausify(3117)]. 21.74/21.68 5429 -class_Groups_Ogroup__add(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C) | B != C # label(fact_right__minus__eq) # label(axiom). [clausify(3117)]. 21.74/21.68 5430 -class_Groups_Ogroup__add(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),hAPP(c_Groups_Ouminus__class_Ouminus(A),C)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C) # label(fact_diff__def) # label(axiom). [clausify(3269)]. 21.74/21.68 5431 -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(3360)]. 21.74/21.68 5432 -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(3580)]. 21.74/21.68 5433 -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(3855)]. 21.74/21.68 5434 -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(3971)]. 21.74/21.68 5435 -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(3971)]. 21.74/21.68 5436 -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(4167)]. 21.74/21.68 5437 -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(4376)]. 21.74/21.68 5438 -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(4466)]. 21.74/21.68 5439 -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(4466)]. 21.74/21.68 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(5411,a,5412,a)]. 21.74/21.68 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(5411,a,5413,a)]. 21.74/21.68 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(5411,a,5414,a)]. 21.74/21.68 Derived: A != B | hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B). [resolve(5411,a,5415,a)]. 21.74/21.68 Derived: A = B | hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A) != hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B). [resolve(5411,a,5416,a)]. 21.74/21.68 Derived: hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A) != B | hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B) = A. [resolve(5411,a,5417,a)]. 23.02/22.98 Derived: hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint)) = A. [resolve(5411,a,5419,a)]. 23.02/22.98 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(5411,a,5420,a)]. 23.02/22.98 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(5411,a,5421,a)]. 23.02/22.98 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(5411,a,5422,a)]. 23.02/22.98 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A. [resolve(5411,a,5423,a)]. 23.02/22.98 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A. [resolve(5411,a,5424,a)]. 23.02/22.98 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(5411,a,5425,a)]. 23.02/22.98 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),A). [resolve(5411,a,5426,a)]. 23.02/22.98 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B)). [resolve(5411,a,5427,a)]. 23.02/22.98 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B) | A = B. [resolve(5411,a,5428,a)]. 23.02/22.98 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B) | A != B. [resolve(5411,a,5429,a)]. 23.02/22.98 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(5411,a,5431,a)]. 23.02/22.98 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(5411,a,5432,a)]. 23.02/22.98 Derived: hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)) = A. [resolve(5411,a,5433,a)]. 23.02/22.98 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(5411,a,5436,a)]. 23.02/22.98 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(5411,a,5437,a)]. 23.02/22.98 5440 -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(4721)]. 23.02/22.98 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(5440,a,5411,a)]. 23.02/22.98 5441 -class_Nat_Osize(A) | c_FunDef_Ois__measure(A,c_Nat_Osize__class_Osize(A)) # label(fact_measure__size) # label(axiom). [clausify(334)]. 23.02/22.98 5442 class_Nat_Osize(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Nat_Osize) # label(axiom). [assumption]. 23.02/22.98 5443 class_Nat_Osize(tc_Product__Type_Ounit) # label(arity_Product____Type__Ounit__Nat_Osize) # label(axiom). [assumption]. 23.02/22.98 5444 class_Nat_Osize(tc_Nitpick_Opair__box(A,B)) # label(arity_Nitpick__Opair____box__Nat_Osize) # label(axiom). [clausify(427)]. 23.02/22.98 5445 class_Nat_Osize(tc_Nat_Onat) # label(arity_Nat__Onat__Nat_Osize) # label(axiom). [assumption]. 23.02/22.98 5446 class_Nat_Osize(tc_Com_Ocom) # label(arity_Com__Ocom__Nat_Osize) # label(axiom). [assumption]. 25.79/25.72 5447 class_Nat_Osize(tc_Code__Evaluation_Oterm) # label(arity_Code____Evaluation__Oterm__Nat_Osize) # label(axiom). [assumption]. 25.79/25.72 5448 class_Nat_Osize(tc_prod(A,B)) # label(arity_prod__Nat_Osize) # label(axiom). [clausify(1619)]. 25.79/25.72 5449 class_Nat_Osize(tc_HOL_Obool) # label(arity_HOL__Obool__Nat_Osize) # label(axiom). [assumption]. 25.79/25.72 5450 class_Nat_Osize(tc_List_Olist(A)) # label(arity_List__Olist__Nat_Osize) # label(axiom). [clausify(2164)]. 25.79/25.72 5451 class_Nat_Osize(tc_Hoare__Mirabelle_Otriple(A)) # label(arity_Hoare____Mirabelle__Otriple__Nat_Osize) # label(axiom). [clausify(3145)]. 25.79/25.72 5452 class_Nat_Osize(tc_Option_Ooption(A)) # label(arity_Option__Ooption__Nat_Osize) # label(axiom). [clausify(3658)]. 25.79/25.72 5453 class_Nat_Osize(tc_sum(A,B)) # label(arity_sum__Nat_Osize) # label(axiom). [clausify(3774)]. 25.79/25.72 5454 class_Nat_Osize(tc_Com_Ostate) # label(arity_Com__Ostate__Nat_Osize) # label(axiom). [assumption]. 25.79/25.72 Derived: c_FunDef_Ois__measure(tc_Com_Ostate,c_Nat_Osize__class_Osize(tc_Com_Ostate)). [resolve(5454,a,5441,a)]. 25.79/25.72 5455 class_Nat_Osize(tc_Lazy__Sequence_Olazy__sequence(A)) # label(arity_Lazy____Sequence__Olazy____sequence__Nat_Osize) # label(axiom). [clausify(4654)]. 25.79/25.72 Derived: c_FunDef_Ois__measure(tc_Lazy__Sequence_Olazy__sequence(A),c_Nat_Osize__class_Osize(tc_Lazy__Sequence_Olazy__sequence(A))). [resolve(5455,a,5441,a)]. 25.79/25.72 5456 class_Rings_Olinordered__ring__strict(tc_Int_Oint) # label(arity_Int__Oint__Rings_Olinordered__ring__strict) # label(axiom). [assumption]. 25.79/25.72 5457 -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(66)]. 25.79/25.72 5458 -class_Rings_Olinordered__ring__strict(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),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),C))),c_Groups_Ozero__class_Ozero(A))) | c_Groups_Ozero__class_Ozero(A) = B # label(fact_sum__squares__le__zero__iff) # label(axiom). [clausify(279)]. 25.79/25.72 5459 -class_Rings_Olinordered__ring__strict(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),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),C))),c_Groups_Ozero__class_Ozero(A))) | c_Groups_Ozero__class_Ozero(A) = C # label(fact_sum__squares__le__zero__iff) # label(axiom). [clausify(279)]. 25.79/25.72 5460 -class_Rings_Olinordered__ring__strict(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),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),C))),c_Groups_Ozero__class_Ozero(A))) | c_Groups_Ozero__class_Ozero(A) != B | c_Groups_Ozero__class_Ozero(A) != C # label(fact_sum__squares__le__zero__iff) # label(axiom). [clausify(279)]. 25.79/25.72 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),C),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),A))). [resolve(5456,a,5457,a)]. 25.79/25.72 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),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),B))),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A. [resolve(5456,a,5458,a)]. 25.79/25.72 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),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),B))),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = B. [resolve(5456,a,5459,a)]. 25.79/25.76 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),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),B))),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != B. [resolve(5456,a,5460,a)]. 25.79/25.76 5461 -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(1079)]. 25.79/25.76 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(5461,a,5456,a)]. 25.79/25.76 5462 -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(1079)]. 25.79/25.76 5463 -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(1079)]. 25.79/25.76 5464 -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(1079)]. 25.79/25.76 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(5464,a,5456,a)]. 25.79/25.76 5465 -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(1079)]. 25.79/25.76 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(5465,a,5456,a)]. 25.79/25.76 5466 -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(1079)]. 25.86/25.80 5467 -class_Rings_Olinordered__ring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),c_Groups_Ozero__class_Ozero(A))) # label(fact_mult__le__0__iff) # label(axiom). [clausify(1268)]. 25.86/25.80 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(5467,a,5456,a)]. 25.86/25.80 5468 -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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),c_Groups_Ozero__class_Ozero(A))) # label(fact_mult__le__0__iff) # label(axiom). [clausify(1268)]. 25.86/25.80 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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5468,a,5456,a)]. 25.86/25.80 5469 -class_Rings_Olinordered__ring__strict(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),c_Groups_Ozero__class_Ozero(A))) # label(fact_mult__le__0__iff) # label(axiom). [clausify(1268)]. 25.86/25.80 5470 -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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),c_Groups_Ozero__class_Ozero(A))) # label(fact_mult__le__0__iff) # label(axiom). [clausify(1268)]. 25.86/25.80 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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5470,a,5456,a)]. 25.86/25.80 5471 -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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),B)),c_Groups_Ozero__class_Ozero(A))) # label(fact_mult__le__0__iff) # label(axiom). [clausify(1268)]. 25.86/25.80 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_Otimes__class_Otimes(tc_Int_Oint),B),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5471,a,5456,a)]. 25.86/25.84 5472 -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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),B)),c_Groups_Ozero__class_Ozero(A))) # label(fact_mult__le__0__iff) # label(axiom). [clausify(1268)]. 25.86/25.84 5473 -class_Rings_Olinordered__ring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),D),C)) # label(fact_mult__le__cancel__left__neg) # label(axiom). [clausify(1307)]. 25.86/25.84 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),C),B)). [resolve(5473,a,5456,a)]. 25.86/25.84 5474 -class_Rings_Olinordered__ring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),D),C)) # label(fact_mult__le__cancel__left__neg) # label(axiom). [clausify(1307)]. 25.86/25.84 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),C),B)). [resolve(5474,a,5456,a)]. 25.86/25.84 5475 -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(1506)]. 25.86/25.84 5476 -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),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C))) # label(fact_mult__less__cancel__left__disj) # label(axiom). [clausify(1506)]. 25.86/25.84 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),B))). [resolve(5476,a,5456,a)]. 25.86/25.84 5477 -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(1506)]. 25.86/25.84 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(5477,a,5456,a)]. 25.99/25.91 5478 -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(1506)]. 25.99/25.91 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(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(5478,a,5456,a)]. 25.99/25.91 5479 -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(1506)]. 25.99/25.91 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)) | -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(5479,a,5456,a)]. 25.99/25.91 5480 -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(1506)]. 25.99/25.91 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),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(5480,a,5456,a)]. 25.99/25.91 5481 -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(1876)]. 25.99/25.91 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),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(5481,a,5456,a)]. 25.99/25.91 5482 -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(1876)]. 25.99/25.91 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),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(5482,a,5456,a)]. 26.32/26.30 5483 -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(1876)]. 26.32/26.30 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(5483,a,5456,a)]. 26.32/26.30 5484 -class_Rings_Olinordered__ring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D))) # label(fact_mult__less__cancel__left__pos) # label(axiom). [clausify(2327)]. 26.32/26.30 5485 -class_Rings_Olinordered__ring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D))) # label(fact_mult__less__cancel__left__pos) # label(axiom). [clausify(2327)]. 26.32/26.30 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(5485,a,5456,a)]. 26.32/26.30 5486 -class_Rings_Olinordered__ring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C))) # label(fact_mult__less__cancel__left__neg) # label(axiom). [clausify(2415)]. 26.32/26.30 5487 -class_Rings_Olinordered__ring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C))) # label(fact_mult__less__cancel__left__neg) # label(axiom). [clausify(2415)]. 26.32/26.30 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(5487,a,5456,a)]. 26.32/26.30 5488 -class_Rings_Olinordered__ring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),D)) # label(fact_mult__less__cancel__right__disj) # label(axiom). [clausify(3207)]. 26.32/26.30 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),C),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),C)). [resolve(5488,a,5456,a)]. 26.32/26.30 5489 -class_Rings_Olinordered__ring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),C)) # label(fact_mult__less__cancel__right__disj) # label(axiom). [clausify(3207)]. 26.32/26.30 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),C),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)). [resolve(5489,a,5456,a)]. 26.32/26.30 5490 -class_Rings_Olinordered__ring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),D)) # label(fact_mult__less__cancel__right__disj) # label(axiom). [clausify(3207)]. 26.32/26.30 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),C)). [resolve(5490,a,5456,a)]. 26.32/26.30 5491 -class_Rings_Olinordered__ring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),C)) # label(fact_mult__less__cancel__right__disj) # label(axiom). [clausify(3207)]. 26.32/26.30 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)). [resolve(5491,a,5456,a)]. 26.32/26.30 5492 -class_Rings_Olinordered__ring__strict(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),c_Groups_Ozero__class_Ozero(A))) # label(fact_mult__less__cancel__right__disj) # label(axiom). [clausify(3207)]. 26.32/26.30 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),C),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5492,a,5456,a)]. 26.32/26.30 5493 -class_Rings_Olinordered__ring__strict(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),C)) # label(fact_mult__less__cancel__right__disj) # label(axiom). [clausify(3207)]. 26.32/26.30 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)). [resolve(5493,a,5456,a)]. 26.93/26.89 5494 -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),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),D)) # label(fact_mult__le__cancel__left__pos) # label(axiom). [clausify(3554)]. 26.93/26.89 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),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),B),C)). [resolve(5494,a,5456,a)]. 26.93/26.89 5495 -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),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),D)) # label(fact_mult__le__cancel__left__pos) # label(axiom). [clausify(3554)]. 26.93/26.89 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),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),B),C)). [resolve(5495,a,5456,a)]. 26.93/26.89 5496 -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(3986)]. 26.93/26.89 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(5496,a,5456,a)]. 26.93/26.89 5497 -class_Rings_Olinordered__ring__strict(A) | c_Groups_Ozero__class_Ozero(A) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),C)) | c_Groups_Ozero__class_Ozero(A) = B # label(fact_sum__squares__eq__zero__iff) # label(axiom). [clausify(4509)]. 26.93/26.89 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),B)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A. [resolve(5497,a,5456,a)]. 26.93/26.89 5498 -class_Rings_Olinordered__ring__strict(A) | c_Groups_Ozero__class_Ozero(A) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),C)) | c_Groups_Ozero__class_Ozero(A) = C # label(fact_sum__squares__eq__zero__iff) # label(axiom). [clausify(4509)]. 26.93/26.89 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),B)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = B. [resolve(5498,a,5456,a)]. 26.93/26.89 5499 -class_Rings_Olinordered__ring__strict(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),C)) | c_Groups_Ozero__class_Ozero(A) != B | c_Groups_Ozero__class_Ozero(A) != C # label(fact_sum__squares__eq__zero__iff) # label(axiom). [clausify(4509)]. 28.50/28.41 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),B)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != B. [resolve(5499,a,5456,a)]. 28.50/28.41 5500 -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(4843)]. 28.50/28.41 5501 class_Groups_Omonoid__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Omonoid__add) # label(axiom). [assumption]. 28.50/28.41 5502 -class_Groups_Omonoid__add(A) | c_List_Ofoldr(A,A,c_Groups_Oplus__class_Oplus(A),B,c_Groups_Ozero__class_Ozero(A)) = hAPP(c_List_Ofoldl(A,A,c_Groups_Oplus__class_Oplus(A),c_Groups_Ozero__class_Ozero(A)),B) # label(fact_foldl__foldr1) # label(axiom). [clausify(68)]. 28.50/28.41 5503 -class_Groups_Omonoid__add(A) | -class_Groups_Oordered__ab__semigroup__add(A) | hBOOL(hAPP(hAPP(c_member(B),f136(C,D,E,B,A)),hAPP(c_List_Oset(B),E))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(B,A),D),E))),hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(B,A),C),E)))) # label(fact_listsum__mono) # label(axiom). [clausify(1173)]. 28.50/28.41 5504 -class_Groups_Omonoid__add(A) | -class_Groups_Oordered__ab__semigroup__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(B,f136(C,B,D,E,A))),hAPP(C,f136(C,B,D,E,A)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(E,A),B),D))),hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(E,A),C),D)))) # label(fact_listsum__mono) # label(axiom). [clausify(1173)]. 28.50/28.41 5505 -class_Groups_Omonoid__add(A) | hBOOL(hAPP(hAPP(c_member(B),f159(C,D,E,B,A)),hAPP(c_List_Oset(B),E))) | hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(B,A),C),hAPP(c_List_Ofilter(B,D),E))) = hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(B,A),C),E)) # label(fact_listsum__map__filter) # label(axiom). [clausify(1383)]. 28.50/28.41 5506 -class_Groups_Omonoid__add(A) | -hBOOL(hAPP(B,f159(C,B,D,E,A))) | hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(E,A),C),hAPP(c_List_Ofilter(E,B),D))) = hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(E,A),C),D)) # label(fact_listsum__map__filter) # label(axiom). [clausify(1383)]. 28.50/28.41 5507 -class_Groups_Omonoid__add(A) | c_Groups_Ozero__class_Ozero(A) != hAPP(B,f159(B,C,D,E,A)) | hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(E,A),B),hAPP(c_List_Ofilter(E,C),D))) = hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(E,A),B),D)) # label(fact_listsum__map__filter) # label(axiom). [clausify(1383)]. 28.50/28.41 5508 -class_Groups_Omonoid__add(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),c_Groups_Ozero__class_Ozero(A)) = B # label(fact_add__0__right) # label(axiom). [clausify(2004)]. 28.50/28.41 Derived: c_List_Ofoldr(tc_Int_Oint,tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint),A,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) = hAPP(c_List_Ofoldl(tc_Int_Oint,tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A). [resolve(5501,a,5502,a)]. 28.50/28.41 Derived: -class_Groups_Oordered__ab__semigroup__add(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_member(A),f136(B,C,D,A,tc_Int_Oint)),hAPP(c_List_Oset(A),D))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),C),D))),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),B),D)))). [resolve(5501,a,5503,a)]. 28.68/28.60 Derived: -class_Groups_Oordered__ab__semigroup__add(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(A,f136(B,A,C,D,tc_Int_Oint))),hAPP(B,f136(B,A,C,D,tc_Int_Oint)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Omap(D,tc_Int_Oint),A),C))),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Omap(D,tc_Int_Oint),B),C)))). [resolve(5501,a,5504,a)]. 28.68/28.60 Derived: hBOOL(hAPP(hAPP(c_member(A),f159(B,C,D,A,tc_Int_Oint)),hAPP(c_List_Oset(A),D))) | hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),B),hAPP(c_List_Ofilter(A,C),D))) = hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),B),D)). [resolve(5501,a,5505,a)]. 28.68/28.60 Derived: -hBOOL(hAPP(A,f159(B,A,C,D,tc_Int_Oint))) | hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Omap(D,tc_Int_Oint),B),hAPP(c_List_Ofilter(D,A),C))) = hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Omap(D,tc_Int_Oint),B),C)). [resolve(5501,a,5506,a)]. 28.68/28.60 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(A,f159(A,B,C,D,tc_Int_Oint)) | hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Omap(D,tc_Int_Oint),A),hAPP(c_List_Ofilter(D,B),C))) = hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Omap(D,tc_Int_Oint),A),C)). [resolve(5501,a,5507,a)]. 28.68/28.60 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint)) = A. [resolve(5501,a,5508,a)]. 28.68/28.60 5509 class_Groups_Omonoid__add(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Groups_Omonoid__add) # label(axiom). [assumption]. 28.68/28.60 Derived: c_List_Ofoldr(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A,c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)) = hAPP(c_List_Ofoldl(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),A). [resolve(5509,a,5502,a)]. 28.68/28.60 Derived: -class_Groups_Oordered__ab__semigroup__add(tc_Code__Numeral_Ocode__numeral) | hBOOL(hAPP(hAPP(c_member(A),f136(B,C,D,A,tc_Code__Numeral_Ocode__numeral)),hAPP(c_List_Oset(A),D))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Omap(A,tc_Code__Numeral_Ocode__numeral),C),D))),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Omap(A,tc_Code__Numeral_Ocode__numeral),B),D)))). [resolve(5509,a,5503,a)]. 28.68/28.60 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,f136(B,A,C,D,tc_Code__Numeral_Ocode__numeral))),hAPP(B,f136(B,A,C,D,tc_Code__Numeral_Ocode__numeral)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Omap(D,tc_Code__Numeral_Ocode__numeral),A),C))),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Omap(D,tc_Code__Numeral_Ocode__numeral),B),C)))). [resolve(5509,a,5504,a)]. 28.68/28.60 Derived: hBOOL(hAPP(hAPP(c_member(A),f159(B,C,D,A,tc_Code__Numeral_Ocode__numeral)),hAPP(c_List_Oset(A),D))) | hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Omap(A,tc_Code__Numeral_Ocode__numeral),B),hAPP(c_List_Ofilter(A,C),D))) = hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Omap(A,tc_Code__Numeral_Ocode__numeral),B),D)). [resolve(5509,a,5505,a)]. 28.68/28.60 Derived: -hBOOL(hAPP(A,f159(B,A,C,D,tc_Code__Numeral_Ocode__numeral))) | hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Omap(D,tc_Code__Numeral_Ocode__numeral),B),hAPP(c_List_Ofilter(D,A),C))) = hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Omap(D,tc_Code__Numeral_Ocode__numeral),B),C)). [resolve(5509,a,5506,a)]. 28.87/28.84 Derived: c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != hAPP(A,f159(A,B,C,D,tc_Code__Numeral_Ocode__numeral)) | hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Omap(D,tc_Code__Numeral_Ocode__numeral),A),hAPP(c_List_Ofilter(D,B),C))) = hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Omap(D,tc_Code__Numeral_Ocode__numeral),A),C)). [resolve(5509,a,5507,a)]. 28.87/28.84 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(5509,a,5508,a)]. 28.87/28.84 5510 -class_Groups_Omonoid__add(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(c_List_Omonoid__add__class_Olistsum(A),c_List_Olist_ONil(A)) # label(fact_listsum__simps_I1_J) # label(axiom). [clausify(2952)]. 28.87/28.84 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),c_List_Olist_ONil(tc_Int_Oint)). [resolve(5510,a,5501,a)]. 28.87/28.84 Derived: c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),c_List_Olist_ONil(tc_Code__Numeral_Ocode__numeral)). [resolve(5510,a,5509,a)]. 28.87/28.84 5511 -class_Groups_Omonoid__add(A) | hAPP(c_List_Ofoldl(A,A,c_Groups_Oplus__class_Oplus(A),B),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),c_List_Ofoldr(A,A,c_Groups_Oplus__class_Oplus(A),C,c_Groups_Ozero__class_Ozero(A))) # label(fact_foldl__foldr1__lemma) # label(axiom). [clausify(3040)]. 28.87/28.84 Derived: hAPP(c_List_Ofoldl(tc_Int_Oint,tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),c_List_Ofoldr(tc_Int_Oint,tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint),B,c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5511,a,5501,a)]. 28.87/28.84 Derived: hAPP(c_List_Ofoldl(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),c_List_Ofoldr(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),B,c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral))). [resolve(5511,a,5509,a)]. 28.87/28.84 5512 -class_Groups_Omonoid__add(A) | hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Oappend(A),B),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(c_List_Omonoid__add__class_Olistsum(A),B)),hAPP(c_List_Omonoid__add__class_Olistsum(A),C)) # label(fact_listsum__append) # label(axiom). [clausify(3251)]. 28.87/28.84 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Oappend(tc_Int_Oint),A),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),A)),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),B)). [resolve(5512,a,5501,a)]. 28.87/28.84 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Oappend(tc_Code__Numeral_Ocode__numeral),A),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),A)),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),B)). [resolve(5512,a,5509,a)]. 28.87/28.84 5513 -class_Groups_Omonoid__add(A) | hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Olist_OCons(A),B),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),hAPP(c_List_Omonoid__add__class_Olistsum(A),C)) # label(fact_listsum__simps_I2_J) # label(axiom). [clausify(3295)]. 28.87/28.84 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Olist_OCons(tc_Int_Oint),A),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),B)). [resolve(5513,a,5501,a)]. 29.02/28.93 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Olist_OCons(tc_Code__Numeral_Ocode__numeral),A),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),B)). [resolve(5513,a,5509,a)]. 29.02/28.93 5514 class_Groups_Omonoid__add(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Omonoid__add) # label(axiom). [assumption]. 29.02/28.93 Derived: c_List_Ofoldr(tc_Nat_Onat,tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = hAPP(c_List_Ofoldl(tc_Nat_Onat,tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A). [resolve(5514,a,5502,a)]. 29.02/28.93 Derived: -class_Groups_Oordered__ab__semigroup__add(tc_Nat_Onat) | hBOOL(hAPP(hAPP(c_member(A),f136(B,C,D,A,tc_Nat_Onat)),hAPP(c_List_Oset(A),D))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(A,tc_Nat_Onat),C),D))),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(A,tc_Nat_Onat),B),D)))). [resolve(5514,a,5503,a)]. 29.02/28.93 Derived: -class_Groups_Oordered__ab__semigroup__add(tc_Nat_Onat) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(A,f136(B,A,C,D,tc_Nat_Onat))),hAPP(B,f136(B,A,C,D,tc_Nat_Onat)))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(D,tc_Nat_Onat),A),C))),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(D,tc_Nat_Onat),B),C)))). [resolve(5514,a,5504,a)]. 29.02/28.93 Derived: hBOOL(hAPP(hAPP(c_member(A),f159(B,C,D,A,tc_Nat_Onat)),hAPP(c_List_Oset(A),D))) | hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(A,tc_Nat_Onat),B),hAPP(c_List_Ofilter(A,C),D))) = hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(A,tc_Nat_Onat),B),D)). [resolve(5514,a,5505,a)]. 29.02/28.93 Derived: -hBOOL(hAPP(A,f159(B,A,C,D,tc_Nat_Onat))) | hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(D,tc_Nat_Onat),B),hAPP(c_List_Ofilter(D,A),C))) = hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(D,tc_Nat_Onat),B),C)). [resolve(5514,a,5506,a)]. 29.02/28.93 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(A,f159(A,B,C,D,tc_Nat_Onat)) | hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(D,tc_Nat_Onat),A),hAPP(c_List_Ofilter(D,B),C))) = hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(D,tc_Nat_Onat),A),C)). [resolve(5514,a,5507,a)]. 29.02/28.93 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = A. [resolve(5514,a,5508,a)]. 29.02/28.93 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),c_List_Olist_ONil(tc_Nat_Onat)). [resolve(5514,a,5510,a)]. 29.02/28.93 Derived: hAPP(c_List_Ofoldl(tc_Nat_Onat,tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),c_List_Ofoldr(tc_Nat_Onat,tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B,c_Groups_Ozero__class_Ozero(tc_Nat_Onat))). [resolve(5514,a,5511,a)]. 29.02/28.93 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Oappend(tc_Nat_Onat),A),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),A)),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),B)). [resolve(5514,a,5512,a)]. 29.02/28.93 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Olist_OCons(tc_Nat_Onat),A),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),B)). [resolve(5514,a,5513,a)]. 29.02/28.93 5515 -class_Groups_Omonoid__add(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(c_List_Omonoid__add__class_Olistsum(A),hAPP(hAPP(c_List_Omap(B,A),hAPP(c_COMBK(A,B),c_Groups_Ozero__class_Ozero(A))),C)) # label(fact_listsum__0) # label(axiom). [clausify(3480)]. 30.85/30.79 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(c_List_Omonoid__add__class_Olistsum(tc_Int_Oint),hAPP(hAPP(c_List_Omap(A,tc_Int_Oint),hAPP(c_COMBK(tc_Int_Oint,A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))),B)). [resolve(5515,a,5501,a)]. 30.85/30.79 Derived: c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = hAPP(c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_List_Omap(A,tc_Code__Numeral_Ocode__numeral),hAPP(c_COMBK(tc_Code__Numeral_Ocode__numeral,A),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral))),B)). [resolve(5515,a,5509,a)]. 30.85/30.79 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(A,tc_Nat_Onat),hAPP(c_COMBK(tc_Nat_Onat,A),c_Groups_Ozero__class_Ozero(tc_Nat_Onat))),B)). [resolve(5515,a,5514,a)]. 30.85/30.79 5516 -class_Groups_Omonoid__add(A) | hAPP(c_List_Ofoldl(A,A,c_Groups_Oplus__class_Oplus(A),B),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),hAPP(c_List_Ofoldl(A,A,c_Groups_Oplus__class_Oplus(A),c_Groups_Ozero__class_Ozero(A)),C)) # label(fact_foldl__absorb0) # label(axiom). [clausify(3934)]. 30.85/30.79 Derived: hAPP(c_List_Ofoldl(tc_Int_Oint,tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),hAPP(c_List_Ofoldl(tc_Int_Oint,tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)). [resolve(5516,a,5501,a)]. 30.85/30.79 Derived: hAPP(c_List_Ofoldl(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),hAPP(c_List_Ofoldl(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),B)). [resolve(5516,a,5509,a)]. 30.85/30.79 Derived: hAPP(c_List_Ofoldl(tc_Nat_Onat,tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),hAPP(c_List_Ofoldl(tc_Nat_Onat,tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B)). [resolve(5516,a,5514,a)]. 30.85/30.79 5517 -class_Groups_Omonoid__add(A) | c_List_Omonoid__add__class_Olistsum(A) = c_List_Ofoldl(A,A,c_Groups_Oplus__class_Oplus(A),c_Groups_Ozero__class_Ozero(A)) # label(fact_listsum__foldl) # label(axiom). [clausify(4102)]. 30.85/30.79 Derived: c_List_Omonoid__add__class_Olistsum(tc_Int_Oint) = c_List_Ofoldl(tc_Int_Oint,tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)). [resolve(5517,a,5501,a)]. 30.85/30.79 Derived: c_List_Omonoid__add__class_Olistsum(tc_Code__Numeral_Ocode__numeral) = c_List_Ofoldl(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)). [resolve(5517,a,5509,a)]. 30.85/30.79 Derived: c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat) = 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)). [resolve(5517,a,5514,a)]. 30.85/30.79 5518 -class_Groups_Omonoid__add(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),c_Groups_Ozero__class_Ozero(A)),B) = B # label(fact_add__0__left) # label(axiom). [clausify(4525)]. 30.85/30.79 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A) = A. [resolve(5518,a,5501,a)]. 30.85/30.79 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(5518,a,5509,a)]. 30.85/30.79 5519 class_Groups_Omonoid__mult(tc_Int_Oint) # label(arity_Int__Oint__Groups_Omonoid__mult) # label(axiom). [assumption]. 30.85/30.79 5520 -class_Groups_Omonoid__mult(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),c_Groups_Oone__class_Oone(A)) = B # label(fact_mult__1__right) # label(axiom). [clausify(78)]. 31.09/30.99 5521 -class_Groups_Omonoid__mult(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)) # label(fact_power__commutes) # label(axiom). [clausify(326)]. 31.09/30.99 5522 -class_Groups_Omonoid__mult(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)),D) = hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),D)) # label(fact_power__mult) # label(axiom). [clausify(595)]. 31.09/30.99 5523 -class_Groups_Omonoid__mult(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),C)) # label(fact_power__even__eq) # label(axiom). [clausify(749)]. 31.09/30.99 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),B)). [resolve(5523,a,5519,a)]. 31.09/30.99 5524 class_Groups_Omonoid__mult(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Groups_Omonoid__mult) # label(axiom). [assumption]. 31.09/30.99 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)) = A. [resolve(5524,a,5520,a)]. 31.09/30.99 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),B)),A) = 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(5524,a,5521,a)]. 31.09/30.99 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),B)),C) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),C)). [resolve(5524,a,5522,a)]. 31.09/30.99 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),B)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),B)). [resolve(5524,a,5523,a)]. 31.09/30.99 5525 -class_Groups_Omonoid__mult(A) | c_Groups_Oone__class_Oone(A) = hAPP(hAPP(c_Power_Opower__class_Opower(A),c_Groups_Oone__class_Oone(A)),B) # label(fact_power__one) # label(axiom). [clausify(1157)]. 31.09/30.99 Derived: c_Groups_Oone__class_Oone(tc_Int_Oint) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),A). [resolve(5525,a,5519,a)]. 31.09/30.99 Derived: c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),A). [resolve(5525,a,5524,a)]. 31.09/30.99 5526 -class_Groups_Omonoid__mult(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(hAPP(c_Power_Opower__class_Opower(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) = 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))) # label(fact_power__odd__eq) # label(axiom). [clausify(1908)]. 31.38/31.31 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) = 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))). [resolve(5526,a,5519,a)]. 31.38/31.31 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),B)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),B))). [resolve(5526,a,5524,a)]. 31.38/31.31 5527 class_Groups_Omonoid__mult(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Omonoid__mult) # label(axiom). [assumption]. 31.38/31.31 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = A. [resolve(5527,a,5520,a)]. 31.38/31.31 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B)),A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B)). [resolve(5527,a,5521,a)]. 31.38/31.31 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B)),C) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),C)). [resolve(5527,a,5522,a)]. 31.38/31.31 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),B)). [resolve(5527,a,5523,a)]. 31.38/31.31 Derived: c_Groups_Oone__class_Oone(tc_Nat_Onat) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),A). [resolve(5527,a,5525,a)]. 31.38/31.31 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),B))). [resolve(5527,a,5526,a)]. 31.38/31.31 5528 -class_Groups_Omonoid__mult(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),c_Groups_Oone__class_Oone(A)),B) = B # label(fact_mult__1__left) # label(axiom). [clausify(3076)]. 31.38/31.31 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)),A) = A. [resolve(5528,a,5524,a)]. 31.38/31.31 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),c_Groups_Oone__class_Oone(tc_Nat_Onat)),A) = A. [resolve(5528,a,5527,a)]. 31.38/31.31 5529 -class_Groups_Omonoid__mult(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),D)) = hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),D)) # label(fact_power__add) # label(axiom). [clausify(3079)]. 31.38/31.31 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),C)) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),C)). [resolve(5529,a,5519,a)]. 31.69/31.62 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),C)) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),C)). [resolve(5529,a,5524,a)]. 31.69/31.62 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),C)) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),C)). [resolve(5529,a,5527,a)]. 31.69/31.62 5530 -class_Groups_Omonoid__mult(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),C)),B) = hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Nat_OSuc,C)) # label(fact_power__Suc2) # label(axiom). [clausify(3437)]. 31.69/31.62 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B)),A) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Nat_OSuc,B)). [resolve(5530,a,5519,a)]. 31.69/31.62 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),B)),A) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),hAPP(c_Nat_OSuc,B)). [resolve(5530,a,5524,a)]. 31.69/31.62 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),B)),A) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),hAPP(c_Nat_OSuc,B)). [resolve(5530,a,5527,a)]. 31.69/31.62 5531 -class_Groups_Omonoid__mult(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),B),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = B # label(fact_power__one__right) # label(axiom). [clausify(3675)]. 31.69/31.62 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = A. [resolve(5531,a,5519,a)]. 31.69/31.62 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = A. [resolve(5531,a,5524,a)]. 31.69/31.62 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = A. [resolve(5531,a,5527,a)]. 31.69/31.62 5532 -class_Groups_Omonoid__mult(A) | -class_Int_Onumber(A) | 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)) = 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)))) # label(fact_power2__eq__square__number__of) # label(axiom). [clausify(3851)]. 31.69/31.62 Derived: -class_Int_Onumber(tc_Int_Oint) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)) = 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)))). [resolve(5532,a,5519,a)]. 31.69/31.62 Derived: -class_Int_Onumber(tc_Code__Numeral_Ocode__numeral) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A)) = hAPP(hAPP(c_Power_Opower__class_Opower(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_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))). [resolve(5532,a,5524,a)]. 31.69/31.62 Derived: -class_Int_Onumber(tc_Nat_Onat) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))). [resolve(5532,a,5527,a)]. 33.56/33.54 5533 -class_Groups_Omonoid__mult(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),B) = hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) # label(fact_power2__eq__square) # label(axiom). [clausify(4177)]. 33.56/33.54 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),A) = 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(5533,a,5519,a)]. 33.56/33.54 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),A) = 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)))). [resolve(5533,a,5524,a)]. 33.56/33.54 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),A) = 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)))). [resolve(5533,a,5527,a)]. 33.56/33.54 5534 -class_Groups_Omonoid__mult(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),B)),B) = hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(c_Int_OBit1(c_Int_OPls)))) # label(fact_power3__eq__cube) # label(axiom). [clausify(4300)]. 33.56/33.54 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),A)),A) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(c_Int_OBit1(c_Int_OPls)))). [resolve(5534,a,5519,a)]. 33.56/33.54 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),A)),A) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Code__Numeral_Ocode__numeral),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(c_Int_OBit1(c_Int_OPls)))). [resolve(5534,a,5524,a)]. 33.56/33.54 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),A)),A) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Nat_Onat),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(c_Int_OBit1(c_Int_OPls)))). [resolve(5534,a,5527,a)]. 33.56/33.54 5535 class_Divides_Osemiring__div(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Divides_Osemiring__div) # label(axiom). [assumption]. 33.56/33.54 5536 -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(79)]. 33.56/33.54 5537 -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),hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),C),B)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),D),B))) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),C),D)) # label(fact_div__dvd__div) # label(axiom). [clausify(101)]. 33.56/33.54 5538 -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),hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),C),B)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),D),B))) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),C),D)) # label(fact_div__dvd__div) # label(axiom). [clausify(101)]. 33.56/33.54 5539 -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(116)]. 33.56/33.54 5540 -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(221)]. 33.56/33.54 5541 -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(268)]. 33.56/33.54 5542 -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(308)]. 33.56/33.54 5543 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Omod(A,B,C) != c_Divides_Odiv__class_Omod(A,D,C) | c_Divides_Odiv__class_Omod(A,E,C) != c_Divides_Odiv__class_Omod(A,F,C) | c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),E),C) = c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),F),C) # label(fact_mod__mult__cong) # label(axiom). [clausify(419)]. 33.56/33.54 5544 -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(467)]. 33.56/33.54 5545 -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(616)]. 33.56/33.54 5546 -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(702)]. 33.56/33.54 5547 -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(808)]. 33.56/33.54 5548 -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(846)]. 33.56/33.54 5549 -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(868)]. 33.56/33.54 5550 -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(893)]. 33.56/33.54 5551 -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(925)]. 33.56/33.54 5552 -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(1126)]. 33.56/33.54 5553 -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(1128)]. 33.56/33.54 5554 -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(1128)]. 33.56/33.54 5555 -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(1310)]. 33.56/33.54 5556 -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(1324)]. 33.56/33.54 5557 -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(1364)]. 33.56/33.54 5558 -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(1637)]. 33.56/33.54 5559 -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(1717)]. 33.56/33.54 5560 -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(1717)]. 33.56/33.54 5561 -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(1826)]. 33.56/33.54 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(5535,a,5536,a)]. 33.56/33.54 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),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))) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),B),C)). [resolve(5535,a,5537,a)]. 33.56/33.54 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),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))) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),B),C)). [resolve(5535,a,5538,a)]. 33.56/33.54 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(5535,a,5539,a)]. 33.56/33.54 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(5535,a,5540,a)]. 33.56/33.54 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(5535,a,5541,a)]. 33.65/33.56 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(5535,a,5542,a)]. 33.65/33.56 Derived: c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,A,B) != c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,C,B) | c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,D,B) != c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,E,B) | c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),D),B) = c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),C),E),B). [resolve(5535,a,5543,a)]. 33.65/33.56 Derived: c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,A,A) = c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral). [resolve(5535,a,5544,a)]. 33.65/33.56 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(5535,a,5545,a)]. 33.65/33.56 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(5535,a,5547,a)]. 33.65/33.56 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(5535,a,5548,a)]. 33.65/33.56 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(5535,a,5549,a)]. 33.65/33.56 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(5535,a,5550,a)]. 33.65/33.56 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(5535,a,5551,a)]. 33.65/33.56 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(5535,a,5552,a)]. 33.65/33.56 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(5535,a,5553,a)]. 33.65/33.56 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(5535,a,5555,a)]. 33.65/33.56 Derived: c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,A,c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)) = A. [resolve(5535,a,5556,a)]. 33.75/33.67 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(5535,a,5557,a)]. 33.75/33.67 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(5535,a,5558,a)]. 33.75/33.67 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(5535,a,5559,a)]. 33.75/33.67 5562 -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),c_Divides_Odiv__class_Omod(A,D,C))) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),D)) # label(fact_dvd__mod__iff) # label(axiom). [clausify(2061)]. 33.75/33.67 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_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,C,B))) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),A),C)). [resolve(5562,a,5535,a)]. 33.75/33.67 5563 -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),c_Divides_Odiv__class_Omod(A,D,C))) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),D)) # label(fact_dvd__mod__iff) # label(axiom). [clausify(2061)]. 33.75/33.67 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_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,C,B))) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),A),C)). [resolve(5563,a,5535,a)]. 33.75/33.67 5564 -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(2062)]. 33.75/33.67 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(5564,a,5535,a)]. 33.75/33.67 5565 -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(2173)]. 33.75/33.67 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(5565,a,5535,a)]. 33.75/33.67 5566 -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(2253)]. 33.75/33.67 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(5566,a,5535,a)]. 33.75/33.67 5567 -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(2297)]. 33.86/33.79 Derived: c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = A | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),B),A)),A) = B. [resolve(5567,a,5535,a)]. 33.86/33.79 5568 -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(2311)]. 33.86/33.79 Derived: c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = A | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),B)),A) = B. [resolve(5568,a,5535,a)]. 33.86/33.79 5569 -class_Divides_Osemiring__div(A) | c_Divides_Odiv__class_Omod(A,B,C) != c_Divides_Odiv__class_Omod(A,D,C) | c_Divides_Odiv__class_Omod(A,E,C) != c_Divides_Odiv__class_Omod(A,F,C) | c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),E),C) = c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),F),C) # label(fact_mod__add__cong) # label(axiom). [clausify(2597)]. 33.86/33.79 Derived: c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,A,B) != c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,C,B) | c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,D,B) != c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,E,B) | c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),D),B) = c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),C),E),B). [resolve(5569,a,5535,a)]. 33.86/33.79 5570 -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(2610)]. 33.86/33.79 Derived: c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,A,B)),c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,C,B)),B) = c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),C),B). [resolve(5570,a,5535,a)]. 33.86/33.79 5571 -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(2620)]. 33.86/33.79 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(5571,a,5535,a)]. 33.86/33.79 5572 -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(2645)]. 33.86/33.79 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(5572,a,5535,a)]. 33.86/33.79 5573 -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(2645)]. 33.86/33.79 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(5573,a,5535,a)]. 34.15/34.05 5574 -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(2720)]. 34.15/34.05 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(5574,a,5535,a)]. 34.15/34.05 5575 -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(2766)]. 34.15/34.05 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),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(5575,a,5535,a)]. 34.15/34.05 5576 -class_Divides_Osemiring__div(A) | c_Groups_Ozero__class_Ozero(A) = B | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),D),B)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C))),B) # label(fact_div__mult__self2) # label(axiom). [clausify(3087)]. 34.15/34.05 Derived: c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = A | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),B),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),C),A)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),B))),A). [resolve(5576,a,5535,a)]. 34.15/34.05 5577 -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(3118)]. 34.15/34.05 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(5577,a,5535,a)]. 34.15/34.05 5578 -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(3211)]. 34.15/34.05 5579 -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(3237)]. 34.15/34.05 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(5579,a,5535,a)]. 34.45/34.37 5580 -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(3284)]. 34.45/34.37 5581 -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_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)) = 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)) # label(fact_div__mult__div__if__dvd) # label(axiom). [clausify(3492)]. 34.45/34.37 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_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)) = 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)). [resolve(5581,a,5535,a)]. 34.45/34.37 5582 -class_Divides_Osemiring__div(A) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),C)) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),C),B)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C)),B) # label(fact_div__mult__swap) # label(axiom). [clausify(3567)]. 34.45/34.37 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),C),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),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),C),B)),A). [resolve(5582,a,5535,a)]. 34.45/34.37 5583 -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(3640)]. 34.45/34.37 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(5583,a,5535,a)]. 34.45/34.37 5584 -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(3692)]. 34.45/34.37 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(5584,a,5535,a)]. 34.45/34.37 5585 -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(3872)]. 34.45/34.37 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(5585,a,5535,a)]. 34.62/34.53 5586 -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(3873)]. 34.62/34.53 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(5586,a,5535,a)]. 34.62/34.53 5587 -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(3921)]. 34.62/34.53 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(5587,a,5535,a)]. 34.62/34.53 5588 -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)),D) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D)),B) # label(fact_dvd__div__mult) # label(axiom). [clausify(3925)]. 34.62/34.53 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)),C) = 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). [resolve(5588,a,5535,a)]. 34.62/34.53 5589 -class_Divides_Osemiring__div(A) | c_Groups_Ozero__class_Ozero(A) = B | c_Groups_Ozero__class_Ozero(A) = C | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),D)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),C),E)) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),E) != hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C) | hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),D),B) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),E),C) # label(fact_dvd__div__div__eq__mult) # label(axiom). [clausify(4018)]. 34.62/34.53 Derived: c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = A | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = B | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),A),C)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),B),D)) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),D) != hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),C),B) | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),C),A) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),D),B). [resolve(5589,a,5535,a)]. 34.62/34.53 5590 -class_Divides_Osemiring__div(A) | c_Groups_Ozero__class_Ozero(A) = B | c_Groups_Ozero__class_Ozero(A) = C | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),D)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),C),E)) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),E) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C) | hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),D),B) != hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),E),C) # label(fact_dvd__div__div__eq__mult) # label(axiom). [clausify(4018)]. 34.62/34.53 Derived: c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = A | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = B | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),A),C)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),B),D)) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),D) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),C),B) | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),C),A) != hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),D),B). [resolve(5590,a,5535,a)]. 34.62/34.53 5591 -class_Divides_Osemiring__div(A) | c_Groups_Ozero__class_Ozero(A) = 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)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),B) # label(fact_div__add__self1) # label(axiom). [clausify(4091)]. 34.66/34.59 Derived: c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = 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)) = 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). [resolve(5591,a,5535,a)]. 34.66/34.59 5592 -class_Divides_Osemiring__div(A) | c_Groups_Ozero__class_Ozero(A) = 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)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),B)),B) # label(fact_div__add__self2) # label(axiom). [clausify(4132)]. 34.66/34.59 Derived: c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = 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)) = 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). [resolve(5592,a,5535,a)]. 34.66/34.59 5593 class_Divides_Osemiring__div(tc_Int_Oint) # label(arity_Int__Oint__Divides_Osemiring__div) # label(axiom). [assumption]. 34.66/34.59 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(5593,a,5536,a)]. 34.66/34.59 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),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),B),A)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),C),A))) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),B),C)). [resolve(5593,a,5537,a)]. 34.66/34.59 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),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),B),A)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),C),A))) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),B),C)). [resolve(5593,a,5538,a)]. 34.66/34.59 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(5593,a,5539,a)]. 34.66/34.59 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(5593,a,5540,a)]. 34.66/34.59 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(5593,a,5541,a)]. 34.66/34.59 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(5593,a,5542,a)]. 34.66/34.59 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,A,B) != c_Divides_Odiv__class_Omod(tc_Int_Oint,C,B) | c_Divides_Odiv__class_Omod(tc_Int_Oint,D,B) != c_Divides_Odiv__class_Omod(tc_Int_Oint,E,B) | c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),D),B) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),E),B). [resolve(5593,a,5543,a)]. 34.66/34.60 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,A,A) = c_Groups_Ozero__class_Ozero(tc_Int_Oint). [resolve(5593,a,5544,a)]. 34.66/34.60 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(5593,a,5545,a)]. 34.66/34.60 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(5593,a,5547,a)]. 34.66/34.60 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(5593,a,5548,a)]. 34.66/34.60 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(5593,a,5549,a)]. 34.66/34.60 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(5593,a,5550,a)]. 34.66/34.60 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(5593,a,5551,a)]. 34.66/34.60 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(5593,a,5552,a)]. 34.66/34.60 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(5593,a,5553,a)]. 34.66/34.60 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(5593,a,5555,a)]. 34.66/34.60 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,A,c_Groups_Ozero__class_Ozero(tc_Int_Oint)) = A. [resolve(5593,a,5556,a)]. 34.66/34.60 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(5593,a,5557,a)]. 34.66/34.60 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(5593,a,5558,a)]. 34.66/34.60 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(5593,a,5559,a)]. 34.66/34.60 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_Divides_Odiv__class_Omod(tc_Int_Oint,C,B))) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),C)). [resolve(5593,a,5562,a)]. 34.66/34.60 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_Divides_Odiv__class_Omod(tc_Int_Oint,C,B))) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),C)). [resolve(5593,a,5563,a)]. 34.66/34.60 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(5593,a,5564,a)]. 34.66/34.60 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(5593,a,5565,a)]. 34.66/34.60 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(5593,a,5566,a)]. 34.66/34.60 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(5593,a,5567,a)]. 34.66/34.60 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(5593,a,5568,a)]. 34.66/34.60 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,A,B) != c_Divides_Odiv__class_Omod(tc_Int_Oint,C,B) | c_Divides_Odiv__class_Omod(tc_Int_Oint,D,B) != c_Divides_Odiv__class_Omod(tc_Int_Oint,E,B) | c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),D),B) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),C),E),B). [resolve(5593,a,5569,a)]. 34.66/34.60 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(5593,a,5570,a)]. 34.66/34.60 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(5593,a,5571,a)]. 34.66/34.60 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(5593,a,5572,a)]. 34.66/34.60 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(5593,a,5573,a)]. 34.66/34.60 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,A,B)),B). [resolve(5593,a,5574,a)]. 34.66/34.60 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),C),A)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B))),A). [resolve(5593,a,5576,a)]. 34.66/34.60 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(5593,a,5577,a)]. 34.66/34.60 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(5593,a,5579,a)]. 34.66/34.60 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_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)) = 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)). [resolve(5593,a,5581,a)]. 34.66/34.60 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),B)) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),B),A)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),B)),A). [resolve(5593,a,5582,a)]. 34.66/34.60 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(5593,a,5583,a)]. 34.66/34.60 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(5593,a,5584,a)]. 34.87/34.81 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(5593,a,5585,a)]. 34.87/34.81 Derived: hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),A),c_Groups_Oone__class_Oone(tc_Int_Oint)) = A. [resolve(5593,a,5586,a)]. 34.87/34.81 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(5593,a,5587,a)]. 34.87/34.81 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)),C) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),C)),A). [resolve(5593,a,5588,a)]. 34.87/34.81 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = B | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),C)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),B),D)) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),D) != hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),B) | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),C),A) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),D),B). [resolve(5593,a,5589,a)]. 34.87/34.81 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = B | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),C)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),B),D)) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),D) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),B) | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),C),A) != hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),D),B). [resolve(5593,a,5590,a)]. 34.87/34.81 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = 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)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B)),A). [resolve(5593,a,5591,a)]. 34.87/34.81 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = 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)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),A)),A). [resolve(5593,a,5592,a)]. 34.87/34.81 5594 -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(4225)]. 34.87/34.81 5595 -class_Divides_Osemiring__div(A) | c_Groups_Ozero__class_Ozero(A) = B | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),D),B)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),B))),B) # label(fact_div__mult__self1) # label(axiom). [clausify(4473)]. 34.87/34.81 Derived: c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = A | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),B),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),C),A)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),B),A))),A). [resolve(5595,a,5535,a)]. 34.87/34.81 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),C),A)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),A))),A). [resolve(5595,a,5593,a)]. 35.09/35.05 5596 -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(4649)]. 35.09/35.05 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(5596,a,5535,a)]. 35.09/35.05 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(5596,a,5593,a)]. 35.09/35.05 5597 -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(4652)]. 35.09/35.05 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(5597,a,5535,a)]. 35.09/35.05 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(5597,a,5593,a)]. 35.09/35.05 5598 -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(4713)]. 35.09/35.05 5599 -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(4780)]. 35.09/35.05 Derived: c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),B),A) = c_Divides_Odiv__class_Omod(tc_Code__Numeral_Ocode__numeral,B,A). [resolve(5599,a,5535,a)]. 35.09/35.05 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(5599,a,5593,a)]. 35.09/35.05 5600 -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(4823)]. 35.09/35.05 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(5600,a,5535,a)]. 35.09/35.05 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(5600,a,5593,a)]. 35.09/35.05 5601 class_Divides_Osemiring__div(tc_Nat_Onat) # label(arity_Nat__Onat__Divides_Osemiring__div) # label(axiom). [assumption]. 35.16/35.06 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(5601,a,5536,a)]. 35.16/35.06 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),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),B),A)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),C),A))) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),B),C)). [resolve(5601,a,5537,a)]. 35.16/35.06 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),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),B),A)),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),C),A))) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),B),C)). [resolve(5601,a,5538,a)]. 35.16/35.06 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(5601,a,5539,a)]. 35.16/35.06 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(5601,a,5540,a)]. 35.16/35.06 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(5601,a,5541,a)]. 35.16/35.06 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(5601,a,5542,a)]. 35.16/35.06 Derived: c_Divides_Odiv__class_Omod(tc_Nat_Onat,A,B) != c_Divides_Odiv__class_Omod(tc_Nat_Onat,C,B) | c_Divides_Odiv__class_Omod(tc_Nat_Onat,D,B) != c_Divides_Odiv__class_Omod(tc_Nat_Onat,E,B) | c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),D),B) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),E),B). [resolve(5601,a,5543,a)]. 35.16/35.06 Derived: c_Divides_Odiv__class_Omod(tc_Nat_Onat,A,A) = c_Groups_Ozero__class_Ozero(tc_Nat_Onat). [resolve(5601,a,5544,a)]. 35.16/35.06 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(5601,a,5545,a)]. 35.16/35.06 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(5601,a,5547,a)]. 35.16/35.06 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(5601,a,5548,a)]. 35.16/35.06 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(5601,a,5549,a)]. 35.16/35.06 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(5601,a,5550,a)]. 35.16/35.06 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(5601,a,5551,a)]. 35.16/35.06 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(5601,a,5552,a)]. 35.16/35.06 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(5601,a,5553,a)]. 35.16/35.06 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(5601,a,5555,a)]. 35.16/35.06 Derived: c_Divides_Odiv__class_Omod(tc_Nat_Onat,A,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)) = A. [resolve(5601,a,5556,a)]. 35.16/35.06 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(5601,a,5557,a)]. 35.16/35.06 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(5601,a,5558,a)]. 35.16/35.06 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(5601,a,5559,a)]. 35.16/35.06 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_Divides_Odiv__class_Omod(tc_Nat_Onat,C,B))) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),A),C)). [resolve(5601,a,5562,a)]. 35.16/35.06 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_Divides_Odiv__class_Omod(tc_Nat_Onat,C,B))) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),A),C)). [resolve(5601,a,5563,a)]. 35.16/35.06 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(5601,a,5564,a)]. 35.16/35.06 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(5601,a,5565,a)]. 35.16/35.06 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(5601,a,5566,a)]. 35.16/35.06 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(5601,a,5567,a)]. 35.16/35.06 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(5601,a,5568,a)]. 35.16/35.06 Derived: c_Divides_Odiv__class_Omod(tc_Nat_Onat,A,B) != c_Divides_Odiv__class_Omod(tc_Nat_Onat,C,B) | c_Divides_Odiv__class_Omod(tc_Nat_Onat,D,B) != c_Divides_Odiv__class_Omod(tc_Nat_Onat,E,B) | c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),D),B) = c_Divides_Odiv__class_Omod(tc_Nat_Onat,hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),E),B). [resolve(5601,a,5569,a)]. 35.16/35.06 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(5601,a,5570,a)]. 35.16/35.06 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(5601,a,5571,a)]. 35.16/35.06 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(5601,a,5572,a)]. 35.16/35.07 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),A),B)) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),A) = B | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),B),A) != C. [resolve(5601,a,5573,a)]. 35.16/35.07 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(5601,a,5574,a)]. 35.16/35.07 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(5601,a,5575,a)]. 35.16/35.07 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),C),A)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B))),A). [resolve(5601,a,5576,a)]. 35.16/35.07 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(5601,a,5577,a)]. 35.16/35.07 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(5601,a,5579,a)]. 35.16/35.07 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_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)) = 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)). [resolve(5601,a,5581,a)]. 35.16/35.07 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),A),B)) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),B),A)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),B)),A). [resolve(5601,a,5582,a)]. 35.16/35.07 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(5601,a,5583,a)]. 35.16/35.07 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(5601,a,5584,a)]. 35.16/35.07 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(5601,a,5585,a)]. 35.16/35.07 Derived: hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),A),c_Groups_Oone__class_Oone(tc_Nat_Onat)) = A. [resolve(5601,a,5586,a)]. 35.16/35.07 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(5601,a,5587,a)]. 35.16/35.07 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)),C) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),C)),A). [resolve(5601,a,5588,a)]. 38.86/38.75 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = B | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),A),C)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),B),D)) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),D) != hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),B) | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),C),A) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),D),B). [resolve(5601,a,5589,a)]. 38.86/38.75 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = B | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),A),C)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),B),D)) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),D) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),B) | hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),C),A) != hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),D),B). [resolve(5601,a,5590,a)]. 38.86/38.75 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = 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)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B)),A). [resolve(5601,a,5591,a)]. 38.86/38.75 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = 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)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),A)),A). [resolve(5601,a,5592,a)]. 38.86/38.75 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),C),A)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),A))),A). [resolve(5601,a,5595,a)]. 38.86/38.75 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(5601,a,5596,a)]. 38.86/38.75 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(5601,a,5597,a)]. 38.86/38.75 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(5601,a,5599,a)]. 38.86/38.75 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(5601,a,5600,a)]. 38.86/38.75 5602 -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(741)]. 38.86/38.75 5603 class_Groups_Oordered__comm__monoid__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Oordered__comm__monoid__add) # label(axiom). [assumption]. 38.86/38.75 5604 -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(1010)]. 38.94/38.87 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(5604,a,5603,a)]. 38.94/38.87 5605 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]. 38.94/38.87 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_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),B))). [resolve(5605,a,5602,a)]. 38.94/38.87 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(5605,a,5604,a)]. 38.94/38.87 5606 -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(1341)]. 38.94/38.87 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(5606,a,5603,a)]. 38.94/38.87 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(5606,a,5605,a)]. 38.94/38.87 5607 -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(1764)]. 38.94/38.87 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(5607,a,5603,a)]. 38.94/38.87 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(5607,a,5605,a)]. 39.23/39.16 5608 -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(1831)]. 39.23/39.16 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(5608,a,5603,a)]. 39.23/39.16 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(5608,a,5605,a)]. 39.23/39.16 5609 -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(2573)]. 39.23/39.16 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(5609,a,5603,a)]. 39.23/39.16 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(5609,a,5605,a)]. 39.23/39.16 5610 -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(2811)]. 39.23/39.16 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),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(5610,a,5603,a)]. 39.32/39.28 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),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(5610,a,5605,a)]. 39.32/39.28 5611 -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(2816)]. 39.32/39.28 5612 class_Groups_Oordered__comm__monoid__add(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Oordered__comm__monoid__add) # label(axiom). [assumption]. 39.32/39.28 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(5612,a,5609,a)]. 39.32/39.28 5613 -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(2992)]. 39.32/39.28 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(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(5613,a,5603,a)]. 39.32/39.28 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(5613,a,5605,a)]. 39.32/39.28 5614 -class_Groups_Oordered__comm__monoid__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),C)) | c_Groups_Ozero__class_Ozero(A) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C) | c_Groups_Ozero__class_Ozero(A) = B # label(fact_add__nonneg__eq__0__iff) # label(axiom). [clausify(3100)]. 39.32/39.28 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A. [resolve(5614,a,5603,a)]. 39.32/39.28 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),B)) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),B) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = A. [resolve(5614,a,5605,a)]. 39.32/39.28 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A. [resolve(5614,a,5612,a)]. 39.32/39.28 5615 -class_Groups_Oordered__comm__monoid__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),C)) | c_Groups_Ozero__class_Ozero(A) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C) | c_Groups_Ozero__class_Ozero(A) = C # label(fact_add__nonneg__eq__0__iff) # label(axiom). [clausify(3100)]. 39.32/39.28 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = B. [resolve(5615,a,5603,a)]. 39.32/39.28 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),B)) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),B) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = B. [resolve(5615,a,5605,a)]. 39.32/39.28 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = B. [resolve(5615,a,5612,a)]. 39.32/39.28 5616 -class_Groups_Oordered__comm__monoid__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),C)) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C) | c_Groups_Ozero__class_Ozero(A) != B | c_Groups_Ozero__class_Ozero(A) != C # label(fact_add__nonneg__eq__0__iff) # label(axiom). [clausify(3100)]. 39.32/39.28 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),B)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != B. [resolve(5616,a,5603,a)]. 39.32/39.28 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),B)) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),B) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != A | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != B. [resolve(5616,a,5605,a)]. 39.32/39.28 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != A | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != B. [resolve(5616,a,5612,a)]. 39.39/39.29 5617 -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),f349(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(3111)]. 39.39/39.29 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),f349(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(5617,a,5603,a)]. 39.39/39.29 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),f349(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(5617,a,5605,a)]. 39.39/39.29 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),f349(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(5617,a,5612,a)]. 39.39/39.29 5618 -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,f349(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(3111)]. 39.39/39.29 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,f349(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(5618,a,5603,a)]. 39.39/39.29 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,f349(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(5618,a,5605,a)]. 40.25/40.20 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,f349(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(5618,a,5612,a)]. 40.25/40.20 5619 -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(3141)]. 40.25/40.20 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(5619,a,5603,a)]. 40.25/40.20 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(5619,a,5605,a)]. 40.25/40.20 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(5619,a,5612,a)]. 40.25/40.20 5620 -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(4080)]. 40.25/40.20 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(5620,a,5603,a)]. 40.25/40.20 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(5620,a,5605,a)]. 40.25/40.20 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),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),A),C))). [resolve(5620,a,5612,a)]. 40.25/40.20 5621 -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(4778)]. 40.25/40.20 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(5621,a,5603,a)]. 41.64/41.55 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(5621,a,5605,a)]. 41.64/41.55 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(5621,a,5612,a)]. 41.64/41.55 5622 -class_Rings_Ocomm__ring(A) | -class_Rings_Odvd(A) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),C)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),E))) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),D),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),F),C))),E))) # label(fact_inf__period_I4_J) # label(axiom). [clausify(1580)]. 41.64/41.55 5623 class_Rings_Odvd(tc_Int_Oint) # label(arity_Int__Oint__Rings_Odvd) # label(axiom). [assumption]. 41.64/41.55 5624 class_Rings_Odvd(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Odvd) # label(axiom). [assumption]. 41.64/41.55 5625 class_Rings_Odvd(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Rings_Odvd) # label(axiom). [assumption]. 41.64/41.55 Derived: -class_Rings_Ocomm__ring(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),C),D))) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),E),B))),D))). [resolve(5622,b,5623,a)]. 41.64/41.55 Derived: -class_Rings_Ocomm__ring(tc_Nat_Onat) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),D))) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),E),B))),D))). [resolve(5622,b,5624,a)]. 41.64/41.55 Derived: -class_Rings_Ocomm__ring(tc_Code__Numeral_Ocode__numeral) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),A),B)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),C),D))) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Code__Numeral_Ocode__numeral),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),E),B))),D))). [resolve(5622,b,5625,a)]. 41.64/41.55 5626 -class_Rings_Ocomm__ring(A) | -class_Rings_Odvd(A) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),C)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),E))) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),D),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),F),C))),E))) # label(fact_inf__period_I4_J) # label(axiom). [clausify(1580)]. 41.64/41.55 Derived: -class_Rings_Ocomm__ring(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),B)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),C),D))) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),E),B))),D))). [resolve(5626,b,5623,a)]. 42.43/42.38 Derived: -class_Rings_Ocomm__ring(tc_Nat_Onat) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),A),B)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),D))) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),E),B))),D))). [resolve(5626,b,5624,a)]. 42.43/42.38 Derived: -class_Rings_Ocomm__ring(tc_Code__Numeral_Ocode__numeral) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),A),B)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),C),D))) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Code__Numeral_Ocode__numeral),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),E),B))),D))). [resolve(5626,b,5625,a)]. 42.43/42.38 5627 -class_Rings_Odvd(A) | -class_Rings_Ocomm__ring(A) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),C)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),E))) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),D),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),F),C))),E))) # label(fact_inf__period_I3_J) # label(axiom). [clausify(2221)]. 42.43/42.38 5628 -class_Rings_Odvd(A) | -class_Rings_Ocomm__ring(A) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),C)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),E))) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),D),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),F),C))),E))) # label(fact_inf__period_I3_J) # label(axiom). [clausify(2221)]. 42.43/42.38 5629 -class_Rings_Osemiring__0(A) | -class_Rings_Odvd(A) | -hBOOL(hAPP(B,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D))) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),C),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),f481(C,B,A)),c_Groups_Ozero__class_Ozero(A)))) # label(fact_unity__coeff__ex) # label(axiom). [clausify(4192)]. 42.43/42.38 Derived: -class_Rings_Osemiring__0(tc_Int_Oint) | -hBOOL(hAPP(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),C))) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),f481(B,A,tc_Int_Oint)),c_Groups_Ozero__class_Ozero(tc_Int_Oint)))). [resolve(5629,b,5623,a)]. 42.43/42.38 Derived: -class_Rings_Osemiring__0(tc_Nat_Onat) | -hBOOL(hAPP(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),C))) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),f481(B,A,tc_Nat_Onat)),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)))). [resolve(5629,b,5624,a)]. 42.43/42.38 Derived: -class_Rings_Osemiring__0(tc_Code__Numeral_Ocode__numeral) | -hBOOL(hAPP(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),B),C))) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),f481(B,A,tc_Code__Numeral_Ocode__numeral)),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)))). [resolve(5629,b,5625,a)]. 42.43/42.38 5630 -class_Rings_Osemiring__0(A) | -class_Rings_Odvd(A) | -hBOOL(hAPP(B,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D))) | hBOOL(hAPP(B,f481(C,B,A))) # label(fact_unity__coeff__ex) # label(axiom). [clausify(4192)]. 42.43/42.38 Derived: -class_Rings_Osemiring__0(tc_Int_Oint) | -hBOOL(hAPP(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),C))) | hBOOL(hAPP(A,f481(B,A,tc_Int_Oint))). [resolve(5630,b,5623,a)]. 42.73/42.61 Derived: -class_Rings_Osemiring__0(tc_Nat_Onat) | -hBOOL(hAPP(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),C))) | hBOOL(hAPP(A,f481(B,A,tc_Nat_Onat))). [resolve(5630,b,5624,a)]. 42.73/42.61 Derived: -class_Rings_Osemiring__0(tc_Code__Numeral_Ocode__numeral) | -hBOOL(hAPP(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),B),C))) | hBOOL(hAPP(A,f481(B,A,tc_Code__Numeral_Ocode__numeral))). [resolve(5630,b,5625,a)]. 42.73/42.61 5631 -class_Rings_Osemiring__0(A) | -class_Rings_Odvd(A) | hBOOL(hAPP(B,hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),f482(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(4192)]. 42.73/42.61 Derived: -class_Rings_Osemiring__0(tc_Int_Oint) | hBOOL(hAPP(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),f482(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(5631,b,5623,a)]. 42.73/42.61 Derived: -class_Rings_Osemiring__0(tc_Nat_Onat) | hBOOL(hAPP(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),f482(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(5631,b,5624,a)]. 42.73/42.61 Derived: -class_Rings_Osemiring__0(tc_Code__Numeral_Ocode__numeral) | hBOOL(hAPP(A,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),B),f482(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(5631,b,5625,a)]. 42.73/42.61 5632 -class_Rings_Odvd(A) | -class_Groups_Ocomm__monoid__mult(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(B,tc_HOL_Obool)),D),C)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),E),D)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(B,A),E),C))) # label(fact_setprod__dvd__setprod__subset) # label(axiom). [clausify(4535)]. 42.73/42.61 Derived: -class_Groups_Ocomm__monoid__mult(tc_Int_Oint) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),C),B)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),D),C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),D),B))). [resolve(5632,a,5623,a)]. 42.73/42.61 Derived: -class_Groups_Ocomm__monoid__mult(tc_Nat_Onat) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),C),B)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),D),C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),D),B))). [resolve(5632,a,5624,a)]. 42.73/42.61 Derived: -class_Groups_Ocomm__monoid__mult(tc_Code__Numeral_Ocode__numeral) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_fun(A,tc_HOL_Obool)),C),B)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),D),C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Code__Numeral_Ocode__numeral),D),B))). [resolve(5632,a,5625,a)]. 42.73/42.61 5633 -class_Rings_Odvd(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C) != D | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),D)) # label(fact_dvdI) # label(axiom). [clausify(4655)]. 44.70/44.61 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B) != C | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),C)). [resolve(5633,a,5623,a)]. 44.70/44.61 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),B) != C | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Nat_Onat),A),C)). [resolve(5633,a,5624,a)]. 44.70/44.61 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),B) != C | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Code__Numeral_Ocode__numeral),A),C)). [resolve(5633,a,5625,a)]. 44.70/44.61 5634 class_Int_Onumber__ring(tc_Int_Oint) # label(arity_Int__Oint__Int_Onumber__ring) # label(axiom). [assumption]. 44.70/44.61 5635 -class_Int_Onumber__ring(A) | hAPP(c_Int_Oring__1__class_Oof__int(A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin)) = hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OMin) # label(fact_of__int__m1) # label(axiom). [clausify(96)]. 44.70/44.61 5636 -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(102)]. 44.70/44.61 5637 -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(102)]. 44.70/44.61 5638 -class_Fields_Ofield__inverse__zero(A) | -class_Int_Onumber__ring(A) | hAPP(c_Groups_Ouminus__class_Ouminus(A),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),c_Groups_Oone__class_Oone(A)),B)) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OMin)),B) # label(fact_minus1__divide) # label(axiom). [clausify(104)]. 44.70/44.61 5639 -class_Int_Onumber__ring(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),hAPP(c_Int_Onumber__class_Onumber__of(A),C)),D)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),hAPP(c_Int_Onumber__class_Onumber__of(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),C))),D) # label(fact_add__number__of__diff1) # label(axiom). [clausify(105)]. 44.70/44.61 5640 -class_Int_Onumber__ring(A) | -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),c_Int_OBit1(c_Int_OPls))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),c_Groups_Oone__class_Oone(A))) # label(fact_le__special_I4_J) # label(axiom). [clausify(156)]. 44.70/44.61 5641 -class_Int_Onumber__ring(A) | -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),B),c_Int_OBit1(c_Int_OPls))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),c_Groups_Oone__class_Oone(A))) # label(fact_le__special_I4_J) # label(axiom). [clausify(156)]. 44.70/44.61 5642 -class_Rings_Olinordered__idom(A) | -class_Int_Onumber__ring(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(c_Int_Onumber__class_Onumber__of(A),C))) # label(fact_less__number__of) # label(axiom). [clausify(209)]. 44.70/44.61 5643 -class_Rings_Olinordered__idom(A) | -class_Int_Onumber__ring(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(c_Int_Onumber__class_Onumber__of(A),C))) # label(fact_less__number__of) # label(axiom). [clausify(209)]. 44.70/44.61 5644 -class_Int_Onumber__ring(A) | -class_Rings_Olinordered__idom(A) | c_Groups_Oone__class_Oone(A) = hAPP(c_Groups_Oabs__class_Oabs(A),hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OMin)) # label(fact_abs__minus__one) # label(axiom). [clausify(211)]. 44.70/44.61 5645 -class_Int_Onumber__ring(A) | hAPP(c_Groups_Ouminus__class_Ouminus(A),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OMin)),B) # label(fact_mult__minus1) # label(axiom). [clausify(215)]. 44.70/44.61 5646 -class_Int_Onumber__ring(A) | hAPP(c_Int_Onumber__class_Onumber__of(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),C))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(c_Int_Onumber__class_Onumber__of(A),C)) # label(fact_diff__number__of__eq) # label(axiom). [clausify(255)]. 44.70/44.61 5647 -class_Int_Onumber__ring(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),C))),D) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),C)),D)) # label(fact_mult__number__of__left) # label(axiom). [clausify(294)]. 44.70/44.61 5648 -class_Int_Onumber__ring(A) | -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),c_Groups_Ozero__class_Ozero(A))) | hAPP(c_Groups_Oabs__class_Oabs(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)) = hAPP(c_Groups_Ouminus__class_Ouminus(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)) # label(fact_abs__number__of) # label(axiom). [clausify(384)]. 44.70/44.61 5649 -class_Int_Onumber__ring(A) | -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),c_Groups_Ozero__class_Ozero(A))) | hAPP(c_Groups_Oabs__class_Oabs(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)) = hAPP(c_Int_Onumber__class_Onumber__of(A),B) # label(fact_abs__number__of) # label(axiom). [clausify(384)]. 44.70/44.61 5650 -class_Int_Onumber__ring(A) | c_Groups_Oone__class_Oone(A) = hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OBit1(c_Int_OPls)) # label(fact_numeral__1__eq__1) # label(axiom). [clausify(478)]. 44.70/44.61 5651 -class_Int_Onumber__ring(A) | hAPP(c_Int_Onumber__class_Onumber__of(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(c_Int_Onumber__class_Onumber__of(A),C)) # label(fact_add__number__of__eq) # label(axiom). [clausify(501)]. 44.70/44.61 5652 -class_Int_Onumber__ring(A) | hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_Osucc(B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),c_Groups_Oone__class_Oone(A)),hAPP(c_Int_Onumber__class_Onumber__of(A),B)) # label(fact_number__of__succ) # label(axiom). [clausify(560)]. 44.70/44.61 5653 -class_Int_Onumber__ring(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),B) # label(fact_mult__2) # label(axiom). [clausify(658)]. 44.70/44.61 5654 -class_Int_Onumber__ring(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))) | c_Groups_Oone__class_Oone(A) = hAPP(hAPP(c_Power_Opower__class_Opower(A),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(B))) # label(fact_power__number__of__odd) # label(axiom). [clausify(801)]. 44.70/44.61 5655 -class_Int_Onumber__ring(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))) | hAPP(hAPP(c_HOL_OLet(A,A),hAPP(hAPP(c_Power_Opower__class_Opower(A),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B))),hAPP(hAPP(c_COMBS(A,A,A),hAPP(hAPP(c_COMBB(A,tc_fun(A,A),A),c_Groups_Otimes__class_Otimes(A)),hAPP(c_Groups_Otimes__class_Otimes(A),C))),c_COMBI(A))) = hAPP(hAPP(c_Power_Opower__class_Opower(A),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(B))) # label(fact_power__number__of__odd) # label(axiom). [clausify(801)]. 44.70/44.61 5656 -class_Int_Onumber__ring(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),B) # label(fact_mult__2__right) # label(axiom). [clausify(908)]. 44.70/44.61 5657 -class_Int_Onumber__ring(A) | hAPP(c_Int_Onumber__class_Onumber__of(A),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B)) = hAPP(c_Groups_Ouminus__class_Ouminus(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)) # label(fact_number__of__minus) # label(axiom). [clausify(954)]. 44.70/44.61 5658 -class_Int_Onumber__ring(A) | hAPP(c_Int_Onumber__class_Onumber__of(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),c_Groups_Oone__class_Oone(A)),hAPP(c_Int_Onumber__class_Onumber__of(A),B)) # label(fact_add__special_I2_J) # label(axiom). [clausify(1084)]. 44.70/44.61 5659 -class_Int_Onumber__ring(A) | hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OBit1(B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),c_Groups_Oone__class_Oone(A)),hAPP(c_Int_Onumber__class_Onumber__of(A),B))),hAPP(c_Int_Onumber__class_Onumber__of(A),B)) # label(fact_number__of__Bit1) # label(axiom). [clausify(1093)]. 44.70/44.61 5660 -class_Int_Oring__char__0(A) | -class_Int_Onumber__ring(A) | -c_Int_Oiszero(A,hAPP(c_Int_Onumber__class_Onumber__of(A),B)) | c_Int_Oiszero(A,hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OBit0(B))) # label(fact_iszero__number__of__Bit0) # label(axiom). [clausify(1114)]. 44.70/44.61 5661 -class_Int_Oring__char__0(A) | -class_Int_Onumber__ring(A) | c_Int_Oiszero(A,hAPP(c_Int_Onumber__class_Onumber__of(A),B)) | -c_Int_Oiszero(A,hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OBit0(B))) # label(fact_iszero__number__of__Bit0) # label(axiom). [clausify(1114)]. 44.70/44.61 5662 -class_Int_Onumber__ring(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B))),C) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B))),C) # label(fact_minus__number__of__mult) # label(axiom). [clausify(1120)]. 44.70/44.61 5663 -class_Int_Onumber__ring(A) | -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),c_Groups_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(1177)]. 44.70/44.61 5664 -class_Int_Onumber__ring(A) | -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),c_Groups_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(1177)]. 44.70/44.61 5665 -class_Int_Onumber__ring(A) | hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_Opred(B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),c_Groups_Oone__class_Oone(A))),hAPP(c_Int_Onumber__class_Onumber__of(A),B)) # label(fact_number__of__pred) # label(axiom). [clausify(1265)]. 44.70/44.61 5666 -class_Int_Onumber__ring(A) | hAPP(c_Int_Onumber__class_Onumber__of(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B),C)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(c_Int_Onumber__class_Onumber__of(A),C)) # label(fact_number__of__diff) # label(axiom). [clausify(1302)]. 44.70/44.61 5667 -class_Int_Onumber__ring(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OPls)) = B # label(fact_add__numeral__0__right) # label(axiom). [clausify(1484)]. 44.70/44.61 5668 -class_Int_Onumber__ring(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))) | c_Groups_Ozero__class_Ozero(A) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)) # label(fact_of__nat__number__of__lemma) # label(axiom). [clausify(1511)]. 44.70/44.61 5669 -class_Int_Onumber__ring(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))) | hAPP(c_Int_Onumber__class_Onumber__of(A),B) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)) # label(fact_of__nat__number__of__lemma) # label(axiom). [clausify(1511)]. 44.70/44.61 5670 -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(1552)]. 44.70/44.61 5671 -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(1552)]. 44.70/44.61 5672 -class_Int_Onumber__ring(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OPls)),B) = B # label(fact_add__numeral__0) # label(axiom). [clausify(1593)]. 44.70/44.61 5673 -class_Int_Onumber__ring(A) | -c_Int_Oiszero(A,hAPP(c_Int_Onumber__class_Onumber__of(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),C)))) | hAPP(c_Int_Onumber__class_Onumber__of(A),C) = hAPP(c_Int_Onumber__class_Onumber__of(A),B) # label(fact_eq__number__of__eq) # label(axiom). [clausify(1694)]. 44.70/44.61 5674 -class_Int_Onumber__ring(A) | c_Int_Oiszero(A,hAPP(c_Int_Onumber__class_Onumber__of(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),C)))) | hAPP(c_Int_Onumber__class_Onumber__of(A),C) != hAPP(c_Int_Onumber__class_Onumber__of(A),B) # label(fact_eq__number__of__eq) # label(axiom). [clausify(1694)]. 44.70/44.61 5675 -class_Int_Onumber__ring(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(c_Int_Onumber__class_Onumber__of(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),C))),D) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(c_Int_Onumber__class_Onumber__of(A),C)),D)) # label(fact_add__number__of__left) # label(axiom). [clausify(1761)]. 44.70/44.61 5676 -class_Int_Onumber__ring(A) | hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OBit0(B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Int_Onumber__class_Onumber__of(A),B))),hAPP(c_Int_Onumber__class_Onumber__of(A),B)) # label(fact_number__of__Bit0) # label(axiom). [clausify(1779)]. 44.70/44.61 5677 -class_Fields_Olinordered__field__inverse__zero(A) | -class_Int_Onumber__ring(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) # label(fact_half__gt__zero__iff) # label(axiom). [clausify(1843)]. 44.70/44.61 5678 -class_Fields_Olinordered__field__inverse__zero(A) | -class_Int_Onumber__ring(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) # label(fact_half__gt__zero__iff) # label(axiom). [clausify(1843)]. 44.70/44.61 5679 -class_Int_Onumber__ring(A) | -class_Int_Oring__char__0(A) | -c_Int_Oiszero(A,hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OBit1(B))) # label(fact_iszero__number__of__Bit1) # label(axiom). [clausify(2014)]. 44.70/44.61 5680 -class_Rings_Olinordered__idom(A) | -class_Int_Onumber__ring(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Int_Onumber__class_Onumber__of(A),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OPls),B)) # label(fact_le__special_I1_J) # label(axiom). [clausify(2025)]. 44.70/44.61 5681 -class_Rings_Olinordered__idom(A) | -class_Int_Onumber__ring(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Int_Onumber__class_Onumber__of(A),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OPls),B)) # label(fact_le__special_I1_J) # label(axiom). [clausify(2025)]. 44.70/44.61 5682 -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(2027)]. 44.70/44.61 5683 -class_Int_Onumber__ring(A) | hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),c_Groups_Oone__class_Oone(A)),c_Groups_Oone__class_Oone(A)) # label(fact_one__add__one__is__two) # label(axiom). [clausify(2100)]. 44.70/44.61 5684 -class_Int_Onumber__ring(A) | hAPP(c_Int_Onumber__class_Onumber__of(A),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B)) = hAPP(c_Groups_Ouminus__class_Ouminus(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)) # label(fact_arith__simps_I30_J) # label(axiom). [clausify(2116)]. 44.70/44.61 5685 -class_Int_Onumber__ring(A) | hAPP(hAPP(c_Power_Opower__class_Opower(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(A),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),B)),C)) # label(fact_power2__diff) # label(axiom). [clausify(2119)]. 44.70/44.61 5686 -class_Int_Onumber__ring(A) | -c_Int_Oiszero(A,hAPP(c_Int_Onumber__class_Onumber__of(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Int_OBit1(c_Int_OPls))))) | c_Groups_Oone__class_Oone(A) = hAPP(c_Int_Onumber__class_Onumber__of(A),B) # label(fact_eq__special_I4_J) # label(axiom). [clausify(2179)]. 44.70/44.61 5687 -class_Int_Onumber__ring(A) | c_Int_Oiszero(A,hAPP(c_Int_Onumber__class_Onumber__of(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Int_OBit1(c_Int_OPls))))) | c_Groups_Oone__class_Oone(A) != hAPP(c_Int_Onumber__class_Onumber__of(A),B) # label(fact_eq__special_I4_J) # label(axiom). [clausify(2179)]. 44.70/44.61 5688 -class_Int_Onumber__ring(A) | hAPP(c_Int_Onumber__class_Onumber__of(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(c_Int_Onumber__class_Onumber__of(A),C)) # label(fact_arith__simps_I32_J) # label(axiom). [clausify(2260)]. 44.70/44.61 5689 -class_Int_Onumber__ring(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OPls),B)) | hBOOL(hAPP(hAPP(c_member(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),c_Nat_Osemiring__1__class_ONats(A))) # label(fact_Nats__number__of) # label(axiom). [clausify(2451)]. 44.70/44.61 5690 -class_Int_Onumber__ring(A) | -class_Fields_Ofield__inverse__zero(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OPls)) # label(fact_divide__Numeral0) # label(axiom). [clausify(2570)]. 44.70/44.61 5691 -class_Int_Onumber__ring(A) | -class_Int_Oring__char__0(A) | hAPP(c_Int_Onumber__class_Onumber__of(A),B) != hAPP(c_Int_Onumber__class_Onumber__of(A),C) | B = C # label(fact_eq__number__of) # label(axiom). [clausify(2583)]. 44.70/44.61 5692 -class_Int_Onumber__ring(A) | -class_Int_Oring__char__0(A) | hAPP(c_Int_Onumber__class_Onumber__of(A),B) = hAPP(c_Int_Onumber__class_Onumber__of(A),C) | B != C # label(fact_eq__number__of) # label(axiom). [clausify(2583)]. 44.70/44.61 5693 -class_Int_Onumber__ring(A) | hAPP(hAPP(c_HOL_OLet(A,A),hAPP(hAPP(c_Power_Opower__class_Opower(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),C))),hAPP(hAPP(c_COMBS(A,A,A),c_Groups_Otimes__class_Otimes(A)),c_COMBI(A))) = hAPP(hAPP(c_Power_Opower__class_Opower(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(C))) # label(fact_power__number__of__even__number__of) # label(axiom). [clausify(2594)]. 44.70/44.61 5694 -class_Int_Onumber__ring(A) | hAPP(hAPP(c_HOL_OLet(A,A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),C))),hAPP(hAPP(c_COMBS(A,A,A),c_Groups_Otimes__class_Otimes(A)),c_COMBI(A))) = hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(C))) # label(fact_power__number__of__even) # label(axiom). [clausify(2622)]. 44.70/44.61 5695 -class_Int_Onumber__ring(A) | -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),hAPP(c_Int_Onumber__class_Onumber__of(A),B))) # label(fact_less__special_I2_J) # label(axiom). [clausify(2760)]. 44.70/44.61 5696 -class_Int_Onumber__ring(A) | -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Oone__class_Oone(A)),hAPP(c_Int_Onumber__class_Onumber__of(A),B))) # label(fact_less__special_I2_J) # label(axiom). [clausify(2760)]. 44.70/44.61 5697 -class_Int_Onumber__ring(A) | -class_Rings_Olinordered__idom(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),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(2913)]. 44.70/44.61 5698 -class_Int_Onumber__ring(A) | -class_Rings_Olinordered__idom(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),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(2913)]. 44.70/44.61 5699 -class_Int_Onumber__ring(A) | -c_Int_Oiszero(A,hAPP(c_Int_Onumber__class_Onumber__of(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B)))) | c_Groups_Oone__class_Oone(A) = hAPP(c_Int_Onumber__class_Onumber__of(A),B) # label(fact_eq__special_I2_J) # label(axiom). [clausify(2991)]. 44.70/44.61 5700 -class_Int_Onumber__ring(A) | c_Int_Oiszero(A,hAPP(c_Int_Onumber__class_Onumber__of(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B)))) | c_Groups_Oone__class_Oone(A) != hAPP(c_Int_Onumber__class_Onumber__of(A),B) # label(fact_eq__special_I2_J) # label(axiom). [clausify(2991)]. 44.70/44.61 5701 -class_Int_Onumber__ring(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(c_Int_Onumber__class_Onumber__of(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),C)))),D) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),D),hAPP(c_Int_Onumber__class_Onumber__of(A),C))) # label(fact_add__number__of__diff2) # label(axiom). [clausify(3004)]. 44.70/44.61 5702 -class_Int_Onumber__ring(A) | hAPP(c_Int_Onumber__class_Onumber__of(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),c_Int_OBit1(c_Int_OPls))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),c_Groups_Oone__class_Oone(A)) # label(fact_add__special_I3_J) # label(axiom). [clausify(3041)]. 44.70/44.62 5703 -class_Int_Onumber__ring(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OPls) # label(fact_number__of__Pls) # label(axiom). [clausify(3088)]. 44.70/44.62 Derived: hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin)) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin). [resolve(5634,a,5635,a)]. 44.70/44.62 Derived: -class_Rings_Olinordered__idom(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OPls),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A))). [resolve(5634,a,5636,a)]. 44.70/44.62 Derived: -class_Rings_Olinordered__idom(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OPls),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A))). [resolve(5634,a,5637,a)]. 44.70/44.62 Derived: -class_Fields_Ofield__inverse__zero(tc_Int_Oint) | hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),A)) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin)),A). [resolve(5634,a,5638,b)]. 44.70/44.62 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)),C)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B))),C). [resolve(5634,a,5639,a)]. 44.70/44.62 Derived: -class_Rings_Olinordered__idom(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),c_Int_OBit1(c_Int_OPls))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),c_Groups_Oone__class_Oone(tc_Int_Oint))). [resolve(5634,a,5640,a)]. 44.70/44.62 Derived: -class_Rings_Olinordered__idom(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),c_Int_OBit1(c_Int_OPls))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),c_Groups_Oone__class_Oone(tc_Int_Oint))). [resolve(5634,a,5641,a)]. 44.70/44.62 Derived: -class_Rings_Olinordered__idom(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))). [resolve(5634,a,5642,b)]. 44.70/44.62 Derived: -class_Rings_Olinordered__idom(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))). [resolve(5634,a,5643,b)]. 44.70/44.62 Derived: -class_Rings_Olinordered__idom(tc_Int_Oint) | c_Groups_Oone__class_Oone(tc_Int_Oint) = hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin)). [resolve(5634,a,5644,a)]. 44.70/44.62 Derived: hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin)),A). [resolve(5634,a,5645,a)]. 44.70/44.62 Derived: hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)). [resolve(5634,a,5646,a)]. 44.70/44.62 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B))),C) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)),C)). [resolve(5634,a,5647,a)]. 44.70/44.63 Derived: -class_Rings_Olinordered__idom(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)). [resolve(5634,a,5648,a)]. 44.70/44.63 Derived: -class_Rings_Olinordered__idom(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A). [resolve(5634,a,5649,a)]. 44.70/44.63 Derived: c_Groups_Oone__class_Oone(tc_Int_Oint) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)). [resolve(5634,a,5650,a)]. 44.70/44.63 Derived: hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)). [resolve(5634,a,5651,a)]. 44.70/44.63 Derived: hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_Osucc(A)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)). [resolve(5634,a,5652,a)]. 44.70/44.63 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),A) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),A). [resolve(5634,a,5653,a)]. 44.70/44.63 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A))) | c_Groups_Oone__class_Oone(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_OBit1(A))). [resolve(5634,a,5654,a)]. 44.70/44.63 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A))) | hAPP(hAPP(c_HOL_OLet(tc_Int_Oint,tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A))),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),B))),c_COMBI(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_OBit1(A))). [resolve(5634,a,5655,a)]. 44.70/44.63 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),A). [resolve(5634,a,5656,a)]. 44.70/44.63 Derived: hAPP(c_Int_Onumber__class_Onumber__of(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_Onumber__class_Onumber__of(tc_Int_Oint),A)). [resolve(5634,a,5657,a)]. 44.70/44.63 Derived: hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)),A)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)). [resolve(5634,a,5658,a)]. 44.70/44.63 Derived: hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit1(A)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A))),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)). [resolve(5634,a,5659,a)]. 44.70/44.64 Derived: -class_Int_Oring__char__0(tc_Int_Oint) | -c_Int_Oiszero(tc_Int_Oint,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)) | c_Int_Oiszero(tc_Int_Oint,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(A))). [resolve(5634,a,5660,b)]. 44.70/44.64 Derived: -class_Int_Oring__char__0(tc_Int_Oint) | c_Int_Oiszero(tc_Int_Oint,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)) | -c_Int_Oiszero(tc_Int_Oint,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(A))). [resolve(5634,a,5661,b)]. 44.70/44.64 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A))),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A))),B). [resolve(5634,a,5662,a)]. 44.70/44.64 Derived: -class_Rings_Olinordered__idom(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),c_Groups_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(5634,a,5663,a)]. 44.70/44.64 Derived: -class_Rings_Olinordered__idom(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),c_Groups_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(5634,a,5664,a)]. 44.70/44.64 Derived: hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_Opred(A)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint))),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)). [resolve(5634,a,5665,a)]. 44.70/44.64 Derived: hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)). [resolve(5634,a,5666,a)]. 44.70/44.64 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OPls)) = A. [resolve(5634,a,5667,a)]. 44.70/44.64 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A))) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)). [resolve(5634,a,5668,a)]. 44.70/44.64 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A))) | hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)). [resolve(5634,a,5669,a)]. 44.70/44.64 Derived: -class_Rings_Olinordered__idom(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(5634,a,5670,b)]. 44.70/44.64 Derived: -class_Rings_Olinordered__idom(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(5634,a,5671,b)]. 44.70/44.64 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OPls)),A) = A. [resolve(5634,a,5672,a)]. 44.70/44.65 Derived: -c_Int_Oiszero(tc_Int_Oint,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B)))) | hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A). [resolve(5634,a,5673,a)]. 44.70/44.65 Derived: c_Int_Oiszero(tc_Int_Oint,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B)))) | hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B) != hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A). [resolve(5634,a,5674,a)]. 44.70/44.65 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B))),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)),C)). [resolve(5634,a,5675,a)]. 44.70/44.65 Derived: hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(A)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A))),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)). [resolve(5634,a,5676,a)]. 44.70/44.65 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(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)). [resolve(5634,a,5677,b)]. 44.70/44.65 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(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)). [resolve(5634,a,5678,b)]. 44.70/44.65 Derived: -class_Int_Oring__char__0(tc_Int_Oint) | -c_Int_Oiszero(tc_Int_Oint,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit1(A))). [resolve(5634,a,5679,a)]. 44.70/44.65 Derived: -class_Rings_Olinordered__idom(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OPls),A)). [resolve(5634,a,5680,b)]. 44.70/44.65 Derived: -class_Rings_Olinordered__idom(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OPls),A)). [resolve(5634,a,5681,b)]. 44.70/44.65 Derived: -class_Rings_Olinordered__idom(tc_Int_Oint) | c_Groups_Oone__class_Oone(tc_Int_Oint) = hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin)),A)). [resolve(5634,a,5682,a)]. 44.70/44.65 Derived: hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),c_Groups_Oone__class_Oone(tc_Int_Oint)). [resolve(5634,a,5683,a)]. 44.70/44.65 Derived: hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),A)),B)). [resolve(5634,a,5685,a)]. 44.78/44.66 Derived: -c_Int_Oiszero(tc_Int_Oint,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Int_OBit1(c_Int_OPls))))) | c_Groups_Oone__class_Oone(tc_Int_Oint) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A). [resolve(5634,a,5686,a)]. 44.78/44.66 Derived: c_Int_Oiszero(tc_Int_Oint,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Int_OBit1(c_Int_OPls))))) | c_Groups_Oone__class_Oone(tc_Int_Oint) != hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A). [resolve(5634,a,5687,a)]. 44.78/44.66 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OPls),A)) | hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),c_Nat_Osemiring__1__class_ONats(tc_Int_Oint))). [resolve(5634,a,5689,a)]. 44.78/44.66 Derived: -class_Fields_Ofield__inverse__zero(tc_Int_Oint) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OPls)). [resolve(5634,a,5690,a)]. 44.78/44.66 Derived: -class_Int_Oring__char__0(tc_Int_Oint) | hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A) != hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B) | A = B. [resolve(5634,a,5691,a)]. 44.78/44.66 Derived: -class_Int_Oring__char__0(tc_Int_Oint) | hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B) | A != B. [resolve(5634,a,5692,a)]. 44.78/44.66 Derived: -class_Rings_Olinordered__idom(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A))). [resolve(5634,a,5695,a)]. 44.78/44.66 Derived: -class_Rings_Olinordered__idom(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A))). [resolve(5634,a,5696,a)]. 44.78/44.66 Derived: -c_Int_Oiszero(tc_Int_Oint,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)))) | c_Groups_Oone__class_Oone(tc_Int_Oint) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A). [resolve(5634,a,5699,a)]. 44.78/44.66 Derived: c_Int_Oiszero(tc_Int_Oint,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)))) | c_Groups_Oone__class_Oone(tc_Int_Oint) != hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A). [resolve(5634,a,5700,a)]. 44.78/44.66 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B)))),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))). [resolve(5634,a,5701,a)]. 44.78/44.66 Derived: hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),c_Int_OBit1(c_Int_OPls))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),c_Groups_Oone__class_Oone(tc_Int_Oint)). [resolve(5634,a,5702,a)]. 45.06/44.96 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OPls). [resolve(5634,a,5703,a)]. 45.06/44.96 5704 -class_Int_Onumber__ring(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))) | hAPP(hAPP(c_HOL_OLet(A,A),hAPP(hAPP(c_Power_Opower__class_Opower(A),hAPP(c_Int_Onumber__class_Onumber__of(A),C)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B))),hAPP(hAPP(c_COMBS(A,A,A),hAPP(hAPP(c_COMBB(A,tc_fun(A,A),A),c_Groups_Otimes__class_Otimes(A)),hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),C)))),c_COMBI(A))) = hAPP(hAPP(c_Power_Opower__class_Opower(A),hAPP(c_Int_Onumber__class_Onumber__of(A),C)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(B))) # label(fact_power__number__of__odd__number__of) # label(axiom). [clausify(3347)]. 45.06/44.96 5705 -class_Int_Onumber__ring(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))) | c_Groups_Oone__class_Oone(A) = hAPP(hAPP(c_Power_Opower__class_Opower(A),hAPP(c_Int_Onumber__class_Onumber__of(A),C)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit1(B))) # label(fact_power__number__of__odd__number__of) # label(axiom). [clausify(3347)]. 45.06/44.96 5706 -class_Int_Onumber__ring(A) | hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OMin) = hAPP(c_Groups_Ouminus__class_Ouminus(A),c_Groups_Oone__class_Oone(A)) # label(fact_number__of__Min) # label(axiom). [clausify(3406)]. 45.06/44.96 Derived: hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)). [resolve(5706,a,5634,a)]. 45.06/44.96 5707 -class_Int_Onumber__ring(A) | hAPP(c_Int_Onumber__class_Onumber__of(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(c_Int_Onumber__class_Onumber__of(A),C)) # label(fact_number__of__mult) # label(axiom). [clausify(3421)]. 45.06/44.96 5708 -class_Int_Onumber__ring(A) | hAPP(c_Int_Onumber__class_Onumber__of(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(c_Int_Onumber__class_Onumber__of(A),C)) # label(fact_number__of__add) # label(axiom). [clausify(3489)]. 45.06/44.96 5709 -class_Int_Onumber__ring(A) | c_Int_Oiszero(A,hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OPls)) # label(fact_iszero__Numeral0) # label(axiom). [clausify(3598)]. 45.06/44.96 Derived: c_Int_Oiszero(tc_Int_Oint,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OPls)). [resolve(5709,a,5634,a)]. 45.06/44.96 5710 -class_Int_Onumber__ring(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OBit1(c_Int_OPls))),B) = B # label(fact_mult__numeral__1) # label(axiom). [clausify(3600)]. 45.06/44.96 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit1(c_Int_OPls))),A) = A. [resolve(5710,a,5634,a)]. 45.06/44.96 5711 -class_Int_Onumber__ring(A) | hAPP(c_Int_Oring__1__class_Oof__int(A),B) = hAPP(c_Int_Onumber__class_Onumber__of(A),B) # label(fact_number__of__eq) # label(axiom). [clausify(3603)]. 45.06/44.96 Derived: hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),A) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A). [resolve(5711,a,5634,a)]. 45.06/44.96 5712 -class_Int_Onumber__ring(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OPls) # label(fact_semiring__norm_I112_J) # label(axiom). [clausify(3695)]. 45.06/44.96 5713 -class_Int_Onumber__ring(A) | hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OBit0(B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),c_Groups_Oone__class_Oone(A)),c_Groups_Oone__class_Oone(A))),hAPP(c_Int_Onumber__class_Onumber__of(A),B)) # label(fact_double__number__of__Bit0) # label(axiom). [clausify(3719)]. 45.45/45.34 Derived: hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(A)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),c_Groups_Oone__class_Oone(tc_Int_Oint))),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)). [resolve(5713,a,5634,a)]. 45.45/45.34 5714 -class_Int_Onumber__ring(A) | c_Groups_Oone__class_Oone(A) = hAPP(hAPP(c_Power_Opower__class_Opower(A),hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OMin)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),B)) # label(fact_power__m1__even) # label(axiom). [clausify(3765)]. 45.45/45.34 Derived: c_Groups_Oone__class_Oone(tc_Int_Oint) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),A)). [resolve(5714,a,5634,a)]. 45.45/45.34 5715 -class_Int_Onumber__ring(A) | -c_Int_Oiszero(A,hAPP(c_Int_Onumber__class_Onumber__of(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Int_OPls)))) | c_Groups_Ozero__class_Ozero(A) = hAPP(c_Int_Onumber__class_Onumber__of(A),B) # label(fact_eq__special_I3_J) # label(axiom). [clausify(3866)]. 45.45/45.34 Derived: -c_Int_Oiszero(tc_Int_Oint,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Int_OPls)))) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A). [resolve(5715,a,5634,a)]. 45.45/45.34 5716 -class_Int_Onumber__ring(A) | c_Int_Oiszero(A,hAPP(c_Int_Onumber__class_Onumber__of(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Int_OPls)))) | c_Groups_Ozero__class_Ozero(A) != hAPP(c_Int_Onumber__class_Onumber__of(A),B) # label(fact_eq__special_I3_J) # label(axiom). [clausify(3866)]. 45.45/45.34 Derived: c_Int_Oiszero(tc_Int_Oint,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Int_OPls)))) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A). [resolve(5716,a,5634,a)]. 45.45/45.34 5717 -class_Int_Onumber__ring(A) | hAPP(c_Int_Oring__1__class_Oof__int(A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B)) = hAPP(c_Int_Onumber__class_Onumber__of(A),B) # label(fact_of__int__number__of__eq) # label(axiom). [clausify(3904)]. 45.45/45.34 Derived: hAPP(c_Int_Oring__1__class_Oof__int(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A). [resolve(5717,a,5634,a)]. 45.45/45.34 5718 -class_Int_Onumber__ring(A) | hBOOL(hAPP(hAPP(c_member(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),c_Int_Oring__1__class_OInts(A))) # label(fact_Ints__number__of) # label(axiom). [clausify(4042)]. 45.45/45.34 Derived: hBOOL(hAPP(hAPP(c_member(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),c_Int_Oring__1__class_OInts(tc_Int_Oint))). [resolve(5718,a,5634,a)]. 45.45/45.34 5719 -class_Int_Onumber__ring(A) | c_Groups_Oone__class_Oone(A) = hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OBit1(c_Int_OPls)) # label(fact_semiring__norm_I110_J) # label(axiom). [clausify(4161)]. 45.45/45.34 5720 -class_Int_Onumber__ring(A) | hAPP(c_Groups_Ouminus__class_Ouminus(A),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OMin)) # label(fact_mult__minus1__right) # label(axiom). [clausify(4242)]. 45.45/45.34 Derived: hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin)). [resolve(5720,a,5634,a)]. 45.53/45.42 5721 -class_Int_Onumber__ring(A) | -hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))) | c_Groups_Ozero__class_Ozero(A) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)) # label(fact_of__nat__number__of__eq) # label(axiom). [clausify(4310)]. 45.53/45.42 Derived: -hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A))) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)). [resolve(5721,a,5634,a)]. 45.53/45.42 5722 -class_Int_Onumber__ring(A) | hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),B))) | hAPP(c_Int_Onumber__class_Onumber__of(A),B) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),B)) # label(fact_of__nat__number__of__eq) # label(axiom). [clausify(4310)]. 45.53/45.42 Derived: hBOOL(hAPP(c_Nat__Numeral_Oneg,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A))) | hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)). [resolve(5722,a,5634,a)]. 45.53/45.42 5723 -class_Int_Onumber__ring(A) | c_Groups_Ozero__class_Ozero(A) != hAPP(c_Int_Onumber__class_Onumber__of(A),B) | c_Int_Oiszero(A,hAPP(c_Int_Onumber__class_Onumber__of(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Int_OPls),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B)))) # label(fact_eq__special_I1_J) # label(axiom). [clausify(4400)]. 45.53/45.42 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A) | c_Int_Oiszero(tc_Int_Oint,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Int_OPls),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)))). [resolve(5723,a,5634,a)]. 45.53/45.42 5724 -class_Int_Onumber__ring(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(c_Int_Onumber__class_Onumber__of(A),B) | -c_Int_Oiszero(A,hAPP(c_Int_Onumber__class_Onumber__of(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Int_OPls),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B)))) # label(fact_eq__special_I1_J) # label(axiom). [clausify(4400)]. 45.53/45.42 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A) | -c_Int_Oiszero(tc_Int_Oint,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Int_OPls),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)))). [resolve(5724,a,5634,a)]. 45.53/45.42 5725 -class_Int_Onumber__ring(A) | hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OMin) = hAPP(hAPP(c_Power_Opower__class_Opower(A),hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OMin)),hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),B))) # label(fact_power__m1__odd) # label(axiom). [clausify(4407)]. 45.53/45.42 Derived: hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin)),hAPP(c_Nat_OSuc,hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),A))). [resolve(5725,a,5634,a)]. 45.53/45.42 5726 -class_Fields_Ofield__inverse__zero(A) | -class_Int_Onumber__ring(A) | hAPP(c_Groups_Ouminus__class_Ouminus(A),B) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OMin)) # label(fact_divide__minus1) # label(axiom). [clausify(4428)]. 45.53/45.42 Derived: -class_Fields_Ofield__inverse__zero(tc_Int_Oint) | hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin)). [resolve(5726,b,5634,a)]. 45.53/45.42 5727 -class_Int_Onumber__ring(A) | hAPP(c_Int_Onumber__class_Onumber__of(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),c_Groups_Oone__class_Oone(A)),hAPP(c_Int_Onumber__class_Onumber__of(A),B)) # label(fact_diff__special_I1_J) # label(axiom). [clausify(4439)]. 45.76/45.70 Derived: hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)). [resolve(5727,a,5634,a)]. 45.76/45.70 5728 -class_Int_Onumber__ring(A) | hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OMin) = hAPP(c_Groups_Ouminus__class_Ouminus(A),c_Groups_Oone__class_Oone(A)) # label(fact_arith__simps_I31_J) # label(axiom). [clausify(4465)]. 45.76/45.70 5729 -class_Fields_Olinordered__field__inverse__zero(A) | -class_Int_Onumber__ring(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))) # label(fact_half__gt__zero) # label(axiom). [clausify(4526)]. 45.76/45.70 5730 -class_Rings_Olinordered__idom(A) | -class_Int_Onumber__ring(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(4582)]. 45.76/45.70 Derived: -class_Rings_Olinordered__idom(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Int_OPls)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5730,b,5634,a)]. 45.76/45.70 5731 -class_Rings_Olinordered__idom(A) | -class_Int_Onumber__ring(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(4582)]. 45.76/45.70 Derived: -class_Rings_Olinordered__idom(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Int_OPls)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5731,b,5634,a)]. 45.76/45.70 5732 -class_Int_Onumber__ring(A) | -c_Int_Oiszero(A,hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OBit1(c_Int_OPls))) # label(fact_not__iszero__Numeral1) # label(axiom). [clausify(4678)]. 45.76/45.70 Derived: -c_Int_Oiszero(tc_Int_Oint,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit1(c_Int_OPls))). [resolve(5732,a,5634,a)]. 45.76/45.70 5733 -class_Int_Onumber__ring(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OBit1(c_Int_OPls))) = B # label(fact_mult__numeral__1__right) # label(axiom). [clausify(4726)]. 45.76/45.70 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit1(c_Int_OPls))) = A. [resolve(5733,a,5634,a)]. 45.76/45.70 5734 -class_Int_Onumber__ring(A) | -c_Int_Oiszero(A,hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OMin)) # label(fact_nonzero__number__of__Min) # label(axiom). [clausify(4796)]. 45.76/45.70 Derived: -c_Int_Oiszero(tc_Int_Oint,hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OMin)). [resolve(5734,a,5634,a)]. 45.76/45.70 5735 -class_Int_Onumber__ring(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Power_Opower__class_Opower(A),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(A),C),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),B)),C)) = hAPP(hAPP(c_Power_Opower__class_Opower(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) # label(fact_power2__sum) # label(axiom). [clausify(4867)]. 47.59/47.51 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls))))),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))),A)),B)) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))). [resolve(5735,a,5634,a)]. 47.59/47.51 5736 -class_Int_Onumber__ring(A) | hAPP(c_Int_Onumber__class_Onumber__of(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),c_Groups_Oone__class_Oone(A)) # label(fact_diff__special_I2_J) # label(axiom). [clausify(4881)]. 47.59/47.51 Derived: hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)))) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),c_Groups_Oone__class_Oone(tc_Int_Oint)). [resolve(5736,a,5634,a)]. 47.59/47.51 5737 -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(4901)]. 47.59/47.51 Derived: -class_Rings_Olinordered__idom(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)),A)). [resolve(5737,a,5634,a)]. 47.59/47.51 5738 -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(4901)]. 47.59/47.51 Derived: -class_Rings_Olinordered__idom(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Int_OBit1(c_Int_OPls)),A)). [resolve(5738,a,5634,a)]. 47.59/47.51 5739 -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(3392)]. 47.59/47.51 5740 class_Groups_Ocancel__ab__semigroup__add(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Ocancel__ab__semigroup__add) # label(axiom). [assumption]. 47.59/47.51 5741 class_Groups_Ocancel__ab__semigroup__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Ocancel__ab__semigroup__add) # label(axiom). [assumption]. 47.59/47.51 5742 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]. 49.73/49.63 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),C) | B = C. [resolve(5739,a,5740,a)]. 49.73/49.63 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),C) | B = C. [resolve(5739,a,5741,a)]. 49.73/49.63 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),B) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),C) | B = C. [resolve(5739,a,5742,a)]. 49.73/49.63 5743 class_Groups_Ocomm__monoid__add(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Groups_Ocomm__monoid__add) # label(axiom). [assumption]. 49.73/49.63 5744 -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(110)]. 49.73/49.63 5745 -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(160)]. 49.73/49.63 5746 -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(174)]. 49.73/49.63 5747 -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(298)]. 49.73/49.63 5748 -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),f43(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(321)]. 49.73/49.63 5749 -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,f43(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(321)]. 49.73/49.63 5750 -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),f54(V6,V7,F,E,D,C,B,A)),E)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,A),V6),F) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),V7),E) # label(fact_setsum__reindex__cong) # label(axiom). [clausify(388)]. 49.73/49.63 5751 -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,f54(V6,V7,F,E,D,C,B,A))) != hAPP(V7,f54(V6,V7,F,E,D,C,B,A)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,A),V6),F) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),V7),E) # label(fact_setsum__reindex__cong) # label(axiom). [clausify(388)]. 49.73/49.63 5752 -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(404)]. 49.73/49.63 5753 -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(406)]. 49.73/49.63 5754 -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(459)]. 49.73/49.63 5755 -class_Groups_Oordered__ab__semigroup__add(A) | -class_Groups_Ocomm__monoid__add(A) | hBOOL(hAPP(hAPP(c_member(B),f65(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(477)]. 49.73/49.63 5756 -class_Groups_Oordered__ab__semigroup__add(A) | -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(B,f65(C,B,D,E,A))),hAPP(C,f65(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(477)]. 49.73/49.63 5757 -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(539)]. 49.73/49.63 5758 -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(623)]. 49.73/49.63 5759 -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(623)]. 49.73/49.63 5760 -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(642)]. 49.73/49.63 5761 -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(642)]. 49.73/49.63 5762 -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(700)]. 49.73/49.63 5763 -class_Groups_Ocomm__monoid__mult(A) | -class_Groups_Ocomm__monoid__add(A) | -c_Nat__Transfer_Onat__set(B) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Int_Oint,A),C),B) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(tc_Nat_Onat,A),hAPP(hAPP(c_COMBB(tc_Int_Oint,A,tc_Nat_Onat),C),c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint))),hAPP(c_Set_Oimage(tc_Int_Oint,tc_Nat_Onat,c_Int_Onat),B)) # label(fact_transfer__int__nat__sum__prod_I2_J) # label(axiom). [clausify(735)]. 49.73/49.63 5764 -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(809)]. 49.73/49.63 5765 -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(844)]. 49.73/49.63 5766 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hBOOL(hAPP(hAPP(c_member(B),f121(D,E,F,C,B,A)),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(D,A),E),hAPP(c_Set_Oimage(B,D,F),C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(c_Fun_Ocomp(D,A,B,E),F)),C) # label(fact_setsum__reindex__nonzero) # label(axiom). [clausify(974)]. 49.73/49.63 5767 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hBOOL(hAPP(hAPP(c_member(B),f122(D,E,F,C,B,A)),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(D,A),E),hAPP(c_Set_Oimage(B,D,F),C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(c_Fun_Ocomp(D,A,B,E),F)),C) # label(fact_setsum__reindex__nonzero) # label(axiom). [clausify(974)]. 49.73/49.63 5768 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | f122(D,E,F,C,B,A) != f121(D,E,F,C,B,A) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(D,A),E),hAPP(c_Set_Oimage(B,D,F),C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(c_Fun_Ocomp(D,A,B,E),F)),C) # label(fact_setsum__reindex__nonzero) # label(axiom). [clausify(974)]. 49.73/49.63 5769 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hAPP(D,f122(E,F,D,C,B,A)) = hAPP(D,f121(E,F,D,C,B,A)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(E,A),F),hAPP(c_Set_Oimage(B,E,D),C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(c_Fun_Ocomp(E,A,B,F),D)),C) # label(fact_setsum__reindex__nonzero) # label(axiom). [clausify(974)]. 49.73/49.63 5770 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | c_Groups_Ozero__class_Ozero(A) != hAPP(D,hAPP(E,f121(F,D,E,C,B,A))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(F,A),D),hAPP(c_Set_Oimage(B,F,E),C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(c_Fun_Ocomp(F,A,B,D),E)),C) # label(fact_setsum__reindex__nonzero) # label(axiom). [clausify(974)]. 49.73/49.63 5771 -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),f126(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(1027)]. 49.73/49.63 5772 -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,f126(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(1027)]. 49.73/49.63 5773 -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(1032)]. 49.73/49.63 5774 -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(1156)]. 49.73/49.63 5775 -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(1497)]. 49.73/49.63 5776 -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(1635)]. 49.73/49.63 5777 -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)),f188(D,C,B,A)),C)) | hBOOL(hAPP(hAPP(c_member(tc_fun(B,tc_HOL_Obool)),f189(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(1724)]. 49.73/49.63 5778 -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)),f188(D,C,B,A)),C)) | hBOOL(hAPP(hAPP(c_member(tc_fun(B,tc_HOL_Obool)),f190(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(1724)]. 49.73/49.63 5779 -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)),f188(D,C,B,A)),C)) | f190(D,C,B,A) != f189(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(1724)]. 49.73/49.63 5780 -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)),f188(D,C,B,A)),C)) | hBOOL(hAPP(hAPP(c_member(B),f191(D,C,B,A)),f189(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(1724)]. 49.73/49.63 5781 -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)),f188(D,C,B,A)),C)) | hBOOL(hAPP(hAPP(c_member(B),f191(D,C,B,A)),f190(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(1724)]. 49.73/49.63 5782 -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)),f188(D,C,B,A)),C)) | c_Groups_Ozero__class_Ozero(A) != hAPP(D,f191(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(1724)]. 49.73/49.63 5783 -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),f188(D,C,B,A))) | hBOOL(hAPP(hAPP(c_member(tc_fun(B,tc_HOL_Obool)),f189(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(1724)]. 49.73/49.63 5784 -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),f188(D,C,B,A))) | hBOOL(hAPP(hAPP(c_member(tc_fun(B,tc_HOL_Obool)),f190(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(1724)]. 49.73/49.63 5785 -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),f188(D,C,B,A))) | f190(D,C,B,A) != f189(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(1724)]. 49.73/49.63 5786 -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),f188(D,C,B,A))) | hBOOL(hAPP(hAPP(c_member(B),f191(D,C,B,A)),f189(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(1724)]. 49.73/49.63 5787 -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),f188(D,C,B,A))) | hBOOL(hAPP(hAPP(c_member(B),f191(D,C,B,A)),f190(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(1724)]. 49.73/49.63 5788 -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),f188(D,C,B,A))) | c_Groups_Ozero__class_Ozero(A) != hAPP(D,f191(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(1724)]. 49.73/49.63 5789 -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(1789)]. 49.73/49.63 5790 -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(1890)]. 49.73/49.63 5791 -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(1945)]. 49.73/49.63 5792 -class_Groups_Ocomm__monoid__add(A) | -c_Fun_Oinj__on(B,A,C,D) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,A),c_Fun_Oid(A)),hAPP(c_Set_Oimage(B,A,C),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),C),D) # label(fact_setsum__reindex__id) # label(axiom). [clausify(1966)]. 49.73/49.63 5793 -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(1990)]. 49.73/49.63 5794 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hBOOL(hAPP(hAPP(c_member(B),f258(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(2187)]. 49.73/49.63 5795 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(D),hAPP(E,f258(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(2187)]. 49.73/49.63 5796 -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),f263(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(2237)]. 49.73/49.63 5797 -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,f263(E,D,C,B,A))),hAPP(E,f263(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(2237)]. 49.73/49.63 5798 -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(2238)]. 49.73/49.63 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(5743,a,5744,a)]. 49.73/49.63 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(5743,a,5745,a)]. 49.76/49.64 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(5743,a,5746,a)]. 49.76/49.64 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(5743,a,5747,a)]. 49.76/49.64 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | hBOOL(hAPP(hAPP(c_member(A),f43(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(5743,a,5748,a)]. 49.76/49.64 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,f43(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(5743,a,5749,a)]. 49.76/49.64 Derived: -c_Fun_Oinj__on(A,B,C,D) | hAPP(c_Set_Oimage(A,B,C),D) != E | hBOOL(hAPP(hAPP(c_member(A),f54(F,V6,E,D,C,B,A,tc_Code__Numeral_Ocode__numeral)),D)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Code__Numeral_Ocode__numeral),F),E) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),V6),D). [resolve(5743,a,5750,a)]. 49.76/49.64 Derived: -c_Fun_Oinj__on(A,B,C,D) | hAPP(c_Set_Oimage(A,B,C),D) != E | hAPP(F,hAPP(C,f54(F,V6,E,D,C,B,A,tc_Code__Numeral_Ocode__numeral))) != hAPP(V6,f54(F,V6,E,D,C,B,A,tc_Code__Numeral_Ocode__numeral)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Code__Numeral_Ocode__numeral),F),E) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),V6),D). [resolve(5743,a,5751,a)]. 49.76/49.64 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(5743,a,5752,a)]. 49.76/49.65 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(5743,a,5753,a)]. 49.76/49.65 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(5743,a,5754,a)]. 49.76/49.65 Derived: -class_Groups_Oordered__ab__semigroup__add(tc_Code__Numeral_Ocode__numeral) | hBOOL(hAPP(hAPP(c_member(A),f65(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(5743,a,5755,b)]. 49.76/49.65 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,f65(B,A,C,D,tc_Code__Numeral_Ocode__numeral))),hAPP(B,f65(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(5743,a,5756,b)]. 49.76/49.65 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(5743,a,5757,a)]. 49.76/49.65 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(5743,a,5758,a)]. 49.76/49.66 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(5743,a,5759,a)]. 49.76/49.66 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(5743,a,5760,a)]. 49.76/49.66 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(5743,a,5761,a)]. 49.76/49.66 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(5743,a,5762,a)]. 49.76/49.66 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(5743,a,5763,b)]. 49.76/49.66 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f121(C,D,E,B,A,tc_Code__Numeral_Ocode__numeral)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Code__Numeral_Ocode__numeral),D),hAPP(c_Set_Oimage(A,C,E),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(c_Fun_Ocomp(C,tc_Code__Numeral_Ocode__numeral,A,D),E)),B). [resolve(5743,a,5766,a)]. 49.76/49.66 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f122(C,D,E,B,A,tc_Code__Numeral_Ocode__numeral)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Code__Numeral_Ocode__numeral),D),hAPP(c_Set_Oimage(A,C,E),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(c_Fun_Ocomp(C,tc_Code__Numeral_Ocode__numeral,A,D),E)),B). [resolve(5743,a,5767,a)]. 49.76/49.66 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | f122(C,D,E,B,A,tc_Code__Numeral_Ocode__numeral) != f121(C,D,E,B,A,tc_Code__Numeral_Ocode__numeral) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Code__Numeral_Ocode__numeral),D),hAPP(c_Set_Oimage(A,C,E),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(c_Fun_Ocomp(C,tc_Code__Numeral_Ocode__numeral,A,D),E)),B). [resolve(5743,a,5768,a)]. 49.76/49.66 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(C,f122(D,E,C,B,A,tc_Code__Numeral_Ocode__numeral)) = hAPP(C,f121(D,E,C,B,A,tc_Code__Numeral_Ocode__numeral)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(D,tc_Code__Numeral_Ocode__numeral),E),hAPP(c_Set_Oimage(A,D,C),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(c_Fun_Ocomp(D,tc_Code__Numeral_Ocode__numeral,A,E),C)),B). [resolve(5743,a,5769,a)]. 49.76/49.66 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != hAPP(C,hAPP(D,f121(E,C,D,B,A,tc_Code__Numeral_Ocode__numeral))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(E,tc_Code__Numeral_Ocode__numeral),C),hAPP(c_Set_Oimage(A,E,D),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(c_Fun_Ocomp(E,tc_Code__Numeral_Ocode__numeral,A,C),D)),B). [resolve(5743,a,5770,a)]. 49.76/49.66 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),f126(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(5743,a,5771,b)]. 49.76/49.66 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,f126(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(5743,a,5772,b)]. 49.76/49.66 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(5743,a,5773,a)]. 49.76/49.66 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(5743,a,5775,a)]. 49.76/49.66 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_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)),C)),D)),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),C),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),D),B)). [resolve(5743,a,5776,a)]. 49.76/49.67 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)),f188(C,B,A,tc_Code__Numeral_Ocode__numeral)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f189(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(5743,a,5777,a)]. 49.76/49.67 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)),f188(C,B,A,tc_Code__Numeral_Ocode__numeral)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f190(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(5743,a,5778,a)]. 49.76/49.67 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)),f188(C,B,A,tc_Code__Numeral_Ocode__numeral)),B)) | f190(C,B,A,tc_Code__Numeral_Ocode__numeral) != f189(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(5743,a,5779,a)]. 49.76/49.67 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)),f188(C,B,A,tc_Code__Numeral_Ocode__numeral)),B)) | hBOOL(hAPP(hAPP(c_member(A),f191(C,B,A,tc_Code__Numeral_Ocode__numeral)),f189(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(5743,a,5780,a)]. 49.76/49.67 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)),f188(C,B,A,tc_Code__Numeral_Ocode__numeral)),B)) | hBOOL(hAPP(hAPP(c_member(A),f191(C,B,A,tc_Code__Numeral_Ocode__numeral)),f190(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(5743,a,5781,a)]. 49.76/49.67 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)),f188(C,B,A,tc_Code__Numeral_Ocode__numeral)),B)) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != hAPP(C,f191(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(5743,a,5782,a)]. 49.76/49.68 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f188(C,B,A,tc_Code__Numeral_Ocode__numeral))) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f189(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(5743,a,5783,a)]. 49.76/49.68 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f188(C,B,A,tc_Code__Numeral_Ocode__numeral))) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f190(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(5743,a,5784,a)]. 49.76/49.68 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f188(C,B,A,tc_Code__Numeral_Ocode__numeral))) | f190(C,B,A,tc_Code__Numeral_Ocode__numeral) != f189(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(5743,a,5785,a)]. 49.76/49.68 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f188(C,B,A,tc_Code__Numeral_Ocode__numeral))) | hBOOL(hAPP(hAPP(c_member(A),f191(C,B,A,tc_Code__Numeral_Ocode__numeral)),f189(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(5743,a,5786,a)]. 49.76/49.68 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f188(C,B,A,tc_Code__Numeral_Ocode__numeral))) | hBOOL(hAPP(hAPP(c_member(A),f191(C,B,A,tc_Code__Numeral_Ocode__numeral)),f190(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(5743,a,5787,a)]. 49.76/49.68 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f188(C,B,A,tc_Code__Numeral_Ocode__numeral))) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != hAPP(C,f191(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(5743,a,5788,a)]. 49.76/49.69 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(5743,a,5789,a)]. 49.76/49.69 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(5743,a,5790,a)]. 49.76/49.69 Derived: -c_Fun_Oinj__on(A,tc_Code__Numeral_Ocode__numeral,B,C) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),c_Fun_Oid(tc_Code__Numeral_Ocode__numeral)),hAPP(c_Set_Oimage(A,tc_Code__Numeral_Ocode__numeral,B),C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),B),C). [resolve(5743,a,5792,a)]. 49.76/49.69 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(5743,a,5793,a)]. 49.76/49.69 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f258(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(5743,a,5794,a)]. 49.76/49.69 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(C),hAPP(D,f258(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(5743,a,5795,a)]. 49.76/49.69 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),f263(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(5743,a,5796,a)]. 49.76/49.69 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,f263(D,C,B,A,tc_Code__Numeral_Ocode__numeral))),hAPP(D,f263(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(5743,a,5797,a)]. 49.76/49.69 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(5743,a,5798,a)]. 49.76/49.69 5799 -class_Groups_Ocomm__monoid__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(D),E)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(D,A),hAPP(hAPP(c_COMBS(D,tc_fun(B,tc_HOL_Obool),A),hAPP(hAPP(c_COMBB(tc_fun(B,A),tc_fun(tc_fun(B,tc_HOL_Obool),A),D),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A)),hAPP(c_COMBC(B,D,A),F))),hAPP(hAPP(c_COMBB(tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_HOL_Obool),D),c_Set_OCollect(B)),hAPP(hAPP(c_COMBB(tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_HOL_Obool),D),hAPP(c_COMBS(B,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),B),c_fconj),hAPP(hAPP(c_COMBC(B,tc_fun(B,tc_HOL_Obool),tc_HOL_Obool),c_member(B)),C)))),hAPP(c_COMBC(B,D,tc_HOL_Obool),V6))))),E) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(hAPP(c_COMBS(B,tc_fun(D,tc_HOL_Obool),A),hAPP(hAPP(c_COMBB(tc_fun(D,A),tc_fun(tc_fun(D,tc_HOL_Obool),A),B),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(D,A)),F)),hAPP(hAPP(c_COMBB(tc_fun(D,tc_HOL_Obool),tc_fun(D,tc_HOL_Obool),B),c_Set_OCollect(D)),hAPP(hAPP(c_COMBB(tc_fun(D,tc_HOL_Obool),tc_fun(D,tc_HOL_Obool),B),hAPP(c_COMBS(D,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),D),c_fconj),hAPP(hAPP(c_COMBC(D,tc_fun(D,tc_HOL_Obool),tc_HOL_Obool),c_member(D)),E)))),V6)))),C) # label(fact_setsum__setsum__restrict) # label(axiom). [clausify(2269)]. 49.76/49.69 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(C),D)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBS(C,tc_fun(A,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBB(tc_fun(A,tc_Code__Numeral_Ocode__numeral),tc_fun(tc_fun(A,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),C),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral)),hAPP(c_COMBC(A,C,tc_Code__Numeral_Ocode__numeral),E))),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool),C),c_Set_OCollect(A)),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool),C),hAPP(c_COMBS(A,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),A),c_fconj),hAPP(hAPP(c_COMBC(A,tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),c_member(A)),B)))),hAPP(c_COMBC(A,C,tc_HOL_Obool),F))))),D) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBS(A,tc_fun(C,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBB(tc_fun(C,tc_Code__Numeral_Ocode__numeral),tc_fun(tc_fun(C,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),A),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Code__Numeral_Ocode__numeral)),E)),hAPP(hAPP(c_COMBB(tc_fun(C,tc_HOL_Obool),tc_fun(C,tc_HOL_Obool),A),c_Set_OCollect(C)),hAPP(hAPP(c_COMBB(tc_fun(C,tc_HOL_Obool),tc_fun(C,tc_HOL_Obool),A),hAPP(c_COMBS(C,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),C),c_fconj),hAPP(hAPP(c_COMBC(C,tc_fun(C,tc_HOL_Obool),tc_HOL_Obool),c_member(C)),D)))),F)))),B). [resolve(5799,a,5743,a)]. 49.84/49.76 5800 -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)),f272(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(2318)]. 49.84/49.76 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)),f272(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(5800,b,5743,a)]. 49.84/49.76 5801 -class_Groups_Ocomm__monoid__mult(A) | -class_Groups_Ocomm__monoid__add(A) | B != C | -c_Nat__Transfer_Onat__set(C) | hAPP(D,f272(D,E,C,B,A)) != hAPP(E,f272(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(2318)]. 49.84/49.76 Derived: -class_Groups_Ocomm__monoid__mult(tc_Code__Numeral_Ocode__numeral) | A != B | -c_Nat__Transfer_Onat__set(B) | hAPP(C,f272(C,D,B,A,tc_Code__Numeral_Ocode__numeral)) != hAPP(D,f272(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(5801,b,5743,a)]. 49.84/49.76 5802 -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(2394)]. 49.84/49.76 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(5802,a,5743,a)]. 49.84/49.76 5803 -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(2495)]. 49.84/49.76 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(5803,a,5743,a)]. 49.92/49.85 5804 -class_Groups_Ocomm__monoid__add(A) | -c_Fun_Oinj__on(B,C,D,E) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,A),F),hAPP(c_Set_Oimage(B,C,D),E)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),hAPP(c_Fun_Ocomp(C,A,B,F),D)),E) # label(fact_setsum__reindex) # label(axiom). [clausify(2539)]. 49.92/49.85 Derived: -c_Fun_Oinj__on(A,B,C,D) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Code__Numeral_Ocode__numeral),E),hAPP(c_Set_Oimage(A,B,C),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),hAPP(c_Fun_Ocomp(B,tc_Code__Numeral_Ocode__numeral,A,E),C)),D). [resolve(5804,a,5743,a)]. 49.92/49.85 5805 -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(2571)]. 49.92/49.85 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(5805,a,5743,a)]. 49.92/49.85 5806 -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(2781)]. 49.92/49.85 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(5806,a,5743,a)]. 49.92/49.85 5807 -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(2804)]. 49.92/49.85 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(5807,a,5743,a)]. 49.99/49.87 5808 -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),f319(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(2845)]. 49.99/49.87 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | hBOOL(hAPP(hAPP(c_member(A),f319(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(5808,a,5743,a)]. 49.99/49.87 5809 -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,f319(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(2845)]. 49.99/49.87 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,f319(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(5809,a,5743,a)]. 49.99/49.87 5810 -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(2850)]. 49.99/49.87 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(5810,a,5743,a)]. 49.99/49.87 5811 class_Groups_Ocomm__monoid__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Ocomm__monoid__add) # label(axiom). [assumption]. 49.99/49.88 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(5811,a,5744,a)]. 49.99/49.88 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(5811,a,5745,a)]. 49.99/49.88 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(5811,a,5746,a)]. 49.99/49.88 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(5811,a,5747,a)]. 49.99/49.88 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | hBOOL(hAPP(hAPP(c_member(A),f43(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(5811,a,5748,a)]. 49.99/49.88 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,f43(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(5811,a,5749,a)]. 49.99/49.88 Derived: -c_Fun_Oinj__on(A,B,C,D) | hAPP(c_Set_Oimage(A,B,C),D) != E | hBOOL(hAPP(hAPP(c_member(A),f54(F,V6,E,D,C,B,A,tc_Int_Oint)),D)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Int_Oint),F),E) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),V6),D). [resolve(5811,a,5750,a)]. 49.99/49.88 Derived: -c_Fun_Oinj__on(A,B,C,D) | hAPP(c_Set_Oimage(A,B,C),D) != E | hAPP(F,hAPP(C,f54(F,V6,E,D,C,B,A,tc_Int_Oint))) != hAPP(V6,f54(F,V6,E,D,C,B,A,tc_Int_Oint)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Int_Oint),F),E) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),V6),D). [resolve(5811,a,5751,a)]. 49.99/49.88 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(5811,a,5752,a)]. 49.99/49.89 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(5811,a,5753,a)]. 49.99/49.89 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(5811,a,5754,a)]. 49.99/49.89 Derived: -class_Groups_Oordered__ab__semigroup__add(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_member(A),f65(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(5811,a,5755,b)]. 49.99/49.89 Derived: -class_Groups_Oordered__ab__semigroup__add(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(A,f65(B,A,C,D,tc_Int_Oint))),hAPP(B,f65(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(5811,a,5756,b)]. 49.99/49.89 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(5811,a,5757,a)]. 49.99/49.89 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(5811,a,5758,a)]. 49.99/49.89 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(5811,a,5759,a)]. 49.99/49.89 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(5811,a,5760,a)]. 49.99/49.90 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(5811,a,5761,a)]. 49.99/49.90 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(5811,a,5762,a)]. 49.99/49.90 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(5811,a,5763,b)]. 49.99/49.90 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f121(C,D,E,B,A,tc_Int_Oint)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Int_Oint),D),hAPP(c_Set_Oimage(A,C,E),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(c_Fun_Ocomp(C,tc_Int_Oint,A,D),E)),B). [resolve(5811,a,5766,a)]. 49.99/49.90 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f122(C,D,E,B,A,tc_Int_Oint)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Int_Oint),D),hAPP(c_Set_Oimage(A,C,E),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(c_Fun_Ocomp(C,tc_Int_Oint,A,D),E)),B). [resolve(5811,a,5767,a)]. 49.99/49.90 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | f122(C,D,E,B,A,tc_Int_Oint) != f121(C,D,E,B,A,tc_Int_Oint) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Int_Oint),D),hAPP(c_Set_Oimage(A,C,E),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(c_Fun_Ocomp(C,tc_Int_Oint,A,D),E)),B). [resolve(5811,a,5768,a)]. 49.99/49.90 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(C,f122(D,E,C,B,A,tc_Int_Oint)) = hAPP(C,f121(D,E,C,B,A,tc_Int_Oint)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(D,tc_Int_Oint),E),hAPP(c_Set_Oimage(A,D,C),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(c_Fun_Ocomp(D,tc_Int_Oint,A,E),C)),B). [resolve(5811,a,5769,a)]. 49.99/49.90 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(C,hAPP(D,f121(E,C,D,B,A,tc_Int_Oint))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(E,tc_Int_Oint),C),hAPP(c_Set_Oimage(A,E,D),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(c_Fun_Ocomp(E,tc_Int_Oint,A,C),D)),B). [resolve(5811,a,5770,a)]. 49.99/49.90 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),f126(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(5811,a,5771,b)]. 49.99/49.90 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,f126(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(5811,a,5772,b)]. 49.99/49.91 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(5811,a,5773,a)]. 49.99/49.91 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(5811,a,5775,a)]. 49.99/49.91 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_COMBB(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint),A),c_Groups_Oplus__class_Oplus(tc_Int_Oint)),C)),D)),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),C),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),B)). [resolve(5811,a,5776,a)]. 49.99/49.91 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)),f188(C,B,A,tc_Int_Oint)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f189(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(5811,a,5777,a)]. 49.99/49.91 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)),f188(C,B,A,tc_Int_Oint)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f190(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(5811,a,5778,a)]. 49.99/49.91 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)),f188(C,B,A,tc_Int_Oint)),B)) | f190(C,B,A,tc_Int_Oint) != f189(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(5811,a,5779,a)]. 49.99/49.91 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)),f188(C,B,A,tc_Int_Oint)),B)) | hBOOL(hAPP(hAPP(c_member(A),f191(C,B,A,tc_Int_Oint)),f189(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(5811,a,5780,a)]. 49.99/49.91 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)),f188(C,B,A,tc_Int_Oint)),B)) | hBOOL(hAPP(hAPP(c_member(A),f191(C,B,A,tc_Int_Oint)),f190(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(5811,a,5781,a)]. 49.99/49.91 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)),f188(C,B,A,tc_Int_Oint)),B)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(C,f191(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(5811,a,5782,a)]. 49.99/49.91 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f188(C,B,A,tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f189(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(5811,a,5783,a)]. 49.99/49.91 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f188(C,B,A,tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f190(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(5811,a,5784,a)]. 49.99/49.91 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f188(C,B,A,tc_Int_Oint))) | f190(C,B,A,tc_Int_Oint) != f189(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(5811,a,5785,a)]. 49.99/49.91 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f188(C,B,A,tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_member(A),f191(C,B,A,tc_Int_Oint)),f189(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(5811,a,5786,a)]. 49.99/49.91 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f188(C,B,A,tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_member(A),f191(C,B,A,tc_Int_Oint)),f190(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(5811,a,5787,a)]. 49.99/49.92 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f188(C,B,A,tc_Int_Oint))) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(C,f191(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(5811,a,5788,a)]. 49.99/49.92 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(5811,a,5789,a)]. 49.99/49.92 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(5811,a,5790,a)]. 49.99/49.92 Derived: -c_Fun_Oinj__on(A,tc_Int_Oint,B,C) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Int_Oint,tc_Int_Oint),c_Fun_Oid(tc_Int_Oint)),hAPP(c_Set_Oimage(A,tc_Int_Oint,B),C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),B),C). [resolve(5811,a,5792,a)]. 49.99/49.92 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(5811,a,5793,a)]. 49.99/49.92 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f258(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(5811,a,5794,a)]. 49.99/49.92 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(C),hAPP(D,f258(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(5811,a,5795,a)]. 49.99/49.92 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),f263(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(5811,a,5796,a)]. 49.99/49.92 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,f263(D,C,B,A,tc_Int_Oint))),hAPP(D,f263(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(5811,a,5797,a)]. 49.99/49.93 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(5811,a,5798,a)]. 49.99/49.93 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(C),D)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Int_Oint),hAPP(hAPP(c_COMBS(C,tc_fun(A,tc_HOL_Obool),tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_fun(A,tc_Int_Oint),tc_fun(tc_fun(A,tc_HOL_Obool),tc_Int_Oint),C),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint)),hAPP(c_COMBC(A,C,tc_Int_Oint),E))),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool),C),c_Set_OCollect(A)),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool),C),hAPP(c_COMBS(A,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),A),c_fconj),hAPP(hAPP(c_COMBC(A,tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),c_member(A)),B)))),hAPP(c_COMBC(A,C,tc_HOL_Obool),F))))),D) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(hAPP(c_COMBS(A,tc_fun(C,tc_HOL_Obool),tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_fun(C,tc_Int_Oint),tc_fun(tc_fun(C,tc_HOL_Obool),tc_Int_Oint),A),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Int_Oint)),E)),hAPP(hAPP(c_COMBB(tc_fun(C,tc_HOL_Obool),tc_fun(C,tc_HOL_Obool),A),c_Set_OCollect(C)),hAPP(hAPP(c_COMBB(tc_fun(C,tc_HOL_Obool),tc_fun(C,tc_HOL_Obool),A),hAPP(c_COMBS(C,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),C),c_fconj),hAPP(hAPP(c_COMBC(C,tc_fun(C,tc_HOL_Obool),tc_HOL_Obool),c_member(C)),D)))),F)))),B). [resolve(5811,a,5799,a)]. 49.99/49.93 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)),f272(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(5811,a,5800,b)]. 49.99/49.93 Derived: -class_Groups_Ocomm__monoid__mult(tc_Int_Oint) | A != B | -c_Nat__Transfer_Onat__set(B) | hAPP(C,f272(C,D,B,A,tc_Int_Oint)) != hAPP(D,f272(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(5811,a,5801,b)]. 49.99/49.93 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(5811,a,5802,a)]. 49.99/49.93 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(5811,a,5803,a)]. 49.99/49.93 Derived: -c_Fun_Oinj__on(A,B,C,D) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Int_Oint),E),hAPP(c_Set_Oimage(A,B,C),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),hAPP(c_Fun_Ocomp(B,tc_Int_Oint,A,E),C)),D). [resolve(5811,a,5804,a)]. 49.99/49.93 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(5811,a,5805,a)]. 49.99/49.94 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(5811,a,5806,a)]. 49.99/49.94 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(5811,a,5807,a)]. 49.99/49.94 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | hBOOL(hAPP(hAPP(c_member(A),f319(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(5811,a,5808,a)]. 49.99/49.94 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,f319(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(5811,a,5809,a)]. 49.99/49.94 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_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(5811,a,5810,a)]. 49.99/49.94 5812 class_Groups_Ocomm__monoid__add(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Ocomm__monoid__add) # label(axiom). [assumption]. 49.99/49.94 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(5812,a,5744,a)]. 49.99/49.94 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(5812,a,5745,a)]. 49.99/49.95 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(5812,a,5746,a)]. 49.99/49.95 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),hAPP(hAPP(c_Set_Oinsert(A),C),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(D,C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),B)). [resolve(5812,a,5747,a)]. 49.99/49.95 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | hBOOL(hAPP(hAPP(c_member(A),f43(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(5812,a,5748,a)]. 49.99/49.95 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(D,f43(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(5812,a,5749,a)]. 49.99/49.95 Derived: -c_Fun_Oinj__on(A,B,C,D) | hAPP(c_Set_Oimage(A,B,C),D) != E | hBOOL(hAPP(hAPP(c_member(A),f54(F,V6,E,D,C,B,A,tc_Nat_Onat)),D)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Nat_Onat),F),E) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),V6),D). [resolve(5812,a,5750,a)]. 49.99/49.95 Derived: -c_Fun_Oinj__on(A,B,C,D) | hAPP(c_Set_Oimage(A,B,C),D) != E | hAPP(F,hAPP(C,f54(F,V6,E,D,C,B,A,tc_Nat_Onat))) != hAPP(V6,f54(F,V6,E,D,C,B,A,tc_Nat_Onat)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Nat_Onat),F),E) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),V6),D). [resolve(5812,a,5751,a)]. 49.99/49.95 Derived: hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(A,tc_Nat_Onat),hAPP(hAPP(c_COMBS(A,tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat),A),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),B)),C)),D)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(A,tc_Nat_Onat),B),D))),hAPP(c_List_Omonoid__add__class_Olistsum(tc_Nat_Onat),hAPP(hAPP(c_List_Omap(A,tc_Nat_Onat),C),D))). [resolve(5812,a,5752,a)]. 49.99/49.95 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(5812,a,5753,a)]. 49.99/49.96 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(5812,a,5754,a)]. 49.99/49.96 Derived: -class_Groups_Oordered__ab__semigroup__add(tc_Nat_Onat) | hBOOL(hAPP(hAPP(c_member(A),f65(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(5812,a,5755,b)]. 49.99/49.96 Derived: -class_Groups_Oordered__ab__semigroup__add(tc_Nat_Onat) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(A,f65(B,A,C,D,tc_Nat_Onat))),hAPP(B,f65(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(5812,a,5756,b)]. 49.99/49.96 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(5812,a,5757,a)]. 49.99/49.96 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(5812,a,5758,a)]. 49.99/49.96 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(5812,a,5759,a)]. 49.99/49.96 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),C),B) = hAPP(hAPP(hAPP(c_Finite__Set_Ofold__image(tc_Nat_Onat,A,c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),C),c_Groups_Ozero__class_Ozero(tc_Nat_Onat)),B). [resolve(5812,a,5760,a)]. 49.99/49.96 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(5812,a,5761,a)]. 49.99/49.96 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),B),C)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),D),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),D),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,A),B))),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),D),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(tc_Nat_Onat,B),C))). [resolve(5812,a,5762,a)]. 49.99/49.97 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(5812,a,5763,b)]. 49.99/49.97 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f121(C,D,E,B,A,tc_Nat_Onat)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Nat_Onat),D),hAPP(c_Set_Oimage(A,C,E),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),hAPP(c_Fun_Ocomp(C,tc_Nat_Onat,A,D),E)),B). [resolve(5812,a,5766,a)]. 49.99/49.97 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f122(C,D,E,B,A,tc_Nat_Onat)),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Nat_Onat),D),hAPP(c_Set_Oimage(A,C,E),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),hAPP(c_Fun_Ocomp(C,tc_Nat_Onat,A,D),E)),B). [resolve(5812,a,5767,a)]. 49.99/49.97 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | f122(C,D,E,B,A,tc_Nat_Onat) != f121(C,D,E,B,A,tc_Nat_Onat) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Nat_Onat),D),hAPP(c_Set_Oimage(A,C,E),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),hAPP(c_Fun_Ocomp(C,tc_Nat_Onat,A,D),E)),B). [resolve(5812,a,5768,a)]. 49.99/49.97 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hAPP(C,f122(D,E,C,B,A,tc_Nat_Onat)) = hAPP(C,f121(D,E,C,B,A,tc_Nat_Onat)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(D,tc_Nat_Onat),E),hAPP(c_Set_Oimage(A,D,C),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),hAPP(c_Fun_Ocomp(D,tc_Nat_Onat,A,E),C)),B). [resolve(5812,a,5769,a)]. 49.99/49.97 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(C,hAPP(D,f121(E,C,D,B,A,tc_Nat_Onat))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(E,tc_Nat_Onat),C),hAPP(c_Set_Oimage(A,E,D),B)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),hAPP(c_Fun_Ocomp(E,tc_Nat_Onat,A,C),D)),B). [resolve(5812,a,5770,a)]. 49.99/49.97 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),f126(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(5812,a,5771,b)]. 49.99/49.97 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,f126(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(5812,a,5772,b)]. 49.99/49.97 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(5812,a,5773,a)]. 50.10/49.98 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(5812,a,5775,a)]. 50.10/49.98 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_COMBB(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat),A),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),C)),D)),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),C),B)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),D),B)). [resolve(5812,a,5776,a)]. 50.10/49.98 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)),f188(C,B,A,tc_Nat_Onat)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f189(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(5812,a,5777,a)]. 50.10/49.98 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)),f188(C,B,A,tc_Nat_Onat)),B)) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f190(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(5812,a,5778,a)]. 50.10/49.98 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)),f188(C,B,A,tc_Nat_Onat)),B)) | f190(C,B,A,tc_Nat_Onat) != f189(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(5812,a,5779,a)]. 50.10/49.98 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)),f188(C,B,A,tc_Nat_Onat)),B)) | hBOOL(hAPP(hAPP(c_member(A),f191(C,B,A,tc_Nat_Onat)),f189(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(5812,a,5780,a)]. 50.10/49.98 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)),f188(C,B,A,tc_Nat_Onat)),B)) | hBOOL(hAPP(hAPP(c_member(A),f191(C,B,A,tc_Nat_Onat)),f190(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(5812,a,5781,a)]. 50.10/49.98 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)),f188(C,B,A,tc_Nat_Onat)),B)) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(C,f191(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(5812,a,5782,a)]. 50.10/49.98 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f188(C,B,A,tc_Nat_Onat))) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f189(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(5812,a,5783,a)]. 50.10/49.98 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f188(C,B,A,tc_Nat_Onat))) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f190(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(5812,a,5784,a)]. 50.10/49.98 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f188(C,B,A,tc_Nat_Onat))) | f190(C,B,A,tc_Nat_Onat) != f189(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(5812,a,5785,a)]. 50.10/49.98 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f188(C,B,A,tc_Nat_Onat))) | hBOOL(hAPP(hAPP(c_member(A),f191(C,B,A,tc_Nat_Onat)),f189(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(5812,a,5786,a)]. 50.10/49.98 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f188(C,B,A,tc_Nat_Onat))) | hBOOL(hAPP(hAPP(c_member(A),f191(C,B,A,tc_Nat_Onat)),f190(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(5812,a,5787,a)]. 50.10/49.98 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_fun(A,tc_HOL_Obool)),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),f188(C,B,A,tc_Nat_Onat))) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != hAPP(C,f191(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(5812,a,5788,a)]. 50.10/49.98 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(5812,a,5789,a)]. 50.10/49.99 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(5812,a,5790,a)]. 50.10/49.99 Derived: -c_Fun_Oinj__on(A,tc_Nat_Onat,B,C) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),c_Fun_Oid(tc_Nat_Onat)),hAPP(c_Set_Oimage(A,tc_Nat_Onat,B),C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),B),C). [resolve(5812,a,5792,a)]. 50.10/49.99 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(5812,a,5793,a)]. 50.10/49.99 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f258(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(5812,a,5794,a)]. 50.10/49.99 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(C),hAPP(D,f258(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(5812,a,5795,a)]. 50.10/49.99 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),f263(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(5812,a,5796,a)]. 50.10/49.99 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,f263(D,C,B,A,tc_Nat_Onat))),hAPP(D,f263(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(5812,a,5797,a)]. 50.10/49.99 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(5812,a,5798,a)]. 50.10/49.99 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(C),D)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Nat_Onat),hAPP(hAPP(c_COMBS(C,tc_fun(A,tc_HOL_Obool),tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_fun(A,tc_Nat_Onat),tc_fun(tc_fun(A,tc_HOL_Obool),tc_Nat_Onat),C),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat)),hAPP(c_COMBC(A,C,tc_Nat_Onat),E))),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool),C),c_Set_OCollect(A)),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool),C),hAPP(c_COMBS(A,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),A),c_fconj),hAPP(hAPP(c_COMBC(A,tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),c_member(A)),B)))),hAPP(c_COMBC(A,C,tc_HOL_Obool),F))))),D) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),hAPP(hAPP(c_COMBS(A,tc_fun(C,tc_HOL_Obool),tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_fun(C,tc_Nat_Onat),tc_fun(tc_fun(C,tc_HOL_Obool),tc_Nat_Onat),A),c_Big__Operators_Ocomm__monoid__add__class_Osetsum(C,tc_Nat_Onat)),E)),hAPP(hAPP(c_COMBB(tc_fun(C,tc_HOL_Obool),tc_fun(C,tc_HOL_Obool),A),c_Set_OCollect(C)),hAPP(hAPP(c_COMBB(tc_fun(C,tc_HOL_Obool),tc_fun(C,tc_HOL_Obool),A),hAPP(c_COMBS(C,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),C),c_fconj),hAPP(hAPP(c_COMBC(C,tc_fun(C,tc_HOL_Obool),tc_HOL_Obool),c_member(C)),D)))),F)))),B). [resolve(5812,a,5799,a)]. 50.13/50.00 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)),f272(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(5812,a,5800,b)]. 50.13/50.00 Derived: -class_Groups_Ocomm__monoid__mult(tc_Nat_Onat) | A != B | -c_Nat__Transfer_Onat__set(B) | hAPP(C,f272(C,D,B,A,tc_Nat_Onat)) != hAPP(D,f272(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(5812,a,5801,b)]. 50.13/50.00 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(5812,a,5802,a)]. 50.13/50.00 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(5812,a,5803,a)]. 50.13/50.00 Derived: -c_Fun_Oinj__on(A,B,C,D) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,tc_Nat_Onat),E),hAPP(c_Set_Oimage(A,B,C),D)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),hAPP(c_Fun_Ocomp(B,tc_Nat_Onat,A,E),C)),D). [resolve(5812,a,5804,a)]. 50.13/50.00 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(5812,a,5805,a)]. 50.13/50.00 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(5812,a,5806,a)]. 50.15/50.06 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(5812,a,5807,a)]. 50.15/50.06 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),C)) | hBOOL(hAPP(hAPP(c_member(A),f319(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(5812,a,5808,a)]. 50.15/50.06 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,f319(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(5812,a,5809,a)]. 50.15/50.06 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(5812,a,5810,a)]. 50.15/50.06 5813 -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),f339(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(3023)]. 50.15/50.06 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),f339(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(5813,a,5743,a)]. 50.15/50.06 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),f339(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(5813,a,5811,a)]. 50.15/50.06 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),f339(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(5813,a,5812,a)]. 50.15/50.07 5814 -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,f339(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(3023)]. 50.15/50.07 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,f339(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(5814,a,5743,a)]. 50.15/50.07 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_Int_Oint) != hAPP(D,f339(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(5814,a,5811,a)]. 50.15/50.07 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_Nat_Onat) != hAPP(D,f339(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(5814,a,5812,a)]. 50.15/50.07 5815 -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(3024)]. 50.15/50.07 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_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(5815,a,5743,a)]. 50.15/50.07 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(D,C)),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),D),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_fun(A,tc_HOL_Obool)),B),hAPP(hAPP(c_Set_Oinsert(A),C),c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)))))). [resolve(5815,a,5811,a)]. 50.15/50.07 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_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(5815,a,5812,a)]. 50.23/50.13 5816 -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(3156)]. 50.23/50.13 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(5816,a,5743,a)]. 50.23/50.13 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(5816,a,5811,a)]. 50.23/50.13 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(5816,a,5812,a)]. 50.23/50.13 5817 -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(3156)]. 50.23/50.13 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(5817,a,5743,a)]. 50.23/50.13 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(5817,a,5811,a)]. 50.23/50.13 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(5817,a,5812,a)]. 50.23/50.13 5818 -class_Groups_Ocomm__monoid__add(A) | -class_Groups_Ocomm__monoid__mult(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(3180)]. 50.30/50.20 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(5818,a,5743,a)]. 50.30/50.20 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(5818,a,5811,a)]. 50.30/50.20 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(5818,a,5812,a)]. 50.30/50.20 5819 -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(3185)]. 50.30/50.20 5820 -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(3225)]. 50.30/50.20 5821 -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(3283)]. 50.30/50.20 5822 -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(3314)]. 50.30/50.20 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(5822,a,5743,a)]. 50.47/50.38 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(5822,a,5811,a)]. 50.47/50.38 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(5822,a,5812,a)]. 50.47/50.38 5823 -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(3412)]. 50.47/50.38 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(5823,a,5743,a)]. 50.47/50.38 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(5823,a,5811,a)]. 50.47/50.38 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(5823,a,5812,a)]. 50.47/50.38 5824 -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)),f426(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(3739)]. 50.47/50.38 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)),f426(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(5824,b,5743,a)]. 50.53/50.40 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)),f426(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(5824,b,5811,a)]. 50.53/50.40 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)),f426(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(5824,b,5812,a)]. 50.53/50.40 5825 -class_Groups_Ocomm__monoid__mult(A) | -class_Groups_Ocomm__monoid__add(A) | B != C | -c_Nat__Transfer_Onat__set(C) | hAPP(D,f426(D,E,C,B,A)) != hAPP(E,f426(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(3739)]. 50.53/50.40 Derived: -class_Groups_Ocomm__monoid__mult(tc_Code__Numeral_Ocode__numeral) | A != B | -c_Nat__Transfer_Onat__set(B) | hAPP(C,f426(C,D,B,A,tc_Code__Numeral_Ocode__numeral)) != hAPP(D,f426(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(5825,b,5743,a)]. 50.53/50.40 Derived: -class_Groups_Ocomm__monoid__mult(tc_Int_Oint) | A != B | -c_Nat__Transfer_Onat__set(B) | hAPP(C,f426(C,D,B,A,tc_Int_Oint)) != hAPP(D,f426(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(5825,b,5811,a)]. 50.53/50.40 Derived: -class_Groups_Ocomm__monoid__mult(tc_Nat_Onat) | A != B | -c_Nat__Transfer_Onat__set(B) | hAPP(C,f426(C,D,B,A,tc_Nat_Onat)) != hAPP(D,f426(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(5825,b,5812,a)]. 50.53/50.40 5826 -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(3762)]. 50.53/50.40 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(5826,a,5743,a)]. 50.53/50.40 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(5826,a,5811,a)]. 50.73/50.61 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(5826,a,5812,a)]. 50.73/50.61 5827 -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(3811)]. 50.73/50.61 5828 -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(4074)]. 50.73/50.61 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(5828,a,5743,a)]. 50.73/50.61 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(5828,a,5811,a)]. 50.73/50.61 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(5828,a,5812,a)]. 50.73/50.61 5829 -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(4146)]. 50.73/50.61 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(5829,a,5743,a)]. 50.73/50.61 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(5829,a,5811,a)]. 50.77/50.65 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(5829,a,5812,a)]. 50.77/50.65 5830 -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),f479(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(4162)]. 50.77/50.65 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),f479(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(5830,a,5743,a)]. 50.77/50.65 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),f479(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(5830,a,5811,a)]. 50.77/50.65 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),f479(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(5830,a,5812,a)]. 50.77/50.65 5831 -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,f479(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(4162)]. 50.77/50.65 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,f479(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(5831,a,5743,a)]. 50.77/50.65 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,f479(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(5831,a,5811,a)]. 50.77/50.65 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,f479(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(5831,a,5812,a)]. 50.77/50.65 5832 -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(4203)]. 50.83/50.71 5833 -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(4295)]. 50.83/50.71 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(5833,a,5743,a)]. 50.83/50.71 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(5833,a,5811,a)]. 50.83/50.71 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(5833,a,5812,a)]. 50.83/50.71 5834 -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(4295)]. 50.83/50.71 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(5834,a,5743,a)]. 50.83/50.71 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(5834,a,5811,a)]. 50.83/50.75 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(5834,a,5812,a)]. 50.83/50.75 5835 -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(4347)]. 50.83/50.75 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Code__Numeral_Ocode__numeral,tc_Nat_Onat),A),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),B))),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(5835,a,5743,a)]. 50.83/50.75 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Int_Oint,tc_Nat_Onat),A),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),B))),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(5835,a,5811,a)]. 50.83/50.75 Derived: hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(tc_Nat_Onat,tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),A),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_Nat_Onat,tc_Nat_Onat),c_Groups_Oplus__class_Oplus(tc_Nat_Onat)),B))),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(5835,a,5812,a)]. 50.83/50.75 5836 -class_Groups_Ocomm__monoid__add(A) | -class_Orderings_Oord(B) | C != D | E != F | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(B),D),f500(V6,V7,F,E,D,C,B,A))) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),V6),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(B,D),F)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),V7),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(B,C),E)) # label(fact_setsum__ivl__cong) # label(axiom). [clausify(4355)]. 50.83/50.75 Derived: -class_Orderings_Oord(A) | B != C | D != E | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),f500(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),F),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,C),E)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),V6),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,B),D)). [resolve(5836,a,5743,a)]. 50.83/50.75 Derived: -class_Orderings_Oord(A) | B != C | D != E | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),f500(F,V6,E,D,C,B,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,C),E)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),V6),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,B),D)). [resolve(5836,a,5811,a)]. 50.83/50.75 Derived: -class_Orderings_Oord(A) | B != C | D != E | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),C),f500(F,V6,E,D,C,B,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,C),E)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),V6),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,B),D)). [resolve(5836,a,5812,a)]. 50.83/50.75 5837 -class_Groups_Ocomm__monoid__add(A) | -class_Orderings_Oord(B) | C != D | E != F | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(B),f500(V6,V7,F,E,D,C,B,A)),F)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),V6),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(B,D),F)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),V7),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(B,C),E)) # label(fact_setsum__ivl__cong) # label(axiom). [clausify(4355)]. 50.83/50.75 Derived: -class_Orderings_Oord(A) | B != C | D != E | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),f500(F,V6,E,D,C,B,A,tc_Code__Numeral_Ocode__numeral)),E)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),F),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,C),E)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),V6),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,B),D)). [resolve(5837,a,5743,a)]. 50.83/50.75 Derived: -class_Orderings_Oord(A) | B != C | D != E | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),f500(F,V6,E,D,C,B,A,tc_Int_Oint)),E)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),F),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,C),E)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),V6),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,B),D)). [resolve(5837,a,5811,a)]. 50.83/50.75 Derived: -class_Orderings_Oord(A) | B != C | D != E | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),f500(F,V6,E,D,C,B,A,tc_Nat_Onat)),E)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),F),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,C),E)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),V6),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,B),D)). [resolve(5837,a,5812,a)]. 50.83/50.75 5838 -class_Groups_Ocomm__monoid__add(A) | -class_Orderings_Oord(B) | C != D | E != F | hAPP(V6,f500(V6,V7,F,E,D,C,B,A)) != hAPP(V7,f500(V6,V7,F,E,D,C,B,A)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),V6),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(B,D),F)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(B,A),V7),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(B,C),E)) # label(fact_setsum__ivl__cong) # label(axiom). [clausify(4355)]. 50.83/50.75 Derived: -class_Orderings_Oord(A) | B != C | D != E | hAPP(F,f500(F,V6,E,D,C,B,A,tc_Code__Numeral_Ocode__numeral)) != hAPP(V6,f500(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),F),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,C),E)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Code__Numeral_Ocode__numeral),V6),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,B),D)). [resolve(5838,a,5743,a)]. 50.83/50.75 Derived: -class_Orderings_Oord(A) | B != C | D != E | hAPP(F,f500(F,V6,E,D,C,B,A,tc_Int_Oint)) != hAPP(V6,f500(F,V6,E,D,C,B,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,C),E)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Int_Oint),V6),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,B),D)). [resolve(5838,a,5811,a)]. 50.89/50.80 Derived: -class_Orderings_Oord(A) | B != C | D != E | hAPP(F,f500(F,V6,E,D,C,B,A,tc_Nat_Onat)) != hAPP(V6,f500(F,V6,E,D,C,B,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,C),E)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__add__class_Osetsum(A,tc_Nat_Onat),V6),hAPP(c_SetInterval_Oord__class_OatLeastLessThan(A,B),D)). [resolve(5838,a,5812,a)]. 50.89/50.80 5839 -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(4399)]. 50.89/50.80 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(5839,a,5743,a)]. 50.89/50.80 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(5839,a,5811,a)]. 50.89/50.80 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(5839,a,5812,a)]. 50.89/50.80 5840 -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(4432)]. 50.89/50.80 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(5840,a,5743,a)]. 50.89/50.83 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(5840,a,5811,a)]. 50.89/50.83 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(5840,a,5812,a)]. 50.89/50.83 5841 -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(4484)]. 50.89/50.83 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(5841,a,5743,a)]. 50.89/50.83 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(5841,a,5811,a)]. 50.89/50.83 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(5841,a,5812,a)]. 50.89/50.83 5842 -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(4484)]. 50.89/50.83 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(5842,a,5743,a)]. 50.89/50.83 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(5842,a,5811,a)]. 51.00/50.88 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(5842,a,5812,a)]. 51.00/50.88 5843 -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(4487)]. 51.00/50.88 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(5843,a,5743,a)]. 51.00/50.88 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(5843,a,5811,a)]. 51.00/50.88 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(5843,a,5812,a)]. 51.00/50.88 5844 -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(4570)]. 51.00/50.88 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(5844,a,5743,a)]. 51.00/50.88 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(5844,a,5811,a)]. 51.00/50.91 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(5844,a,5812,a)]. 51.00/50.91 5845 -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(4591)]. 51.00/50.91 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(5845,a,5743,a)]. 51.00/50.91 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(5845,a,5811,a)]. 51.00/50.91 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(5845,a,5812,a)]. 51.00/50.91 5846 -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(4614)]. 51.00/50.91 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(5846,a,5743,a)]. 51.00/50.91 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(5846,a,5811,a)]. 51.04/50.93 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(5846,a,5812,a)]. 51.04/50.93 5847 -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(4614)]. 51.04/50.93 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__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(5847,a,5743,a)]. 51.04/50.93 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__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(5847,a,5811,a)]. 51.04/50.93 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__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(5847,a,5812,a)]. 51.04/50.93 5848 -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(4647)]. 51.04/50.93 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(5848,a,5743,a)]. 51.04/50.93 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(5848,a,5811,a)]. 51.08/51.02 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(5848,a,5812,a)]. 51.08/51.02 5849 -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(4688)]. 51.08/51.02 5850 -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(4806)]. 51.08/51.02 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(5850,a,5743,a)]. 51.08/51.02 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(5850,a,5811,a)]. 51.08/51.02 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(5850,a,5812,a)]. 51.21/51.10 5851 -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(4852)]. 51.21/51.10 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(5851,a,5743,a)]. 51.21/51.10 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(5851,a,5811,a)]. 51.21/51.10 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(5851,a,5812,a)]. 51.21/51.10 5852 -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(4903)]. 51.21/51.10 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(5852,a,5743,a)]. 51.21/51.10 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(5852,a,5811,a)]. 51.21/51.10 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(5852,a,5812,a)]. 56.65/56.54 5853 -class_Rings_Olinordered__semiring__1(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),D),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),E)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),F)) | c_Groups_Oone__class_Oone(A) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),E),F) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),E),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),F),D))),C)) # label(fact_convex__bound__le) # label(axiom). [clausify(1082)]. 56.65/56.54 5854 class_Rings_Olinordered__semiring__1(tc_Int_Oint) # label(arity_Int__Oint__Rings_Olinordered__semiring__1) # label(axiom). [assumption]. 56.65/56.54 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),C),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),E)) | c_Groups_Oone__class_Oone(tc_Int_Oint) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),D),E) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),D),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),E),C))),B)). [resolve(5853,a,5854,a)]. 56.65/56.54 5855 class_Rings_Olinordered__ring(tc_Int_Oint) # label(arity_Int__Oint__Rings_Olinordered__ring) # label(axiom). [assumption]. 56.65/56.54 5856 -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(129)]. 56.65/56.54 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(5855,a,5856,a)]. 56.65/56.54 5857 -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(1777)]. 56.65/56.54 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(5857,a,5855,a)]. 56.65/56.54 5858 -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(2682)]. 56.65/56.54 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(5858,a,5855,a)]. 56.65/56.54 5859 -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(4795)]. 62.01/61.89 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(5859,a,5855,a)]. 62.01/61.89 5860 -class_Divides_Oring__div(A) | c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),c_Divides_Odiv__class_Omod(A,C,D)),D) = c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C),D) # label(fact_mod__diff__right__eq) # label(axiom). [clausify(208)]. 62.01/61.89 5861 class_Divides_Oring__div(tc_Int_Oint) # label(arity_Int__Oint__Divides_Oring__div) # label(axiom). [assumption]. 62.01/61.89 5862 -class_Divides_Oring__div(A) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),C)) | hAPP(c_Groups_Ouminus__class_Ouminus(A),hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),C),B)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),C)),B) # label(fact_dvd__neg__div) # label(axiom). [clausify(325)]. 62.01/61.89 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),B)) | hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),B),A)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B)),A). [resolve(5862,a,5861,a)]. 62.01/61.89 5863 -class_Divides_Oring__div(A) | c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),c_Divides_Odiv__class_Omod(A,B,C)),D),C) = c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),D),C) # label(fact_mod__diff__left__eq) # label(axiom). [clausify(664)]. 62.01/61.89 5864 -class_Divides_Oring__div(A) | c_Divides_Odiv__class_Omod(A,hAPP(c_Groups_Ouminus__class_Ouminus(A),c_Divides_Odiv__class_Omod(A,B,C)),C) = c_Divides_Odiv__class_Omod(A,hAPP(c_Groups_Ouminus__class_Ouminus(A),B),C) # label(fact_mod__minus__eq) # label(axiom). [clausify(1654)]. 62.01/61.89 5865 -class_Divides_Oring__div(A) | c_Divides_Odiv__class_Omod(A,B,C) != c_Divides_Odiv__class_Omod(A,D,C) | c_Divides_Odiv__class_Omod(A,hAPP(c_Groups_Ouminus__class_Ouminus(A),B),C) = c_Divides_Odiv__class_Omod(A,hAPP(c_Groups_Ouminus__class_Ouminus(A),D),C) # label(fact_mod__minus__cong) # label(axiom). [clausify(2541)]. 62.01/61.89 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,A,B) != c_Divides_Odiv__class_Omod(tc_Int_Oint,C,B) | c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A),B) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),C),B). [resolve(5865,a,5861,a)]. 62.01/61.89 5866 -class_Divides_Oring__div(A) | c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),c_Divides_Odiv__class_Omod(A,B,C)),c_Divides_Odiv__class_Omod(A,D,C)),C) = c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),D),C) # label(fact_mod__diff__eq) # label(axiom). [clausify(2731)]. 62.01/61.89 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),c_Divides_Odiv__class_Omod(tc_Int_Oint,A,B)),c_Divides_Odiv__class_Omod(tc_Int_Oint,C,B)),B) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),C),B). [resolve(5866,a,5861,a)]. 62.01/61.89 5867 -class_Divides_Oring__div(A) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),C)) | hAPP(c_Groups_Ouminus__class_Ouminus(A),hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),C),B)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(A),C),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)) # label(fact_dvd__div__neg) # label(axiom). [clausify(4098)]. 62.01/61.89 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),B)) | hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),B),A)) = hAPP(hAPP(c_Divides_Odiv__class_Odiv(tc_Int_Oint),B),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)). [resolve(5867,a,5861,a)]. 62.01/61.89 5868 -class_Divides_Oring__div(A) | c_Divides_Odiv__class_Omod(A,B,C) != c_Divides_Odiv__class_Omod(A,D,C) | c_Divides_Odiv__class_Omod(A,E,C) != c_Divides_Odiv__class_Omod(A,F,C) | c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),E),C) = c_Divides_Odiv__class_Omod(A,hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),D),F),C) # label(fact_mod__diff__cong) # label(axiom). [clausify(4889)]. 63.35/63.27 Derived: c_Divides_Odiv__class_Omod(tc_Int_Oint,A,B) != c_Divides_Odiv__class_Omod(tc_Int_Oint,C,B) | c_Divides_Odiv__class_Omod(tc_Int_Oint,D,B) != c_Divides_Odiv__class_Omod(tc_Int_Oint,E,B) | c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),D),B) = c_Divides_Odiv__class_Omod(tc_Int_Oint,hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),C),E),B). [resolve(5868,a,5861,a)]. 63.35/63.27 5869 class_Rings_Olinordered__semiring__strict(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Rings_Olinordered__semiring__strict) # label(axiom). [assumption]. 63.35/63.27 5870 -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(169)]. 63.35/63.27 5871 -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(286)]. 63.35/63.27 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(5869,a,5870,a)]. 63.35/63.27 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(5869,a,5871,a)]. 63.35/63.27 5872 -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(576)]. 63.35/63.27 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(5872,a,5869,a)]. 63.35/63.27 5873 -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(691)]. 63.35/63.27 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(5873,a,5869,a)]. 63.51/63.39 5874 -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(990)]. 63.51/63.39 5875 -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(1380)]. 63.51/63.39 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(5875,a,5869,a)]. 63.51/63.39 5876 -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(1614)]. 63.51/63.39 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),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(5876,a,5869,a)]. 63.51/63.39 5877 -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(1708)]. 63.51/63.39 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(5877,a,5869,a)]. 63.55/63.47 5878 -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(1914)]. 63.55/63.47 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(5878,a,5869,a)]. 63.55/63.47 5879 -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(2034)]. 63.55/63.47 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(5879,a,5869,a)]. 63.55/63.47 5880 -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(2046)]. 63.55/63.47 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_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(5880,a,5869,a)]. 63.55/63.47 5881 -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(2095)]. 63.55/63.47 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__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(5881,a,5869,a)]. 63.65/63.54 5882 class_Rings_Olinordered__semiring__strict(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Olinordered__semiring__strict) # label(axiom). [assumption]. 63.65/63.54 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(5882,a,5876,a)]. 63.65/63.54 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(5882,a,5877,a)]. 63.65/63.54 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(5882,a,5878,a)]. 63.65/63.54 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(5882,a,5881,a)]. 63.65/63.54 5883 -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(2401)]. 63.65/63.54 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Code__Numeral_Ocode__numeral),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(5883,a,5869,a)]. 63.87/63.76 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(5883,a,5882,a)]. 63.87/63.76 5884 -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(2508)]. 63.87/63.76 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(5884,a,5869,a)]. 63.87/63.76 5885 -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(3046)]. 63.87/63.76 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(5885,a,5869,a)]. 63.87/63.76 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(5885,a,5882,a)]. 63.87/63.76 5886 -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(3083)]. 63.87/63.76 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(5886,a,5869,a)]. 63.87/63.76 5887 class_Rings_Olinordered__semiring__strict(tc_Int_Oint) # label(arity_Int__Oint__Rings_Olinordered__semiring__strict) # label(axiom). [assumption]. 63.87/63.77 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(5887,a,5870,a)]. 63.87/63.77 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(5887,a,5871,a)]. 63.87/63.77 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(5887,a,5873,a)]. 63.87/63.77 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(5887,a,5875,a)]. 63.87/63.77 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(5887,a,5876,a)]. 63.87/63.77 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(5887,a,5877,a)]. 63.87/63.77 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(5887,a,5878,a)]. 63.87/63.77 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(5887,a,5879,a)]. 63.87/63.77 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(5887,a,5881,a)]. 66.56/66.47 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),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(5887,a,5883,a)]. 66.56/66.47 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(5887,a,5884,a)]. 66.56/66.47 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(5887,a,5886,a)]. 66.56/66.47 5888 class_Groups_Oordered__ab__group__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Oordered__ab__group__add) # label(axiom). [assumption]. 66.56/66.47 5889 -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(185)]. 66.56/66.47 5890 -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(462)]. 66.56/66.47 5891 -class_Groups_Oordered__ab__group__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Groups_Ouminus__class_Ouminus(A),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) # label(fact_neg__0__less__iff__less) # label(axiom). [clausify(526)]. 66.56/66.47 5892 -class_Groups_Oordered__ab__group__add(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Groups_Ouminus__class_Ouminus(A),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) # label(fact_neg__0__less__iff__less) # label(axiom). [clausify(526)]. 66.56/66.47 5893 -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(594)]. 66.56/66.47 5894 -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(756)]. 66.56/66.47 5895 -class_Groups_Oordered__ab__group__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hBOOL(hAPP(hAPP(c_member(B),f94(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(783)]. 66.56/66.47 5896 -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,f94(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(783)]. 66.56/66.47 5897 -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(1025)]. 66.56/66.47 5898 -class_Groups_Oordered__ab__group__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C)),c_Groups_Ozero__class_Ozero(A))) # label(fact_less__iff__diff__less__0) # label(axiom). [clausify(1396)]. 66.56/66.47 5899 -class_Groups_Oordered__ab__group__add(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C)),c_Groups_Ozero__class_Ozero(A))) # label(fact_less__iff__diff__less__0) # label(axiom). [clausify(1396)]. 66.56/66.47 5900 -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(1559)]. 66.56/66.47 5901 -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(1559)]. 66.56/66.47 5902 -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(1817)]. 66.56/66.47 5903 -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(1817)]. 66.56/66.47 5904 -class_Groups_Oordered__ab__group__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Groups_Ouminus__class_Ouminus(A),B))) # label(fact_neg__0__le__iff__le) # label(axiom). [clausify(1889)]. 66.56/66.47 5905 -class_Groups_Oordered__ab__group__add(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),hAPP(c_Groups_Ouminus__class_Ouminus(A),B))) # label(fact_neg__0__le__iff__le) # label(axiom). [clausify(1889)]. 66.56/66.47 5906 -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(2033)]. 66.56/66.47 5907 -class_Groups_Oordered__ab__group__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(B),C)) | hBOOL(hAPP(hAPP(c_member(B),f244(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(2104)]. 66.56/66.48 5908 -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,f244(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(2104)]. 66.56/66.48 5909 -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(2205)]. 66.56/66.48 5910 -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(2205)]. 66.56/66.48 5911 -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(2576)]. 66.56/66.48 5912 -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(2576)]. 66.56/66.48 5913 -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(2813)]. 66.56/66.48 5914 -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(2813)]. 66.56/66.48 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(5888,a,5889,a)]. 66.56/66.48 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(5888,a,5890,a)]. 66.56/66.48 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5888,a,5891,a)]. 66.56/66.48 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5888,a,5892,a)]. 66.56/66.48 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(5888,a,5893,a)]. 66.56/66.48 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(5888,a,5894,a)]. 66.56/66.49 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f94(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(5888,a,5895,a)]. 66.56/66.49 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,f94(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(5888,a,5896,a)]. 66.56/66.49 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(5888,a,5897,a)]. 66.56/66.49 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5888,a,5898,a)]. 66.56/66.49 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5888,a,5899,a)]. 66.56/66.49 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(5888,a,5900,a)]. 66.56/66.49 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(5888,a,5902,a)]. 66.56/66.49 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(5888,a,5903,a)]. 66.56/66.49 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A))). [resolve(5888,a,5904,a)]. 66.56/66.49 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A))). [resolve(5888,a,5905,a)]. 66.56/66.49 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(5888,a,5906,a)]. 66.56/66.49 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),f244(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(5888,a,5907,a)]. 66.56/66.49 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,f244(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(5888,a,5908,a)]. 66.81/66.70 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(5888,a,5909,a)]. 66.81/66.70 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(5888,a,5912,a)]. 66.81/66.70 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(5888,a,5913,a)]. 66.81/66.70 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(5888,a,5914,a)]. 66.81/66.70 5915 -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(3104)]. 66.81/66.70 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(5915,a,5888,a)]. 66.81/66.70 5916 -class_Groups_Oordered__ab__group__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),c_Groups_Ozero__class_Ozero(A))) # label(fact_neg__less__0__iff__less) # label(axiom). [clausify(3212)]. 66.81/66.70 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5916,a,5888,a)]. 66.81/66.70 5917 -class_Groups_Oordered__ab__group__add(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),c_Groups_Ozero__class_Ozero(A))) # label(fact_neg__less__0__iff__less) # label(axiom). [clausify(3212)]. 66.81/66.70 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5917,a,5888,a)]. 66.81/66.70 5918 -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(3472)]. 66.81/66.70 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(5918,a,5888,a)]. 66.81/66.70 5919 -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(3533)]. 67.28/67.15 Derived: c_Fun_Oinj__on(tc_Int_Oint,tc_Int_Oint,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A). [resolve(5919,a,5888,a)]. 67.28/67.15 5920 -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(3685)]. 67.28/67.15 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(5920,a,5888,a)]. 67.28/67.15 5921 -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(3777)]. 67.28/67.15 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(5921,a,5888,a)]. 67.28/67.15 5922 -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(4282)]. 67.28/67.15 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(5922,a,5888,a)]. 67.28/67.15 5923 -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(4282)]. 67.28/67.15 5924 -class_Groups_Oordered__ab__group__add(A) | hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),D),E) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),D),E)) # label(fact_diff__eq__diff__less__eq) # label(axiom). [clausify(4304)]. 67.28/67.15 Derived: hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),C),D) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),C),D)). [resolve(5924,a,5888,a)]. 67.28/67.15 5925 -class_Groups_Oordered__ab__group__add(A) | hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),D),E) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),D),E)) # label(fact_diff__eq__diff__less__eq) # label(axiom). [clausify(4304)]. 67.28/67.15 Derived: hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),C),D) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),C),D)). [resolve(5925,a,5888,a)]. 67.28/67.15 5926 -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(4335)]. 69.86/69.77 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(5926,a,5888,a)]. 69.86/69.77 5927 -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(4335)]. 69.86/69.77 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(5927,a,5888,a)]. 69.86/69.77 5928 -class_Groups_Oordered__ab__group__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C)),c_Groups_Ozero__class_Ozero(A))) # label(fact_le__iff__diff__le__0) # label(axiom). [clausify(4345)]. 69.86/69.77 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5928,a,5888,a)]. 69.86/69.77 5929 -class_Groups_Oordered__ab__group__add(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C)),c_Groups_Ozero__class_Ozero(A))) # label(fact_le__iff__diff__le__0) # label(axiom). [clausify(4345)]. 69.86/69.77 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5929,a,5888,a)]. 69.86/69.77 5930 -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(4360)]. 69.86/69.77 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(5930,a,5888,a)]. 69.86/69.77 5931 -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(4360)]. 69.86/69.77 5932 -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(4850)]. 69.86/69.77 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(5932,a,5888,a)]. 69.86/69.77 5933 class_Groups_Olinordered__ab__group__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Olinordered__ab__group__add) # label(axiom). [assumption]. 69.86/69.77 5934 -class_Groups_Olinordered__ab__group__add(A) | c_Groups_Ozero__class_Ozero(A) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),B) | c_Groups_Ozero__class_Ozero(A) = B # label(fact_double__zero__sym) # label(axiom). [clausify(193)]. 69.86/69.77 5935 -class_Groups_Olinordered__ab__group__add(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),B) | c_Groups_Ozero__class_Ozero(A) != B # label(fact_double__zero__sym) # label(axiom). [clausify(193)]. 69.86/69.77 5936 -class_Groups_Olinordered__ab__group__add(A) | c_Groups_Ozero__class_Ozero(A) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),B) | c_Groups_Ozero__class_Ozero(A) = B # label(fact_double__eq__0__iff) # label(axiom). [clausify(619)]. 69.86/69.77 5937 -class_Groups_Olinordered__ab__group__add(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),B) | c_Groups_Ozero__class_Ozero(A) != B # label(fact_double__eq__0__iff) # label(axiom). [clausify(619)]. 69.86/69.77 5938 -class_Groups_Olinordered__ab__group__add(A) | c_Groups_Ozero__class_Ozero(A) != B | hAPP(c_Groups_Ouminus__class_Ouminus(A),B) = B # label(fact_neg__equal__zero) # label(axiom). [clausify(903)]. 69.86/69.77 5939 -class_Groups_Olinordered__ab__group__add(A) | c_Groups_Ozero__class_Ozero(A) = B | hAPP(c_Groups_Ouminus__class_Ouminus(A),B) != B # label(fact_neg__equal__zero) # label(axiom). [clausify(903)]. 69.86/69.77 5940 -class_Groups_Olinordered__ab__group__add(A) | hAPP(c_Groups_Ouminus__class_Ouminus(A),hAPP(hAPP(c_Orderings_Oord__class_Omax(A),B),C)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),hAPP(c_Groups_Ouminus__class_Ouminus(A),C)) # label(fact_minus__max__eq__min) # label(axiom). [clausify(1057)]. 69.86/69.77 5941 -class_Groups_Olinordered__ab__group__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) # label(fact_zero__less__double__add__iff__zero__less__single__add) # label(axiom). [clausify(1189)]. 69.86/69.77 5942 -class_Groups_Olinordered__ab__group__add(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) # label(fact_zero__less__double__add__iff__zero__less__single__add) # label(axiom). [clausify(1189)]. 69.86/69.77 5943 -class_Groups_Olinordered__ab__group__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) # label(fact_neg__less__nonneg) # label(axiom). [clausify(1281)]. 69.86/69.77 5944 -class_Groups_Olinordered__ab__group__add(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),B)) # label(fact_neg__less__nonneg) # label(axiom). [clausify(1281)]. 69.86/69.77 5945 -class_Groups_Olinordered__ab__group__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),B)) # label(fact_minus__le__self__iff) # label(axiom). [clausify(1297)]. 69.86/69.77 5946 -class_Groups_Olinordered__ab__group__add(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),B)) # label(fact_minus__le__self__iff) # label(axiom). [clausify(1297)]. 69.86/69.77 5947 -class_Groups_Olinordered__ab__group__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),B)),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),c_Groups_Ozero__class_Ozero(A))) # label(fact_double__add__le__zero__iff__single__add__le__zero) # label(axiom). [clausify(1314)]. 69.86/69.77 5948 -class_Groups_Olinordered__ab__group__add(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),B)),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),c_Groups_Ozero__class_Ozero(A))) # label(fact_double__add__le__zero__iff__single__add__le__zero) # label(axiom). [clausify(1314)]. 69.86/69.78 5949 -class_Groups_Olinordered__ab__group__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = B | c_Big__Operators_Olinorder__class_OMax(A,hAPP(c_Set_Oimage(A,A,c_Groups_Ouminus__class_Ouminus(A)),B)) = hAPP(c_Groups_Ouminus__class_Ouminus(A),c_Big__Operators_Olinorder__class_OMin(A,B)) # label(fact_minus__Min__eq__Max) # label(axiom). [clausify(1363)]. 69.86/69.78 5950 -class_Groups_Olinordered__ab__group__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = B | c_Big__Operators_Olinorder__class_OMin(A,hAPP(c_Set_Oimage(A,A,c_Groups_Ouminus__class_Ouminus(A)),B)) = hAPP(c_Groups_Ouminus__class_Ouminus(A),c_Big__Operators_Olinorder__class_OMax(A,B)) # label(fact_minus__Max__eq__Min) # label(axiom). [clausify(1490)]. 69.86/69.78 5951 -class_Groups_Olinordered__ab__group__add(A) | hAPP(c_Groups_Ouminus__class_Ouminus(A),hAPP(hAPP(c_Orderings_Oord__class_Omin(A),B),C)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),hAPP(c_Groups_Ouminus__class_Ouminus(A),C)) # label(fact_minus__min__eq__max) # label(axiom). [clausify(1775)]. 69.86/69.78 5952 -class_Groups_Olinordered__ab__group__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),c_Groups_Ozero__class_Ozero(A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),hAPP(c_Groups_Ouminus__class_Ouminus(A),B))) # label(fact_le__minus__self__iff) # label(axiom). [clausify(2493)]. 69.86/69.78 5953 -class_Groups_Olinordered__ab__group__add(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),c_Groups_Ozero__class_Ozero(A))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),hAPP(c_Groups_Ouminus__class_Ouminus(A),B))) # label(fact_le__minus__self__iff) # label(axiom). [clausify(2493)]. 69.86/69.78 5954 -class_Groups_Olinordered__ab__group__add(A) | hAPP(c_Groups_Ouminus__class_Ouminus(A),B) != B | c_Groups_Ozero__class_Ozero(A) = B # label(fact_equal__neg__zero) # label(axiom). [clausify(4288)]. 69.86/69.78 5955 -class_Groups_Olinordered__ab__group__add(A) | hAPP(c_Groups_Ouminus__class_Ouminus(A),B) = B | c_Groups_Ozero__class_Ozero(A) != B # label(fact_equal__neg__zero) # label(axiom). [clausify(4288)]. 69.86/69.78 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),A) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A. [resolve(5933,a,5934,a)]. 69.86/69.78 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),A) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A. [resolve(5933,a,5935,a)]. 69.86/69.78 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A | hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A) = A. [resolve(5933,a,5938,a)]. 69.86/69.78 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A) != A. [resolve(5933,a,5939,a)]. 69.86/69.78 Derived: hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),A),B)) = hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B)). [resolve(5933,a,5940,a)]. 69.86/69.78 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),A),A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)). [resolve(5933,a,5941,a)]. 69.86/69.78 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)). [resolve(5933,a,5943,a)]. 69.86/69.78 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)). [resolve(5933,a,5944,a)]. 70.12/69.98 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),A)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)). [resolve(5933,a,5945,a)]. 70.12/69.98 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)). [resolve(5933,a,5946,a)]. 70.12/69.98 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))). [resolve(5933,a,5947,a)]. 70.12/69.98 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) = A | c_Big__Operators_Olinorder__class_OMax(tc_Int_Oint,hAPP(c_Set_Oimage(tc_Int_Oint,tc_Int_Oint,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint)),A)) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Big__Operators_Olinorder__class_OMin(tc_Int_Oint,A)). [resolve(5933,a,5949,a)]. 70.12/69.98 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) = A | c_Big__Operators_Olinorder__class_OMin(tc_Int_Oint,hAPP(c_Set_Oimage(tc_Int_Oint,tc_Int_Oint,c_Groups_Ouminus__class_Ouminus(tc_Int_Oint)),A)) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Big__Operators_Olinorder__class_OMax(tc_Int_Oint,A)). [resolve(5933,a,5950,a)]. 70.12/69.98 Derived: hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),hAPP(hAPP(c_Orderings_Oord__class_Omin(tc_Int_Oint),A),B)) = hAPP(hAPP(c_Orderings_Oord__class_Omax(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B)). [resolve(5933,a,5951,a)]. 70.12/69.98 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),A),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A))). [resolve(5933,a,5952,a)]. 70.12/69.98 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),A),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A))). [resolve(5933,a,5953,a)]. 70.12/69.98 5956 -class_Groups_Olinordered__ab__group__add(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),B))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),B)) # label(fact_zero__le__double__add__iff__zero__le__single__add) # label(axiom). [clausify(4781)]. 70.12/69.98 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),A))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)). [resolve(5956,a,5933,a)]. 70.12/69.98 5957 -class_Groups_Olinordered__ab__group__add(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),B))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),B)) # label(fact_zero__le__double__add__iff__zero__le__single__add) # label(axiom). [clausify(4781)]. 70.12/69.98 5958 -class_Groups_Olinordered__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),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),B)),c_Groups_Ozero__class_Ozero(A))) # label(fact_double__add__less__zero__iff__single__add__less__zero) # label(axiom). [clausify(4797)]. 71.92/71.82 5959 -class_Groups_Olinordered__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),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),B)),c_Groups_Ozero__class_Ozero(A))) # label(fact_double__add__less__zero__iff__single__add__less__zero) # label(axiom). [clausify(4797)]. 71.92/71.82 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(5959,a,5933,a)]. 71.92/71.82 5960 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]. 71.92/71.82 5961 -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(194)]. 71.92/71.82 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(5960,a,5961,a)]. 71.92/71.82 5962 -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(1254)]. 71.92/71.82 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(5962,a,5960,a)]. 71.92/71.82 5963 -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(1254)]. 71.92/71.82 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(5963,a,5960,a)]. 71.92/71.82 5964 -c_Typedef_Otype__definition(A,B,C,D,E) | hAPP(D,hAPP(C,F)) = F # label(fact_type__definition_ORep__inverse) # label(axiom). [clausify(1522)]. 71.92/71.82 Derived: hAPP(c_Int_OAbs__Integ,hAPP(c_Int_ORep__Integ,A)) = A. [resolve(5964,a,5960,a)]. 71.92/71.82 5965 -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(1863)]. 71.92/71.82 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(5965,a,5960,a)]. 71.92/71.82 5966 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]. 71.92/71.82 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(5966,a,5961,a)]. 71.92/71.82 Derived: hAPP(c_Code__Numeral_Oof__nat,hAPP(c_Code__Numeral_Onat__of,A)) = A. [resolve(5966,a,5964,a)]. 71.92/71.82 5967 -c_Typedef_Otype__definition(A,B,C,D,E) | F != V6 | hAPP(C,F) = hAPP(C,V6) # label(fact_type__definition_ORep__inject) # label(axiom). [clausify(2936)]. 71.92/71.82 Derived: A != B | hAPP(c_Int_ORep__Integ,A) = hAPP(c_Int_ORep__Integ,B). [resolve(5967,a,5960,a)]. 71.92/71.82 5968 -c_Typedef_Otype__definition(A,B,C,D,E) | F = V6 | hAPP(C,F) != hAPP(C,V6) # label(fact_type__definition_ORep__inject) # label(axiom). [clausify(2936)]. 82.89/82.80 Derived: A = B | hAPP(c_Int_ORep__Integ,A) != hAPP(c_Int_ORep__Integ,B). [resolve(5968,a,5960,a)]. 82.89/82.80 5969 -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(3067)]. 82.89/82.80 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(5969,a,5960,a)]. 82.89/82.80 5970 -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(3234)]. 82.89/82.80 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(5970,a,5960,a)]. 82.89/82.80 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(5970,a,5966,a)]. 82.89/82.80 5971 -c_Nat__Transfer_Onat__set(hAPP(c_List_Oset(tc_Int_Oint),A)) | c_List_Onat__list(A) # label(fact_nat__list__def) # label(axiom). [clausify(875)]. 82.89/82.80 5972 -c_List_Onat__list(A) | c_List_Oembed__list(c_List_Oreturn__list(A)) = A # label(fact_transfer__nat__int__list__return__embed) # label(axiom). [clausify(222)]. 82.89/82.80 Derived: -c_Nat__Transfer_Onat__set(hAPP(c_List_Oset(tc_Int_Oint),A)) | c_List_Oembed__list(c_List_Oreturn__list(A)) = A. [resolve(5971,b,5972,a)]. 82.89/82.80 5973 c_Nat__Transfer_Onat__set(hAPP(c_List_Oset(tc_Int_Oint),A)) | -c_List_Onat__list(A) # label(fact_nat__list__def) # label(axiom). [clausify(875)]. 82.89/82.80 5974 class_Rings_Oring__1__no__zero__divisors(tc_Int_Oint) # label(arity_Int__Oint__Rings_Oring__1__no__zero__divisors) # label(axiom). [assumption]. 82.89/82.80 5975 -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(244)]. 82.89/82.80 5976 -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(669)]. 82.89/82.80 5977 -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(669)]. 82.89/82.80 5978 -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(1311)]. 82.89/82.80 5979 -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(1311)]. 82.89/82.80 5980 -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(1311)]. 82.89/82.80 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),B). [resolve(5974,a,5975,a)]. 82.89/82.80 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(5974,a,5976,a)]. 82.89/82.80 Derived: c_Groups_Oone__class_Oone(tc_Int_Oint) != hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),A) | hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)) = A | c_Groups_Oone__class_Oone(tc_Int_Oint) = A. [resolve(5974,a,5978,a)]. 84.92/84.81 Derived: c_Groups_Oone__class_Oone(tc_Int_Oint) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),A) | hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)) != A. [resolve(5974,a,5979,a)]. 84.92/84.81 Derived: c_Groups_Oone__class_Oone(tc_Int_Oint) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),A) | c_Groups_Oone__class_Oone(tc_Int_Oint) != A. [resolve(5974,a,5980,a)]. 84.92/84.81 5981 -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 | c_Groups_Oone__class_Oone(A) = B # label(fact_power2__eq__1__iff) # label(axiom). [clausify(4482)]. 84.92/84.81 Derived: c_Groups_Oone__class_Oone(tc_Int_Oint) != hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) | hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)) = A | c_Groups_Oone__class_Oone(tc_Int_Oint) = A. [resolve(5981,a,5974,a)]. 84.92/84.81 5982 -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(4482)]. 84.92/84.81 Derived: c_Groups_Oone__class_Oone(tc_Int_Oint) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) | hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint)) != A. [resolve(5982,a,5974,a)]. 84.92/84.81 5983 -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(4482)]. 84.92/84.81 Derived: c_Groups_Oone__class_Oone(tc_Int_Oint) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),c_Int_OBit0(c_Int_OBit1(c_Int_OPls)))) | c_Groups_Oone__class_Oone(tc_Int_Oint) != A. [resolve(5983,a,5974,a)]. 84.92/84.81 5984 class_Groups_Oab__group__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Oab__group__add) # label(axiom). [assumption]. 84.92/84.81 5985 -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(250)]. 84.92/84.81 5986 -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(476)]. 84.92/84.81 5987 -class_Groups_Oab__group__add(A) | c_Groups_Ozero__class_Ozero(A) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C) | B = C # label(fact_eq__iff__diff__eq__0) # label(axiom). [clausify(482)]. 84.92/84.81 5988 -class_Groups_Oab__group__add(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C) | B != C # label(fact_eq__iff__diff__eq__0) # label(axiom). [clausify(482)]. 84.92/84.81 5989 -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(544)]. 84.97/84.83 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(5984,a,5985,a)]. 84.97/84.83 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(5984,a,5986,a)]. 84.97/84.83 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(5984,a,5989,a)]. 84.97/84.83 5990 -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(812)]. 84.97/84.83 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(5990,a,5984,a)]. 84.97/84.83 5991 -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(812)]. 85.21/85.06 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(5991,a,5984,a)]. 85.21/85.06 5992 -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(1578)]. 85.21/85.06 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__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(5992,a,5984,a)]. 85.21/85.06 5993 -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(1578)]. 85.21/85.06 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | -hBOOL(hAPP(hAPP(c_member(A),C),B)) | hAPP(hAPP(c_Big__Operators_Ocomm__monoid__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(5993,a,5984,a)]. 85.21/85.06 5994 -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(1936)]. 85.21/85.06 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(5994,a,5984,a)]. 85.21/85.06 5995 -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(2416)]. 85.21/85.06 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(5995,a,5984,a)]. 85.52/85.42 5996 -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(2432)]. 85.52/85.42 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(5996,a,5984,a)]. 85.52/85.42 5997 -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(2580)]. 85.52/85.42 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(5997,a,5984,a)]. 85.52/85.42 5998 -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(2745)]. 85.52/85.42 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(5998,a,5984,a)]. 85.52/85.42 5999 -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(3108)]. 85.52/85.42 6000 -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(3657)]. 88.80/88.66 6001 -class_Groups_Oab__group__add(A) | hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),D),E) | D != E | B = C # label(fact_diff__eq__diff__eq) # label(axiom). [clausify(4114)]. 88.80/88.66 Derived: hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),C),D) | C != D | A = B. [resolve(6001,a,5984,a)]. 88.80/88.66 6002 -class_Groups_Oab__group__add(A) | hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),D),E) | D = E | B != C # label(fact_diff__eq__diff__eq) # label(axiom). [clausify(4114)]. 88.80/88.66 Derived: hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B) != hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),C),D) | C = D | A != B. [resolve(6002,a,5984,a)]. 88.80/88.66 6003 -class_Groups_Oab__group__add(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),hAPP(c_Groups_Ouminus__class_Ouminus(A),C)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C) # label(fact_ab__diff__minus) # label(axiom). [clausify(4741)]. 88.80/88.66 6004 class_Groups_Ocancel__semigroup__add(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Ocancel__semigroup__add) # label(axiom). [assumption]. 88.80/88.66 6005 -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(271)]. 88.80/88.66 6006 -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(271)]. 88.80/88.66 6007 -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(2634)]. 88.80/88.66 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),B) | A = C. [resolve(6004,a,6005,a)]. 88.80/88.66 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),B) | A != C. [resolve(6004,a,6006,a)]. 88.80/88.66 6008 -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(3725)]. 88.80/88.66 6009 class_Groups_Ocancel__semigroup__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Ocancel__semigroup__add) # label(axiom). [assumption]. 88.80/88.66 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(6009,a,6005,a)]. 88.80/88.66 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(6009,a,6006,a)]. 88.80/88.66 6010 class_Groups_Ocancel__semigroup__add(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Groups_Ocancel__semigroup__add) # label(axiom). [assumption]. 88.80/88.66 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(6010,a,6005,a)]. 88.80/88.66 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(6010,a,6006,a)]. 88.80/88.66 6011 -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(4552)]. 88.80/88.66 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(6011,a,6004,a)]. 88.80/88.66 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(6011,a,6009,a)]. 90.48/90.35 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(6011,a,6010,a)]. 90.48/90.35 6012 -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(4552)]. 90.48/90.35 6013 class_Rings_Osemiring__1(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Osemiring__1) # label(axiom). [assumption]. 90.48/90.35 6014 -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(284)]. 90.48/90.35 6015 -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(408)]. 90.48/90.35 6016 -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(418)]. 90.48/90.35 6017 -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(752)]. 90.48/90.35 6018 -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(763)]. 90.48/90.35 6019 -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(810)]. 90.48/90.35 6020 -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(1065)]. 90.48/90.35 6021 -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(1113)]. 90.48/90.35 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(6013,a,6014,a)]. 90.48/90.35 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),A)) | hAPP(c_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(6013,a,6015,a)]. 90.48/90.35 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(6013,a,6016,a)]. 90.55/90.40 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(6013,a,6017,a)]. 90.55/90.40 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(6013,a,6018,a)]. 90.55/90.40 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(6013,a,6019,a)]. 90.55/90.40 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(6013,a,6020,a)]. 90.55/90.40 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(6013,a,6021,a)]. 90.55/90.40 6022 class_Rings_Osemiring__1(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Rings_Osemiring__1) # label(axiom). [assumption]. 90.55/90.40 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(6022,a,6014,a)]. 90.55/90.40 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_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(6022,a,6015,a)]. 90.55/90.40 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(6022,a,6016,a)]. 90.55/90.40 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(6022,a,6017,a)]. 90.55/90.40 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(6022,a,6018,a)]. 90.55/90.40 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(6022,a,6019,a)]. 90.71/90.59 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(6022,a,6020,a)]. 90.71/90.59 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(6022,a,6021,a)]. 90.71/90.59 6023 -class_Groups_Ocomm__monoid__mult(A) | -class_Rings_Osemiring__1(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(1845)]. 90.71/90.59 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(6023,b,6013,a)]. 90.71/90.59 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(6023,b,6022,a)]. 90.71/90.59 6024 -class_Rings_Osemiring__1(A) | c_Int_Oiszero(A,c_Groups_Ozero__class_Ozero(A)) # label(fact_iszero__0) # label(axiom). [clausify(1906)]. 90.71/90.59 Derived: c_Int_Oiszero(tc_Nat_Onat,c_Groups_Ozero__class_Ozero(tc_Nat_Onat)). [resolve(6024,a,6013,a)]. 90.71/90.59 Derived: c_Int_Oiszero(tc_Code__Numeral_Ocode__numeral,c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)). [resolve(6024,a,6022,a)]. 90.71/90.59 6025 -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(2292)]. 90.71/90.59 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(6025,a,6013,a)]. 90.71/90.59 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(6025,a,6022,a)]. 90.71/90.59 6026 -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(2436)]. 90.71/90.59 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(6026,a,6013,a)]. 90.71/90.59 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(6026,a,6022,a)]. 90.71/90.59 6027 -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(2534)]. 90.95/90.84 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(6027,a,6013,a)]. 90.95/90.84 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(6027,a,6022,a)]. 90.95/90.84 6028 -class_Rings_Osemiring__1(A) | -c_Int_Oiszero(A,B) | c_Groups_Ozero__class_Ozero(A) = B # label(fact_iszero__def) # label(axiom). [clausify(2704)]. 90.95/90.84 Derived: -c_Int_Oiszero(tc_Nat_Onat,A) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A. [resolve(6028,a,6013,a)]. 90.95/90.84 Derived: -c_Int_Oiszero(tc_Code__Numeral_Ocode__numeral,A) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) = A. [resolve(6028,a,6022,a)]. 90.95/90.84 6029 -class_Rings_Osemiring__1(A) | c_Int_Oiszero(A,B) | c_Groups_Ozero__class_Ozero(A) != B # label(fact_iszero__def) # label(axiom). [clausify(2704)]. 90.95/90.84 Derived: c_Int_Oiszero(tc_Nat_Onat,A) | c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != A. [resolve(6029,a,6013,a)]. 90.95/90.84 Derived: c_Int_Oiszero(tc_Code__Numeral_Ocode__numeral,A) | c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral) != A. [resolve(6029,a,6022,a)]. 90.95/90.84 6030 -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(2853)]. 90.95/90.84 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(6030,a,6013,a)]. 90.95/90.84 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(6030,a,6022,a)]. 90.95/90.84 6031 -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(3030)]. 90.95/90.84 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(6031,a,6013,a)]. 90.95/90.84 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(6031,a,6022,a)]. 90.95/90.84 6032 -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(3351)]. 90.95/90.84 Derived: c_Nat_Osemiring__1__class_Oof__nat__aux(tc_Nat_Onat,A,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),B) = B. [resolve(6032,a,6013,a)]. 90.95/90.84 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(6032,a,6022,a)]. 90.95/90.84 6033 -class_Rings_Osemiring__1(A) | -c_Int_Oiszero(A,c_Groups_Oone__class_Oone(A)) # label(fact_not__iszero__1) # label(axiom). [clausify(3403)]. 91.20/91.08 Derived: -c_Int_Oiszero(tc_Nat_Onat,c_Groups_Oone__class_Oone(tc_Nat_Onat)). [resolve(6033,a,6013,a)]. 91.20/91.08 Derived: -c_Int_Oiszero(tc_Code__Numeral_Ocode__numeral,c_Groups_Oone__class_Oone(tc_Code__Numeral_Ocode__numeral)). [resolve(6033,a,6022,a)]. 91.20/91.08 6034 -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(3595)]. 91.20/91.08 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(6034,a,6013,a)]. 91.20/91.08 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(6034,a,6022,a)]. 91.20/91.08 6035 -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(3876)]. 91.20/91.08 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(6035,a,6013,a)]. 91.20/91.08 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(6035,a,6022,a)]. 91.20/91.08 6036 -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(3909)]. 91.20/91.08 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(6036,a,6013,a)]. 91.20/91.08 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(6036,a,6022,a)]. 91.20/91.08 6037 -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(3961)]. 91.20/91.08 6038 -class_Groups_Oordered__ab__semigroup__add(A) | -class_Rings_Osemiring__1(A) | hBOOL(hAPP(hAPP(c_member(B),f464(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(4008)]. 91.20/91.08 Derived: -class_Groups_Oordered__ab__semigroup__add(tc_Nat_Onat) | hBOOL(hAPP(hAPP(c_member(A),f464(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(6038,b,6013,a)]. 91.20/91.08 Derived: -class_Groups_Oordered__ab__semigroup__add(tc_Code__Numeral_Ocode__numeral) | hBOOL(hAPP(hAPP(c_member(A),f464(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(6038,b,6022,a)]. 91.35/91.21 6039 -class_Groups_Oordered__ab__semigroup__add(A) | -class_Rings_Osemiring__1(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(B,f464(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(4008)]. 91.35/91.21 Derived: -class_Groups_Oordered__ab__semigroup__add(tc_Nat_Onat) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),hAPP(A,f464(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(6039,b,6013,a)]. 91.35/91.21 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,f464(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(6039,b,6022,a)]. 91.35/91.21 6040 -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(4318)]. 91.35/91.21 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(6040,a,6013,a)]. 91.35/91.21 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(6040,a,6022,a)]. 91.35/91.21 6041 class_Rings_Osemiring__1(tc_Int_Oint) # label(arity_Int__Oint__Rings_Osemiring__1) # label(axiom). [assumption]. 91.35/91.21 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(6041,a,6014,a)]. 91.35/91.21 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(6041,a,6015,a)]. 91.35/91.21 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(6041,a,6016,a)]. 91.35/91.22 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(6041,a,6017,a)]. 91.35/91.22 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(6041,a,6018,a)]. 91.35/91.22 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(6041,a,6019,a)]. 91.35/91.22 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(6041,a,6020,a)]. 91.35/91.22 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(6041,a,6021,a)]. 91.35/91.22 Derived: -class_Groups_Ocomm__monoid__mult(tc_Int_Oint) | hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Nat_Onat),B),C)) = hAPP(hAPP(c_Big__Operators_Ocomm__monoid__mult__class_Osetprod(A,tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_Nat_Onat,tc_Int_Oint,A),c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint)),B)),C). [resolve(6041,a,6023,b)]. 91.35/91.22 Derived: c_Int_Oiszero(tc_Int_Oint,c_Groups_Ozero__class_Ozero(tc_Int_Oint)). [resolve(6041,a,6024,a)]. 91.35/91.22 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(6041,a,6025,a)]. 91.35/91.22 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(6041,a,6026,a)]. 91.35/91.22 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(6041,a,6027,a)]. 91.35/91.22 Derived: -c_Int_Oiszero(tc_Int_Oint,A) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A. [resolve(6041,a,6028,a)]. 91.35/91.22 Derived: c_Int_Oiszero(tc_Int_Oint,A) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A. [resolve(6041,a,6029,a)]. 91.35/91.22 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(6041,a,6030,a)]. 91.35/91.22 Derived: c_Nat_Osemiring__1__class_Oof__nat__aux(tc_Int_Oint,A,c_Groups_Ozero__class_Ozero(tc_Nat_Onat),B) = B. [resolve(6041,a,6032,a)]. 91.35/91.22 Derived: -c_Int_Oiszero(tc_Int_Oint,c_Groups_Oone__class_Oone(tc_Int_Oint)). [resolve(6041,a,6033,a)]. 91.35/91.22 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(6041,a,6034,a)]. 91.35/91.22 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(6041,a,6035,a)]. 91.35/91.22 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(6041,a,6036,a)]. 96.12/95.97 Derived: -class_Groups_Oordered__ab__semigroup__add(tc_Int_Oint) | hBOOL(hAPP(hAPP(c_member(A),f464(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(6041,a,6038,b)]. 96.12/95.97 Derived: -class_Groups_Oordered__ab__semigroup__add(tc_Int_Oint) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(A,f464(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(6041,a,6039,b)]. 96.12/95.97 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(6041,a,6040,a)]. 96.12/95.97 6042 -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(4785)]. 96.12/95.97 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(6042,a,6013,a)]. 96.12/95.97 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(6042,a,6022,a)]. 96.12/95.97 Derived: hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),A)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),B)). [resolve(6042,a,6041,a)]. 96.12/95.97 6043 class_Rings_Ocomm__ring__1(tc_Int_Oint) # label(arity_Int__Oint__Rings_Ocomm__ring__1) # label(axiom). [assumption]. 96.12/95.97 6044 -class_Rings_Ocomm__ring__1(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),hAPP(c_Groups_Ouminus__class_Ouminus(A),C)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C) # label(fact_comm__ring__1__class_Onormalizing__ring__rules_I2_J) # label(axiom). [clausify(306)]. 96.12/95.97 6045 -class_Rings_Ocomm__ring__1(A) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),hAPP(c_Groups_Ouminus__class_Ouminus(A),C))) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),C)) # label(fact_dvd__minus__iff) # label(axiom). [clausify(785)]. 96.12/95.97 6046 -class_Rings_Ocomm__ring__1(A) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),hAPP(c_Groups_Ouminus__class_Ouminus(A),C))) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),C)) # label(fact_dvd__minus__iff) # label(axiom). [clausify(785)]. 96.12/95.97 6047 -class_Rings_Ocomm__ring__1(A) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),C)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),C)) # label(fact_minus__dvd__iff) # label(axiom). [clausify(1887)]. 96.12/95.97 6048 -class_Rings_Ocomm__ring__1(A) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),C)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),C)) # label(fact_minus__dvd__iff) # label(axiom). [clausify(1887)]. 96.12/95.97 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B))) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),B)). [resolve(6043,a,6045,a)]. 99.21/99.08 Derived: hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B))) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),B)). [resolve(6043,a,6046,a)]. 99.21/99.08 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),B)) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),B)). [resolve(6043,a,6047,a)]. 99.21/99.08 Derived: hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),B)). [resolve(6043,a,6048,a)]. 99.21/99.08 6049 -class_Rings_Ocomm__ring__1(A) | hAPP(c_Groups_Ouminus__class_Ouminus(A),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),c_Groups_Oone__class_Oone(A))),B) # label(fact_comm__ring__1__class_Onormalizing__ring__rules_I1_J) # label(axiom). [clausify(3074)]. 99.21/99.08 Derived: hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),c_Groups_Oone__class_Oone(tc_Int_Oint))),A). [resolve(6049,a,6043,a)]. 99.21/99.08 6050 class_Groups_Oordered__ab__group__add__abs(tc_Int_Oint) # label(arity_Int__Oint__Groups_Oordered__ab__group__add__abs) # label(axiom). [assumption]. 99.21/99.08 6051 -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(361)]. 99.21/99.08 6052 -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(563)]. 99.21/99.08 6053 -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(624)]. 99.21/99.08 6054 -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(624)]. 99.21/99.08 6055 -class_Groups_Oordered__ab__group__add__abs(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),c_Groups_Ozero__class_Ozero(A))) | hAPP(c_Groups_Oabs__class_Oabs(A),B) = hAPP(c_Groups_Ouminus__class_Ouminus(A),B) # label(fact_abs__of__neg) # label(axiom). [clausify(714)]. 99.21/99.08 6056 -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(795)]. 99.21/99.08 6057 -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(840)]. 99.21/99.08 6058 -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(852)]. 99.21/99.08 6059 -class_Groups_Oordered__ab__group__add__abs(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),c_Groups_Ozero__class_Ozero(A))) | hAPP(c_Groups_Oabs__class_Oabs(A),B) = hAPP(c_Groups_Ouminus__class_Ouminus(A),B) # label(fact_abs__of__nonpos) # label(axiom). [clausify(1199)]. 99.21/99.08 6060 -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(1259)]. 99.21/99.08 6061 -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(1474)]. 99.21/99.08 6062 -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(1606)]. 99.21/99.08 6063 -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(1736)]. 99.21/99.08 6064 -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(1759)]. 99.21/99.08 6065 -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(1835)]. 99.21/99.08 6066 -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(1899)]. 99.21/99.08 6067 -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(1955)]. 99.21/99.08 6068 -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(1976)]. 99.21/99.08 6069 -class_Groups_Oordered__ab__group__add__abs(A) | c_Groups_Ozero__class_Ozero(A) != hAPP(c_Groups_Oabs__class_Oabs(A),B) | c_Groups_Ozero__class_Ozero(A) = B # label(fact_abs__eq__0) # label(axiom). [clausify(2037)]. 99.21/99.08 6070 -class_Groups_Oordered__ab__group__add__abs(A) | c_Groups_Ozero__class_Ozero(A) = hAPP(c_Groups_Oabs__class_Oabs(A),B) | c_Groups_Ozero__class_Ozero(A) != B # label(fact_abs__eq__0) # label(axiom). [clausify(2037)]. 99.21/99.08 6071 -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(2239)]. 99.21/99.08 6072 -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(2372)]. 99.21/99.09 6073 -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(2536)]. 99.21/99.09 6074 -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(2578)]. 99.21/99.09 6075 -class_Groups_Oordered__ab__group__add__abs(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Groups_Oabs__class_Oabs(A),B)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) # label(fact_abs__le__iff) # label(axiom). [clausify(3259)]. 99.21/99.09 6076 -class_Groups_Oordered__ab__group__add__abs(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Groups_Oabs__class_Oabs(A),B)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),C)) # label(fact_abs__le__iff) # label(axiom). [clausify(3259)]. 99.21/99.09 6077 -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)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),C)) # label(fact_abs__le__iff) # label(axiom). [clausify(3259)]. 99.21/99.09 6078 -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(3780)]. 99.21/99.09 6079 -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(3810)]. 99.21/99.09 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(6050,a,6051,a)]. 99.21/99.09 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(6050,a,6052,a)]. 99.21/99.09 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(6050,a,6053,a)]. 99.21/99.09 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(6050,a,6054,a)]. 99.21/99.09 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A). [resolve(6050,a,6055,a)]. 99.21/99.09 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(6050,a,6056,a)]. 99.21/99.10 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(6050,a,6057,a)]. 99.21/99.10 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(6050,a,6058,a)]. 99.21/99.10 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A) = hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A). [resolve(6050,a,6059,a)]. 99.21/99.10 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),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(6050,a,6060,a)]. 99.21/99.10 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(6050,a,6061,a)]. 99.21/99.10 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(6050,a,6062,a)]. 99.21/99.10 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(6050,a,6063,a)]. 99.21/99.10 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(6050,a,6064,a)]. 99.21/99.10 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(6050,a,6065,a)]. 99.21/99.10 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(6050,a,6066,a)]. 99.21/99.10 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(6050,a,6067,a)]. 99.21/99.10 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),A),B)). [resolve(6050,a,6068,a)]. 99.21/99.10 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) != hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A. [resolve(6050,a,6069,a)]. 99.21/99.10 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A. [resolve(6050,a,6070,a)]. 99.21/99.10 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(6050,a,6071,a)]. 99.21/99.10 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(6050,a,6072,a)]. 99.73/99.59 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),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(6050,a,6073,a)]. 99.73/99.59 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(6050,a,6074,a)]. 99.73/99.59 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),A),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A))). [resolve(6050,a,6078,a)]. 99.73/99.59 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(6050,a,6079,a)]. 99.73/99.59 6080 -class_Groups_Oordered__ab__group__add__abs(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Groups_Oabs__class_Oabs(A),B)),c_Groups_Ozero__class_Ozero(A))) | c_Groups_Ozero__class_Ozero(A) = B # label(fact_abs__le__zero__iff) # label(axiom). [clausify(4036)]. 99.73/99.59 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A. [resolve(6080,a,6050,a)]. 99.73/99.59 6081 -class_Groups_Oordered__ab__group__add__abs(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(c_Groups_Oabs__class_Oabs(A),B)),c_Groups_Ozero__class_Ozero(A))) | c_Groups_Ozero__class_Ozero(A) != B # label(fact_abs__le__zero__iff) # label(axiom). [clausify(4036)]. 99.73/99.59 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A)),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A. [resolve(6081,a,6050,a)]. 99.73/99.59 6082 -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(4200)]. 99.73/99.59 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(6082,a,6050,a)]. 99.73/99.59 6083 -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(4302)]. 99.73/99.59 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(6083,a,6050,a)]. 99.73/99.59 6084 -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(4683)]. 99.73/99.59 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(6084,a,6050,a)]. 99.73/99.59 6085 -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(4811)]. 104.85/104.73 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(6085,a,6050,a)]. 104.85/104.73 6086 -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(4882)]. 104.85/104.73 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(6086,a,6050,a)]. 104.85/104.73 6087 -class_Lattices_Obounded__lattice__top(A) | hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(A),B),c_Orderings_Otop__class_Otop(A)) = B # label(fact_inf__top__right) # label(axiom). [clausify(1370)]. 104.85/104.73 6088 class_Lattices_Obounded__lattice__top(tc_HOL_Obool) # label(arity_HOL__Obool__Lattices_Obounded__lattice__top) # label(axiom). [assumption]. 104.85/104.73 Derived: hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool),A),c_Orderings_Otop__class_Otop(tc_HOL_Obool)) = A. [resolve(6087,a,6088,a)]. 104.85/104.73 6089 -class_Lattices_Obounded__lattice__top(A) | c_Orderings_Otop__class_Otop(A) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(A),B),c_Orderings_Otop__class_Otop(A)) # label(fact_sup__top__right) # label(axiom). [clausify(1531)]. 104.85/104.73 Derived: c_Orderings_Otop__class_Otop(tc_HOL_Obool) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool),A),c_Orderings_Otop__class_Otop(tc_HOL_Obool)). [resolve(6089,a,6088,a)]. 104.85/104.73 6090 -class_Lattices_Obounded__lattice__top(A) | c_Orderings_Otop__class_Otop(A) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(A),c_Orderings_Otop__class_Otop(A)),B) # label(fact_sup__top__left) # label(axiom). [clausify(1917)]. 104.85/104.73 Derived: c_Orderings_Otop__class_Otop(tc_HOL_Obool) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool),c_Orderings_Otop__class_Otop(tc_HOL_Obool)),A). [resolve(6090,a,6088,a)]. 104.85/104.73 6091 -class_Lattices_Obounded__lattice(A) | class_Lattices_Obounded__lattice__top(tc_fun(B,A)) # label(arity_fun__Lattices_Obounded__lattice__top) # label(axiom). [clausify(2007)]. 104.85/104.73 Derived: -class_Lattices_Obounded__lattice(A) | hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(B,A)),C),c_Orderings_Otop__class_Otop(tc_fun(B,A))) = C. [resolve(6091,b,6087,a)]. 104.85/104.73 Derived: -class_Lattices_Obounded__lattice(A) | c_Orderings_Otop__class_Otop(tc_fun(B,A)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(B,A)),C),c_Orderings_Otop__class_Otop(tc_fun(B,A))). [resolve(6091,b,6089,a)]. 104.85/104.73 Derived: -class_Lattices_Obounded__lattice(A) | c_Orderings_Otop__class_Otop(tc_fun(B,A)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(B,A)),c_Orderings_Otop__class_Otop(tc_fun(B,A))),C). [resolve(6091,b,6090,a)]. 104.85/104.73 6092 -class_Lattices_Obounded__lattice__top(A) | c_Orderings_Otop__class_Otop(A) != B | c_Orderings_Otop__class_Otop(A) != C | c_Orderings_Otop__class_Otop(A) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(A),B),C) # label(fact_inf__eq__top__iff) # label(axiom). [clausify(4566)]. 104.85/104.73 Derived: c_Orderings_Otop__class_Otop(tc_HOL_Obool) != A | c_Orderings_Otop__class_Otop(tc_HOL_Obool) != B | c_Orderings_Otop__class_Otop(tc_HOL_Obool) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool),A),B). [resolve(6092,a,6088,a)]. 104.85/104.73 Derived: c_Orderings_Otop__class_Otop(tc_fun(A,B)) != C | c_Orderings_Otop__class_Otop(tc_fun(A,B)) != D | c_Orderings_Otop__class_Otop(tc_fun(A,B)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,B)),C),D) | -class_Lattices_Obounded__lattice(B). [resolve(6092,a,6091,b)]. 107.51/107.39 6093 -class_Lattices_Obounded__lattice__top(A) | c_Orderings_Otop__class_Otop(A) = B | c_Orderings_Otop__class_Otop(A) != hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(A),B),C) # label(fact_inf__eq__top__iff) # label(axiom). [clausify(4566)]. 107.51/107.39 Derived: c_Orderings_Otop__class_Otop(tc_HOL_Obool) = A | c_Orderings_Otop__class_Otop(tc_HOL_Obool) != hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool),A),B). [resolve(6093,a,6088,a)]. 107.51/107.39 Derived: c_Orderings_Otop__class_Otop(tc_fun(A,B)) = C | c_Orderings_Otop__class_Otop(tc_fun(A,B)) != hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,B)),C),D) | -class_Lattices_Obounded__lattice(B). [resolve(6093,a,6091,b)]. 107.51/107.39 6094 -class_Lattices_Obounded__lattice__top(A) | c_Orderings_Otop__class_Otop(A) = B | c_Orderings_Otop__class_Otop(A) != hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(A),C),B) # label(fact_inf__eq__top__iff) # label(axiom). [clausify(4566)]. 107.51/107.39 Derived: c_Orderings_Otop__class_Otop(tc_HOL_Obool) = A | c_Orderings_Otop__class_Otop(tc_HOL_Obool) != hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool),B),A). [resolve(6094,a,6088,a)]. 107.51/107.39 Derived: c_Orderings_Otop__class_Otop(tc_fun(A,B)) = C | c_Orderings_Otop__class_Otop(tc_fun(A,B)) != hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,B)),D),C) | -class_Lattices_Obounded__lattice(B). [resolve(6094,a,6091,b)]. 107.51/107.39 6095 -class_Lattices_Obounded__lattice__top(A) | hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(A),c_Orderings_Otop__class_Otop(A)),B) = B # label(fact_inf__top__left) # label(axiom). [clausify(4696)]. 107.51/107.39 Derived: hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_HOL_Obool),c_Orderings_Otop__class_Otop(tc_HOL_Obool)),A) = A. [resolve(6095,a,6088,a)]. 107.51/107.39 Derived: hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,B)),c_Orderings_Otop__class_Otop(tc_fun(A,B))),C) = C | -class_Lattices_Obounded__lattice(B). [resolve(6095,a,6091,b)]. 107.51/107.39 6096 -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(692)]. 107.51/107.39 6097 -class_Lattices_Obounded__lattice(A) | class_Lattices_Obounded__lattice__bot(tc_fun(B,A)) # label(arity_fun__Lattices_Obounded__lattice__bot) # label(axiom). [clausify(397)]. 107.51/107.39 Derived: c_Orderings_Obot__class_Obot(tc_fun(A,B)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,B)),c_Orderings_Obot__class_Obot(tc_fun(A,B))),C) | -class_Lattices_Obounded__lattice(B). [resolve(6096,a,6097,b)]. 107.51/107.39 6098 -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(2938)]. 107.51/107.39 Derived: c_Orderings_Obot__class_Obot(tc_fun(A,B)) = hAPP(hAPP(c_Lattices_Osemilattice__inf__class_Oinf(tc_fun(A,B)),C),c_Orderings_Obot__class_Obot(tc_fun(A,B))) | -class_Lattices_Obounded__lattice(B). [resolve(6098,a,6097,b)]. 107.51/107.39 6099 -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(3484)]. 107.51/107.39 Derived: hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B)),c_Orderings_Obot__class_Obot(tc_fun(A,B))),C) = C | -class_Lattices_Obounded__lattice(B). [resolve(6099,a,6097,b)]. 107.51/107.39 6100 -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(3832)]. 107.51/107.39 Derived: hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B)),C),c_Orderings_Obot__class_Obot(tc_fun(A,B))) = C | -class_Lattices_Obounded__lattice(B). [resolve(6100,a,6097,b)]. 107.51/107.39 6101 class_Lattices_Obounded__lattice__bot(tc_HOL_Obool) # label(arity_HOL__Obool__Lattices_Obounded__lattice__bot) # label(axiom). [assumption]. 115.11/115.00 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(6101,a,6096,a)]. 115.11/115.00 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(6101,a,6098,a)]. 115.11/115.00 Derived: hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool),c_Orderings_Obot__class_Obot(tc_HOL_Obool)),A) = A. [resolve(6101,a,6099,a)]. 115.11/115.00 Derived: hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool),A),c_Orderings_Obot__class_Obot(tc_HOL_Obool)) = A. [resolve(6101,a,6100,a)]. 115.11/115.00 6102 -class_Lattices_Obounded__lattice__bot(A) | c_Orderings_Obot__class_Obot(A) != hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(A),B),C) | c_Orderings_Obot__class_Obot(A) = C # label(fact_sup__eq__bot__iff) # label(axiom). [clausify(4840)]. 115.11/115.00 Derived: c_Orderings_Obot__class_Obot(tc_fun(A,B)) != hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B)),C),D) | c_Orderings_Obot__class_Obot(tc_fun(A,B)) = D | -class_Lattices_Obounded__lattice(B). [resolve(6102,a,6097,b)]. 115.11/115.00 Derived: c_Orderings_Obot__class_Obot(tc_HOL_Obool) != hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool),A),B) | c_Orderings_Obot__class_Obot(tc_HOL_Obool) = B. [resolve(6102,a,6101,a)]. 115.11/115.00 6103 -class_Lattices_Obounded__lattice__bot(A) | c_Orderings_Obot__class_Obot(A) != hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(A),B),C) | c_Orderings_Obot__class_Obot(A) = B # label(fact_sup__eq__bot__iff) # label(axiom). [clausify(4840)]. 115.11/115.00 Derived: c_Orderings_Obot__class_Obot(tc_fun(A,B)) != hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B)),C),D) | c_Orderings_Obot__class_Obot(tc_fun(A,B)) = C | -class_Lattices_Obounded__lattice(B). [resolve(6103,a,6097,b)]. 115.11/115.00 Derived: c_Orderings_Obot__class_Obot(tc_HOL_Obool) != hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool),A),B) | c_Orderings_Obot__class_Obot(tc_HOL_Obool) = A. [resolve(6103,a,6101,a)]. 115.11/115.00 6104 -class_Lattices_Obounded__lattice__bot(A) | c_Orderings_Obot__class_Obot(A) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(A),B),C) | c_Orderings_Obot__class_Obot(A) != C | c_Orderings_Obot__class_Obot(A) != B # label(fact_sup__eq__bot__iff) # label(axiom). [clausify(4840)]. 115.11/115.00 Derived: c_Orderings_Obot__class_Obot(tc_fun(A,B)) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_fun(A,B)),C),D) | c_Orderings_Obot__class_Obot(tc_fun(A,B)) != D | c_Orderings_Obot__class_Obot(tc_fun(A,B)) != C | -class_Lattices_Obounded__lattice(B). [resolve(6104,a,6097,b)]. 115.11/115.00 Derived: c_Orderings_Obot__class_Obot(tc_HOL_Obool) = hAPP(hAPP(c_Lattices_Osemilattice__sup__class_Osup(tc_HOL_Obool),A),B) | c_Orderings_Obot__class_Obot(tc_HOL_Obool) != B | c_Orderings_Obot__class_Obot(tc_HOL_Obool) != A. [resolve(6104,a,6101,a)]. 115.11/115.00 6105 hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Nitpick_Onum(A),c_Nitpick_Oplus__frac(A,B,c_Nitpick_Ouminus__frac(A,C)))),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | -c_Nitpick_Oless__eq__frac(A,B,C) # label(fact_Nitpick_Oless__eq__frac__def) # label(axiom). [clausify(432)]. 115.11/115.00 6106 -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),hAPP(c_Nitpick_Onum(A),c_Nitpick_Oplus__frac(A,B,c_Nitpick_Ouminus__frac(A,C)))),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | c_Nitpick_Oless__eq__frac(A,B,C) # label(fact_Nitpick_Oless__eq__frac__def) # label(axiom). [clausify(432)]. 115.11/115.00 6107 -class_Rings_Olinordered__comm__semiring__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),c_Groups_Ozero__class_Ozero(A)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C))) # label(fact_comm__mult__strict__left__mono) # label(axiom). [clausify(3634)]. 115.11/115.00 6108 class_Rings_Olinordered__comm__semiring__strict(tc_Int_Oint) # label(arity_Int__Oint__Rings_Olinordered__comm__semiring__strict) # label(axiom). [assumption]. 119.90/119.80 6109 class_Rings_Olinordered__comm__semiring__strict(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Rings_Olinordered__comm__semiring__strict) # label(axiom). [assumption]. 119.90/119.80 6110 class_Rings_Olinordered__comm__semiring__strict(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Olinordered__comm__semiring__strict) # label(axiom). [assumption]. 119.90/119.80 6111 -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(4540)]. 119.90/119.80 6112 class_Groups_Oabs__if(tc_Int_Oint) # label(arity_Int__Oint__Groups_Oabs__if) # label(axiom). [assumption]. 119.90/119.80 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),c_Groups_Ozero__class_Ozero(tc_Int_Oint))) | hAPP(c_Groups_Oabs__class_Oabs(tc_Int_Oint),A) = A. [resolve(6111,a,6112,a)]. 119.90/119.80 6113 -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) = hAPP(c_Groups_Ouminus__class_Ouminus(A),B) # label(fact_abs__if) # label(axiom). [clausify(4540)]. 119.90/119.80 6114 -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(1208)]. 119.90/119.80 6115 class_Rings_Oordered__cancel__semiring(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Oordered__cancel__semiring) # label(axiom). [assumption]. 119.90/119.80 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(6114,a,6115,a)]. 119.90/119.80 6116 -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(1266)]. 119.90/119.80 6117 -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(1515)]. 119.90/119.80 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(6117,a,6115,a)]. 119.90/119.80 6118 -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(1515)]. 119.90/119.80 6119 -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(1662)]. 122.88/122.73 6120 class_Rings_Oordered__cancel__semiring(tc_Int_Oint) # label(arity_Int__Oint__Rings_Oordered__cancel__semiring) # label(axiom). [assumption]. 122.88/122.73 6121 -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(3511)]. 122.88/122.73 6122 class_Rings_Oordered__cancel__semiring(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Rings_Oordered__cancel__semiring) # label(axiom). [assumption]. 122.88/122.73 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(6122,a,6114,a)]. 122.88/122.73 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(6122,a,6116,a)]. 122.88/122.73 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(6122,a,6117,a)]. 122.88/122.73 6123 c_Equiv__Relations_Opart__equivp(A,B) | -hBOOL(hAPP(hAPP(B,C),C)) | -c_Predicate_Osymp(A,B) | -c_Predicate_Otransp(A,B) # label(fact_part__equivp__refl__symp__transp) # label(axiom). [clausify(505)]. 122.88/122.73 6124 -c_Equiv__Relations_Opart__equivp(A,B) | hBOOL(hAPP(hAPP(B,f67(B,A)),f67(B,A))) # label(fact_part__equivp__refl__symp__transp) # label(axiom). [clausify(505)]. 122.88/122.73 6125 -c_Equiv__Relations_Opart__equivp(A,B) | c_Predicate_Osymp(A,B) # label(fact_part__equivp__refl__symp__transp) # label(axiom). [clausify(505)]. 122.88/122.73 6126 -c_Equiv__Relations_Opart__equivp(A,B) | c_Predicate_Otransp(A,B) # label(fact_part__equivp__refl__symp__transp) # label(axiom). [clausify(505)]. 122.88/122.73 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | -c_Predicate_Osymp(C,A) | -c_Predicate_Otransp(C,A) | hBOOL(hAPP(hAPP(A,f67(A,C)),f67(A,C))). [resolve(6123,a,6124,a)]. 122.88/122.73 6127 -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(1901)]. 122.88/122.73 6128 -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(2122)]. 122.88/122.73 Derived: -hBOOL(hAPP(hAPP(A,B),C)) | -hBOOL(hAPP(hAPP(A,C),D)) | hBOOL(hAPP(hAPP(A,B),D)) | -hBOOL(hAPP(hAPP(A,E),E)) | -c_Predicate_Osymp(F,A) | -c_Predicate_Otransp(F,A). [resolve(6128,a,6123,a)]. 123.63/123.50 6129 -c_Equiv__Relations_Opart__equivp(A,B) | hBOOL(hAPP(hAPP(B,f312(B,A)),f312(B,A))) # label(fact_part__equivpE) # label(axiom). [clausify(2754)]. 123.63/123.50 Derived: hBOOL(hAPP(hAPP(A,f312(A,B)),f312(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -c_Predicate_Osymp(B,A) | -c_Predicate_Otransp(B,A). [resolve(6129,a,6123,a)]. 123.63/123.50 6130 -c_Equiv__Relations_Opart__equivp(A,B) | c_Predicate_Osymp(A,B) # label(fact_part__equivpE) # label(axiom). [clausify(2754)]. 123.63/123.50 6131 -c_Equiv__Relations_Opart__equivp(A,B) | c_Predicate_Otransp(A,B) # label(fact_part__equivpE) # label(axiom). [clausify(2754)]. 123.63/123.50 6132 -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(3643)]. 123.63/123.50 Derived: -hBOOL(hAPP(hAPP(A,B),C)) | hBOOL(hAPP(hAPP(A,C),B)) | -hBOOL(hAPP(hAPP(A,D),D)) | -c_Predicate_Osymp(E,A) | -c_Predicate_Otransp(E,A). [resolve(6132,a,6123,a)]. 123.63/123.50 6133 -c_Equiv__Relations_Opart__equivp(A,B) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f529(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(4561)]. 123.63/123.50 Derived: hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f529(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_Predicate_Osymp(A,B) | -c_Predicate_Otransp(A,B). [resolve(6133,a,6123,a)]. 123.63/123.50 6134 -c_Equiv__Relations_Oequivp(A,B) | c_Equiv__Relations_Opart__equivp(A,B) # label(fact_equivp__implies__part__equivp) # label(axiom). [clausify(4577)]. 123.63/123.50 Derived: -c_Equiv__Relations_Oequivp(A,B) | hBOOL(hAPP(hAPP(B,f67(B,A)),f67(B,A))). [resolve(6134,b,6124,a)]. 123.63/123.50 Derived: -c_Equiv__Relations_Oequivp(A,B) | c_Predicate_Osymp(A,B). [resolve(6134,b,6125,a)]. 123.63/123.50 Derived: -c_Equiv__Relations_Oequivp(A,B) | c_Predicate_Otransp(A,B). [resolve(6134,b,6126,a)]. 123.63/123.50 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(6134,b,6128,a)]. 123.63/123.50 Derived: -c_Equiv__Relations_Oequivp(A,B) | hBOOL(hAPP(hAPP(B,f312(B,A)),f312(B,A))). [resolve(6134,b,6129,a)]. 123.63/123.50 Derived: -c_Equiv__Relations_Oequivp(A,B) | -hBOOL(hAPP(hAPP(B,C),D)) | hBOOL(hAPP(hAPP(B,D),C)). [resolve(6134,b,6132,a)]. 123.63/123.50 Derived: -c_Equiv__Relations_Oequivp(A,B) | hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f529(B,A)),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_HOL_Obool,tc_fun(A,tc_HOL_Obool)),c_HOL_OEx(A)),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool)),hAPP(c_COMBS(A,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),A),c_fconj),hAPP(hAPP(c_COMBS(A,A,tc_HOL_Obool),B),c_COMBI(A))))),hAPP(hAPP(c_COMBC(tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_fun(A,tc_HOL_Obool)),tc_fun(A,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),tc_fun(tc_fun(A,tc_fun(A,tc_HOL_Obool)),tc_fun(A,tc_HOL_Obool)),tc_fun(A,tc_HOL_Obool)),c_COMBB(tc_fun(A,tc_HOL_Obool),tc_HOL_Obool,A)),c_fequal)),B))))). [resolve(6134,b,6133,a)]. 123.63/123.53 6135 -c_Equiv__Relations_Opart__equivp(A,B) | -hBOOL(hAPP(hAPP(B,C),C)) | hAPP(B,C) != hAPP(B,D) | -hBOOL(hAPP(hAPP(B,D),D)) | hBOOL(hAPP(hAPP(B,D),C)) # label(fact_part__equivp__def) # label(axiom). [clausify(4631)]. 123.63/123.53 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hAPP(A,B) != hAPP(A,C) | -hBOOL(hAPP(hAPP(A,C),C)) | hBOOL(hAPP(hAPP(A,C),B)) | -hBOOL(hAPP(hAPP(A,D),D)) | -c_Predicate_Osymp(E,A) | -c_Predicate_Otransp(E,A). [resolve(6135,a,6123,a)]. 123.63/123.53 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hAPP(A,B) != hAPP(A,C) | -hBOOL(hAPP(hAPP(A,C),C)) | hBOOL(hAPP(hAPP(A,C),B)) | -c_Equiv__Relations_Oequivp(D,A). [resolve(6135,a,6134,b)]. 123.63/123.53 6136 -c_Equiv__Relations_Opart__equivp(A,B) | hBOOL(hAPP(hAPP(B,C),C)) | -hBOOL(hAPP(hAPP(B,D),C)) # label(fact_part__equivp__def) # label(axiom). [clausify(4631)]. 123.63/123.53 Derived: hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,C),B)) | -hBOOL(hAPP(hAPP(A,D),D)) | -c_Predicate_Osymp(E,A) | -c_Predicate_Otransp(E,A). [resolve(6136,a,6123,a)]. 123.63/123.53 Derived: hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,C),B)) | -c_Equiv__Relations_Oequivp(D,A). [resolve(6136,a,6134,b)]. 123.63/123.53 6137 -c_Equiv__Relations_Opart__equivp(A,B) | hAPP(B,C) = hAPP(B,D) | -hBOOL(hAPP(hAPP(B,D),C)) # label(fact_part__equivp__def) # label(axiom). [clausify(4631)]. 123.63/123.53 Derived: hAPP(A,B) = hAPP(A,C) | -hBOOL(hAPP(hAPP(A,C),B)) | -hBOOL(hAPP(hAPP(A,D),D)) | -c_Predicate_Osymp(E,A) | -c_Predicate_Otransp(E,A). [resolve(6137,a,6123,a)]. 123.63/123.53 Derived: hAPP(A,B) = hAPP(A,C) | -hBOOL(hAPP(hAPP(A,C),B)) | -c_Equiv__Relations_Oequivp(D,A). [resolve(6137,a,6134,b)]. 123.63/123.53 6138 -c_Equiv__Relations_Opart__equivp(A,B) | hBOOL(hAPP(hAPP(B,C),C)) | -hBOOL(hAPP(hAPP(B,C),D)) # label(fact_part__equivp__def) # label(axiom). [clausify(4631)]. 123.63/123.53 Derived: hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,B),C)) | -hBOOL(hAPP(hAPP(A,D),D)) | -c_Predicate_Osymp(E,A) | -c_Predicate_Otransp(E,A). [resolve(6138,a,6123,a)]. 123.63/123.53 Derived: hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,B),C)) | -c_Equiv__Relations_Oequivp(D,A). [resolve(6138,a,6134,b)]. 123.63/123.53 6139 -c_Equiv__Relations_Opart__equivp(A,B) | hBOOL(hAPP(hAPP(B,f534(B,A)),f534(B,A))) # label(fact_part__equivp__def) # label(axiom). [clausify(4631)]. 123.63/123.53 Derived: hBOOL(hAPP(hAPP(A,f534(A,B)),f534(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -c_Predicate_Osymp(B,A) | -c_Predicate_Otransp(B,A). [resolve(6139,a,6123,a)]. 123.63/123.53 Derived: hBOOL(hAPP(hAPP(A,f534(A,B)),f534(A,B))) | -c_Equiv__Relations_Oequivp(B,A). [resolve(6139,a,6134,b)]. 123.63/123.53 6140 c_Equiv__Relations_Opart__equivp(A,B) | hBOOL(hAPP(hAPP(B,f536(B,A)),f536(B,A))) | hBOOL(hAPP(hAPP(B,f535(B,A)),f536(B,A))) | -hBOOL(hAPP(hAPP(B,C),C)) # label(fact_part__equivp__def) # label(axiom). [clausify(4631)]. 123.63/123.53 Derived: hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | hBOOL(hAPP(hAPP(A,f67(A,B)),f67(A,B))). [resolve(6140,a,6124,a)]. 123.63/123.53 Derived: hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | c_Predicate_Osymp(B,A). [resolve(6140,a,6125,a)]. 123.63/123.53 Derived: hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | c_Predicate_Otransp(B,A). [resolve(6140,a,6126,a)]. 123.63/123.53 Derived: hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),E)) | -hBOOL(hAPP(hAPP(A,E),F)) | hBOOL(hAPP(hAPP(A,D),F)). [resolve(6140,a,6128,a)]. 123.63/123.53 Derived: hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | hBOOL(hAPP(hAPP(A,f312(A,B)),f312(A,B))). [resolve(6140,a,6129,a)]. 123.63/123.53 Derived: hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),E)) | hBOOL(hAPP(hAPP(A,E),D)). [resolve(6140,a,6132,a)]. 123.63/123.54 Derived: hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | hBOOL(hAPP(hAPP(c_member(tc_fun(B,tc_HOL_Obool)),f529(A,B)),hAPP(hAPP(c_COMBB(tc_fun(B,tc_HOL_Obool),tc_HOL_Obool,tc_fun(B,tc_HOL_Obool)),c_HOL_OEx(B)),hAPP(hAPP(c_COMBB(tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_HOL_Obool)),hAPP(c_COMBS(B,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),B),c_fconj),hAPP(hAPP(c_COMBS(B,B,tc_HOL_Obool),A),c_COMBI(B))))),hAPP(hAPP(c_COMBC(tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_fun(B,tc_HOL_Obool)),tc_fun(B,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(B,tc_HOL_Obool),tc_HOL_Obool),tc_fun(tc_fun(B,tc_fun(B,tc_HOL_Obool)),tc_fun(B,tc_HOL_Obool)),tc_fun(B,tc_HOL_Obool)),c_COMBB(tc_fun(B,tc_HOL_Obool),tc_HOL_Obool,B)),c_fequal)),A))))). [resolve(6140,a,6133,a)]. 123.63/123.54 Derived: hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),D)) | hAPP(A,D) != hAPP(A,E) | -hBOOL(hAPP(hAPP(A,E),E)) | hBOOL(hAPP(hAPP(A,E),D)). [resolve(6140,a,6135,a)]. 123.63/123.54 Derived: hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | hBOOL(hAPP(hAPP(A,D),D)) | -hBOOL(hAPP(hAPP(A,E),D)). [resolve(6140,a,6136,a)]. 123.63/123.54 Derived: hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | hAPP(A,D) = hAPP(A,E) | -hBOOL(hAPP(hAPP(A,E),D)). [resolve(6140,a,6137,a)]. 123.63/123.54 Derived: hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | hBOOL(hAPP(hAPP(A,D),D)) | -hBOOL(hAPP(hAPP(A,D),E)). [resolve(6140,a,6138,a)]. 123.63/123.54 Derived: hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | hBOOL(hAPP(hAPP(A,f534(A,B)),f534(A,B))). [resolve(6140,a,6139,a)]. 123.63/123.54 6141 c_Equiv__Relations_Opart__equivp(A,B) | hAPP(B,f536(B,A)) = hAPP(B,f535(B,A)) | hBOOL(hAPP(hAPP(B,f535(B,A)),f536(B,A))) | -hBOOL(hAPP(hAPP(B,C),C)) # label(fact_part__equivp__def) # label(axiom). [clausify(4631)]. 123.63/123.54 Derived: hAPP(A,f536(A,B)) = hAPP(A,f535(A,B)) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | hBOOL(hAPP(hAPP(A,f67(A,B)),f67(A,B))). [resolve(6141,a,6124,a)]. 123.63/123.54 Derived: hAPP(A,f536(A,B)) = hAPP(A,f535(A,B)) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | c_Predicate_Osymp(B,A). [resolve(6141,a,6125,a)]. 123.63/123.54 Derived: hAPP(A,f536(A,B)) = hAPP(A,f535(A,B)) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | c_Predicate_Otransp(B,A). [resolve(6141,a,6126,a)]. 123.63/123.54 Derived: hAPP(A,f536(A,B)) = hAPP(A,f535(A,B)) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),E)) | -hBOOL(hAPP(hAPP(A,E),F)) | hBOOL(hAPP(hAPP(A,D),F)). [resolve(6141,a,6128,a)]. 123.63/123.54 Derived: hAPP(A,f536(A,B)) = hAPP(A,f535(A,B)) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | hBOOL(hAPP(hAPP(A,f312(A,B)),f312(A,B))). [resolve(6141,a,6129,a)]. 123.63/123.54 Derived: hAPP(A,f536(A,B)) = hAPP(A,f535(A,B)) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),E)) | hBOOL(hAPP(hAPP(A,E),D)). [resolve(6141,a,6132,a)]. 123.63/123.54 Derived: hAPP(A,f536(A,B)) = hAPP(A,f535(A,B)) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | hBOOL(hAPP(hAPP(c_member(tc_fun(B,tc_HOL_Obool)),f529(A,B)),hAPP(hAPP(c_COMBB(tc_fun(B,tc_HOL_Obool),tc_HOL_Obool,tc_fun(B,tc_HOL_Obool)),c_HOL_OEx(B)),hAPP(hAPP(c_COMBB(tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_HOL_Obool)),hAPP(c_COMBS(B,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),B),c_fconj),hAPP(hAPP(c_COMBS(B,B,tc_HOL_Obool),A),c_COMBI(B))))),hAPP(hAPP(c_COMBC(tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_fun(B,tc_HOL_Obool)),tc_fun(B,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(B,tc_HOL_Obool),tc_HOL_Obool),tc_fun(tc_fun(B,tc_fun(B,tc_HOL_Obool)),tc_fun(B,tc_HOL_Obool)),tc_fun(B,tc_HOL_Obool)),c_COMBB(tc_fun(B,tc_HOL_Obool),tc_HOL_Obool,B)),c_fequal)),A))))). [resolve(6141,a,6133,a)]. 123.63/123.55 Derived: hAPP(A,f536(A,B)) = hAPP(A,f535(A,B)) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),D)) | hAPP(A,D) != hAPP(A,E) | -hBOOL(hAPP(hAPP(A,E),E)) | hBOOL(hAPP(hAPP(A,E),D)). [resolve(6141,a,6135,a)]. 123.63/123.55 Derived: hAPP(A,f536(A,B)) = hAPP(A,f535(A,B)) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | hBOOL(hAPP(hAPP(A,D),D)) | -hBOOL(hAPP(hAPP(A,E),D)). [resolve(6141,a,6136,a)]. 123.63/123.55 Derived: hAPP(A,f536(A,B)) = hAPP(A,f535(A,B)) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | hAPP(A,D) = hAPP(A,E) | -hBOOL(hAPP(hAPP(A,E),D)). [resolve(6141,a,6137,a)]. 123.63/123.55 Derived: hAPP(A,f536(A,B)) = hAPP(A,f535(A,B)) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | hBOOL(hAPP(hAPP(A,D),D)) | -hBOOL(hAPP(hAPP(A,D),E)). [resolve(6141,a,6138,a)]. 123.63/123.55 Derived: hAPP(A,f536(A,B)) = hAPP(A,f535(A,B)) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | hBOOL(hAPP(hAPP(A,f534(A,B)),f534(A,B))). [resolve(6141,a,6139,a)]. 123.63/123.55 6142 c_Equiv__Relations_Opart__equivp(A,B) | hBOOL(hAPP(hAPP(B,f535(B,A)),f535(B,A))) | hBOOL(hAPP(hAPP(B,f535(B,A)),f536(B,A))) | -hBOOL(hAPP(hAPP(B,C),C)) # label(fact_part__equivp__def) # label(axiom). [clausify(4631)]. 123.63/123.55 Derived: hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | hBOOL(hAPP(hAPP(A,f67(A,B)),f67(A,B))). [resolve(6142,a,6124,a)]. 123.63/123.55 Derived: hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | c_Predicate_Osymp(B,A). [resolve(6142,a,6125,a)]. 123.63/123.55 Derived: hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | c_Predicate_Otransp(B,A). [resolve(6142,a,6126,a)]. 123.63/123.55 Derived: hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),E)) | -hBOOL(hAPP(hAPP(A,E),F)) | hBOOL(hAPP(hAPP(A,D),F)). [resolve(6142,a,6128,a)]. 123.63/123.55 Derived: hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | hBOOL(hAPP(hAPP(A,f312(A,B)),f312(A,B))). [resolve(6142,a,6129,a)]. 123.63/123.55 Derived: hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),E)) | hBOOL(hAPP(hAPP(A,E),D)). [resolve(6142,a,6132,a)]. 123.63/123.55 Derived: hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | hBOOL(hAPP(hAPP(c_member(tc_fun(B,tc_HOL_Obool)),f529(A,B)),hAPP(hAPP(c_COMBB(tc_fun(B,tc_HOL_Obool),tc_HOL_Obool,tc_fun(B,tc_HOL_Obool)),c_HOL_OEx(B)),hAPP(hAPP(c_COMBB(tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_HOL_Obool)),hAPP(c_COMBS(B,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),B),c_fconj),hAPP(hAPP(c_COMBS(B,B,tc_HOL_Obool),A),c_COMBI(B))))),hAPP(hAPP(c_COMBC(tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_fun(B,tc_HOL_Obool)),tc_fun(B,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(B,tc_HOL_Obool),tc_HOL_Obool),tc_fun(tc_fun(B,tc_fun(B,tc_HOL_Obool)),tc_fun(B,tc_HOL_Obool)),tc_fun(B,tc_HOL_Obool)),c_COMBB(tc_fun(B,tc_HOL_Obool),tc_HOL_Obool,B)),c_fequal)),A))))). [resolve(6142,a,6133,a)]. 123.63/123.55 Derived: hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),D)) | hAPP(A,D) != hAPP(A,E) | -hBOOL(hAPP(hAPP(A,E),E)) | hBOOL(hAPP(hAPP(A,E),D)). [resolve(6142,a,6135,a)]. 123.63/123.55 Derived: hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | hBOOL(hAPP(hAPP(A,D),D)) | -hBOOL(hAPP(hAPP(A,E),D)). [resolve(6142,a,6136,a)]. 123.63/123.55 Derived: hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | hAPP(A,D) = hAPP(A,E) | -hBOOL(hAPP(hAPP(A,E),D)). [resolve(6142,a,6137,a)]. 123.63/123.56 Derived: hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | hBOOL(hAPP(hAPP(A,D),D)) | -hBOOL(hAPP(hAPP(A,D),E)). [resolve(6142,a,6138,a)]. 123.63/123.56 Derived: hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | hBOOL(hAPP(hAPP(A,f534(A,B)),f534(A,B))). [resolve(6142,a,6139,a)]. 123.63/123.56 6143 c_Equiv__Relations_Opart__equivp(A,B) | -hBOOL(hAPP(hAPP(B,f536(B,A)),f536(B,A))) | hAPP(B,f536(B,A)) != hAPP(B,f535(B,A)) | -hBOOL(hAPP(hAPP(B,f535(B,A)),f535(B,A))) | -hBOOL(hAPP(hAPP(B,f535(B,A)),f536(B,A))) | -hBOOL(hAPP(hAPP(B,C),C)) # label(fact_part__equivp__def) # label(axiom). [clausify(4631)]. 123.63/123.56 Derived: -hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hAPP(A,f536(A,B)) != hAPP(A,f535(A,B)) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | hBOOL(hAPP(hAPP(A,f67(A,B)),f67(A,B))). [resolve(6143,a,6124,a)]. 123.63/123.56 Derived: -hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hAPP(A,f536(A,B)) != hAPP(A,f535(A,B)) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | c_Predicate_Osymp(B,A). [resolve(6143,a,6125,a)]. 123.63/123.56 Derived: -hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hAPP(A,f536(A,B)) != hAPP(A,f535(A,B)) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | c_Predicate_Otransp(B,A). [resolve(6143,a,6126,a)]. 123.63/123.56 Derived: -hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hAPP(A,f536(A,B)) != hAPP(A,f535(A,B)) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),E)) | -hBOOL(hAPP(hAPP(A,E),F)) | hBOOL(hAPP(hAPP(A,D),F)). [resolve(6143,a,6128,a)]. 123.63/123.56 Derived: -hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hAPP(A,f536(A,B)) != hAPP(A,f535(A,B)) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | hBOOL(hAPP(hAPP(A,f312(A,B)),f312(A,B))). [resolve(6143,a,6129,a)]. 123.63/123.56 Derived: -hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hAPP(A,f536(A,B)) != hAPP(A,f535(A,B)) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),E)) | hBOOL(hAPP(hAPP(A,E),D)). [resolve(6143,a,6132,a)]. 123.63/123.56 Derived: -hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hAPP(A,f536(A,B)) != hAPP(A,f535(A,B)) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | hBOOL(hAPP(hAPP(c_member(tc_fun(B,tc_HOL_Obool)),f529(A,B)),hAPP(hAPP(c_COMBB(tc_fun(B,tc_HOL_Obool),tc_HOL_Obool,tc_fun(B,tc_HOL_Obool)),c_HOL_OEx(B)),hAPP(hAPP(c_COMBB(tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_HOL_Obool)),hAPP(c_COMBS(B,tc_HOL_Obool,tc_HOL_Obool),hAPP(hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),B),c_fconj),hAPP(hAPP(c_COMBS(B,B,tc_HOL_Obool),A),c_COMBI(B))))),hAPP(hAPP(c_COMBC(tc_fun(B,tc_HOL_Obool),tc_fun(B,tc_fun(B,tc_HOL_Obool)),tc_fun(B,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_fun(B,tc_HOL_Obool),tc_HOL_Obool),tc_fun(tc_fun(B,tc_fun(B,tc_HOL_Obool)),tc_fun(B,tc_HOL_Obool)),tc_fun(B,tc_HOL_Obool)),c_COMBB(tc_fun(B,tc_HOL_Obool),tc_HOL_Obool,B)),c_fequal)),A))))). [resolve(6143,a,6133,a)]. 123.63/123.56 Derived: -hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hAPP(A,f536(A,B)) != hAPP(A,f535(A,B)) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),D)) | hAPP(A,D) != hAPP(A,E) | -hBOOL(hAPP(hAPP(A,E),E)) | hBOOL(hAPP(hAPP(A,E),D)). [resolve(6143,a,6135,a)]. 123.63/123.56 Derived: -hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hAPP(A,f536(A,B)) != hAPP(A,f535(A,B)) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | hBOOL(hAPP(hAPP(A,D),D)) | -hBOOL(hAPP(hAPP(A,E),D)). [resolve(6143,a,6136,a)]. 123.63/123.56 Derived: -hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hAPP(A,f536(A,B)) != hAPP(A,f535(A,B)) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | hAPP(A,D) = hAPP(A,E) | -hBOOL(hAPP(hAPP(A,E),D)). [resolve(6143,a,6137,a)]. 128.63/128.49 Derived: -hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hAPP(A,f536(A,B)) != hAPP(A,f535(A,B)) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | hBOOL(hAPP(hAPP(A,D),D)) | -hBOOL(hAPP(hAPP(A,D),E)). [resolve(6143,a,6138,a)]. 128.63/128.49 Derived: -hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hAPP(A,f536(A,B)) != hAPP(A,f535(A,B)) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | hBOOL(hAPP(hAPP(A,f534(A,B)),f534(A,B))). [resolve(6143,a,6139,a)]. 128.63/128.49 6144 class_Rings_Olinordered__semiring__1__strict(tc_Int_Oint) # label(arity_Int__Oint__Rings_Olinordered__semiring__1__strict) # label(axiom). [assumption]. 128.63/128.49 6145 -class_Rings_Olinordered__semiring__1__strict(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),E)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),F)) | c_Groups_Oone__class_Oone(A) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),E),F) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),E),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),F),D))),C)) # label(fact_convex__bound__lt) # label(axiom). [clausify(522)]. 128.63/128.49 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),C),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Int_Oint),c_Groups_Ozero__class_Ozero(tc_Int_Oint)),E)) | c_Groups_Oone__class_Oone(tc_Int_Oint) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),D),E) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),D),A)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),E),C))),B)). [resolve(6144,a,6145,a)]. 128.63/128.49 6146 -class_Groups_Olinordered__ab__semigroup__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = B | c_Big__Operators_Olinorder__class_OMin(A,hAPP(c_Set_OCollect(A),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_HOL_Obool,A),c_HOL_OEx(A)),hAPP(hAPP(c_COMBC(A,tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(A,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool)),A),c_COMBS(A,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),A),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),A),c_fconj)),hAPP(hAPP(c_COMBC(A,tc_fun(A,A),tc_fun(A,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_fun(tc_fun(A,A),tc_fun(A,tc_HOL_Obool)),A),c_COMBB(A,tc_HOL_Obool,A)),c_fequal)),hAPP(c_Groups_Oplus__class_Oplus(A),C))))),hAPP(hAPP(c_COMBC(A,tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),c_member(A)),B))))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),c_Big__Operators_Olinorder__class_OMin(A,B)) # label(fact_add__Min__commute) # label(axiom). [clausify(3386)]. 128.63/128.49 6147 class_Groups_Olinordered__ab__semigroup__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Olinordered__ab__semigroup__add) # label(axiom). [assumption]. 128.63/128.49 6148 class_Groups_Olinordered__ab__semigroup__add(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Groups_Olinordered__ab__semigroup__add) # label(axiom). [assumption]. 128.63/128.49 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_Olinorder__class_OMin(tc_Int_Oint,hAPP(c_Set_OCollect(tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_fun(tc_Int_Oint,tc_HOL_Obool),tc_HOL_Obool,tc_Int_Oint),c_HOL_OEx(tc_Int_Oint)),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_HOL_Obool),tc_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Int_Oint,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_Int_Oint,tc_HOL_Obool),tc_fun(tc_Int_Oint,tc_HOL_Obool)),tc_Int_Oint),c_COMBS(tc_Int_Oint,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Int_Oint,tc_HOL_Obool),tc_fun(tc_Int_Oint,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_Int_Oint),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Int_Oint),c_fconj)),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint),tc_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Int_Oint,tc_HOL_Obool),tc_fun(tc_fun(tc_Int_Oint,tc_Int_Oint),tc_fun(tc_Int_Oint,tc_HOL_Obool)),tc_Int_Oint),c_COMBB(tc_Int_Oint,tc_HOL_Obool,tc_Int_Oint)),c_fequal)),hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B))))),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_HOL_Obool),tc_HOL_Obool),c_member(tc_Int_Oint)),A))))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),c_Big__Operators_Olinorder__class_OMin(tc_Int_Oint,A)). [resolve(6146,a,6147,a)]. 129.02/128.86 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)) = A | c_Big__Operators_Olinorder__class_OMin(tc_Code__Numeral_Ocode__numeral,hAPP(c_Set_OCollect(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBB(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool),tc_HOL_Obool,tc_Code__Numeral_Ocode__numeral),c_HOL_OEx(tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_COMBC(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),tc_Code__Numeral_Ocode__numeral),c_COMBS(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_Code__Numeral_Ocode__numeral),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),c_fconj)),hAPP(hAPP(c_COMBC(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool),tc_fun(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),tc_Code__Numeral_Ocode__numeral),c_COMBB(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool,tc_Code__Numeral_Ocode__numeral)),c_fequal)),hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),B))))),hAPP(hAPP(c_COMBC(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool),tc_HOL_Obool),c_member(tc_Code__Numeral_Ocode__numeral)),A))))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),B),c_Big__Operators_Olinorder__class_OMin(tc_Code__Numeral_Ocode__numeral,A)). [resolve(6146,a,6148,a)]. 129.02/128.86 6149 class_Groups_Olinordered__ab__semigroup__add(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Olinordered__ab__semigroup__add) # label(axiom). [assumption]. 129.02/128.86 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = A | c_Big__Operators_Olinorder__class_OMin(tc_Nat_Onat,hAPP(c_Set_OCollect(tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool,tc_Nat_Onat),c_HOL_OEx(tc_Nat_Onat)),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),tc_Nat_Onat),c_COMBS(tc_Nat_Onat,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_Nat_Onat),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Nat_Onat),c_fconj)),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_fun(tc_Nat_Onat,tc_Nat_Onat),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),tc_Nat_Onat),c_COMBB(tc_Nat_Onat,tc_HOL_Obool,tc_Nat_Onat)),c_fequal)),hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B))))),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool),c_member(tc_Nat_Onat)),A))))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),c_Big__Operators_Olinorder__class_OMin(tc_Nat_Onat,A)). [resolve(6149,a,6146,a)]. 129.10/128.96 6150 -class_Groups_Olinordered__ab__semigroup__add(A) | -hBOOL(hAPP(c_Finite__Set_Ofinite(A),B)) | c_Orderings_Obot__class_Obot(tc_fun(A,tc_HOL_Obool)) = B | c_Big__Operators_Olinorder__class_OMax(A,hAPP(c_Set_OCollect(A),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_HOL_Obool,A),c_HOL_OEx(A)),hAPP(hAPP(c_COMBC(A,tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(A,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_HOL_Obool)),A),c_COMBS(A,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_fun(A,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),A),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),A),c_fconj)),hAPP(hAPP(c_COMBC(A,tc_fun(A,A),tc_fun(A,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(A,tc_HOL_Obool),tc_fun(tc_fun(A,A),tc_fun(A,tc_HOL_Obool)),A),c_COMBB(A,tc_HOL_Obool,A)),c_fequal)),hAPP(c_Groups_Oplus__class_Oplus(A),C))))),hAPP(hAPP(c_COMBC(A,tc_fun(A,tc_HOL_Obool),tc_HOL_Obool),c_member(A)),B))))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),c_Big__Operators_Olinorder__class_OMax(A,B)) # label(fact_add__Max__commute) # label(axiom). [clausify(4632)]. 129.10/128.96 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Int_Oint),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Int_Oint,tc_HOL_Obool)) = A | c_Big__Operators_Olinorder__class_OMax(tc_Int_Oint,hAPP(c_Set_OCollect(tc_Int_Oint),hAPP(hAPP(c_COMBB(tc_fun(tc_Int_Oint,tc_HOL_Obool),tc_HOL_Obool,tc_Int_Oint),c_HOL_OEx(tc_Int_Oint)),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_HOL_Obool),tc_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Int_Oint,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_Int_Oint,tc_HOL_Obool),tc_fun(tc_Int_Oint,tc_HOL_Obool)),tc_Int_Oint),c_COMBS(tc_Int_Oint,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Int_Oint,tc_HOL_Obool),tc_fun(tc_Int_Oint,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_Int_Oint),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Int_Oint),c_fconj)),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_Int_Oint),tc_fun(tc_Int_Oint,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Int_Oint,tc_HOL_Obool),tc_fun(tc_fun(tc_Int_Oint,tc_Int_Oint),tc_fun(tc_Int_Oint,tc_HOL_Obool)),tc_Int_Oint),c_COMBB(tc_Int_Oint,tc_HOL_Obool,tc_Int_Oint)),c_fequal)),hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B))))),hAPP(hAPP(c_COMBC(tc_Int_Oint,tc_fun(tc_Int_Oint,tc_HOL_Obool),tc_HOL_Obool),c_member(tc_Int_Oint)),A))))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),c_Big__Operators_Olinorder__class_OMax(tc_Int_Oint,A)). [resolve(6150,a,6147,a)]. 129.10/128.96 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Code__Numeral_Ocode__numeral),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)) = A | c_Big__Operators_Olinorder__class_OMax(tc_Code__Numeral_Ocode__numeral,hAPP(c_Set_OCollect(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_COMBB(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool),tc_HOL_Obool,tc_Code__Numeral_Ocode__numeral),c_HOL_OEx(tc_Code__Numeral_Ocode__numeral)),hAPP(hAPP(c_COMBC(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),tc_Code__Numeral_Ocode__numeral),c_COMBS(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_Code__Numeral_Ocode__numeral),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Code__Numeral_Ocode__numeral),c_fconj)),hAPP(hAPP(c_COMBC(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool),tc_fun(tc_fun(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral),tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool)),tc_Code__Numeral_Ocode__numeral),c_COMBB(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool,tc_Code__Numeral_Ocode__numeral)),c_fequal)),hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),B))))),hAPP(hAPP(c_COMBC(tc_Code__Numeral_Ocode__numeral,tc_fun(tc_Code__Numeral_Ocode__numeral,tc_HOL_Obool),tc_HOL_Obool),c_member(tc_Code__Numeral_Ocode__numeral)),A))))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),B),c_Big__Operators_Olinorder__class_OMax(tc_Code__Numeral_Ocode__numeral,A)). [resolve(6150,a,6148,a)]. 130.81/130.65 Derived: -hBOOL(hAPP(c_Finite__Set_Ofinite(tc_Nat_Onat),A)) | c_Orderings_Obot__class_Obot(tc_fun(tc_Nat_Onat,tc_HOL_Obool)) = A | c_Big__Operators_Olinorder__class_OMax(tc_Nat_Onat,hAPP(c_Set_OCollect(tc_Nat_Onat),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool,tc_Nat_Onat),c_HOL_OEx(tc_Nat_Onat)),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_fun(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),tc_Nat_Onat),c_COMBS(tc_Nat_Onat,tc_HOL_Obool,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_Nat_Onat,tc_fun(tc_HOL_Obool,tc_HOL_Obool)),tc_Nat_Onat),hAPP(c_COMBB(tc_HOL_Obool,tc_fun(tc_HOL_Obool,tc_HOL_Obool),tc_Nat_Onat),c_fconj)),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_Nat_Onat),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),hAPP(hAPP(c_COMBB(tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_fun(tc_fun(tc_Nat_Onat,tc_Nat_Onat),tc_fun(tc_Nat_Onat,tc_HOL_Obool)),tc_Nat_Onat),c_COMBB(tc_Nat_Onat,tc_HOL_Obool,tc_Nat_Onat)),c_fequal)),hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B))))),hAPP(hAPP(c_COMBC(tc_Nat_Onat,tc_fun(tc_Nat_Onat,tc_HOL_Obool),tc_HOL_Obool),c_member(tc_Nat_Onat)),A))))) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),c_Big__Operators_Olinorder__class_OMax(tc_Nat_Onat,A)). [resolve(6150,a,6149,a)]. 130.81/130.65 6151 -class_Rings_Osemiring(A) | -class_Int_Onumber(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),D)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),D)) # label(fact_right__distrib__number__of) # label(axiom). [clausify(1160)]. 130.81/130.65 6152 class_Rings_Osemiring(tc_Int_Oint) # label(arity_Int__Oint__Rings_Osemiring) # label(axiom). [assumption]. 130.81/130.65 6153 class_Rings_Osemiring(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Rings_Osemiring) # label(axiom). [assumption]. 130.81/130.65 Derived: -class_Int_Onumber(tc_Int_Oint) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),C)). [resolve(6151,a,6152,a)]. 130.81/130.65 Derived: -class_Int_Onumber(tc_Code__Numeral_Ocode__numeral) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),B),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A)),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),A)),C)). [resolve(6151,a,6153,a)]. 131.71/131.55 6154 class_Rings_Osemiring(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Osemiring) # label(axiom). [assumption]. 131.71/131.55 Derived: -class_Int_Onumber(tc_Nat_Onat) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),A)),C)). [resolve(6154,a,6151,a)]. 131.71/131.55 6155 -class_Int_Onumber(A) | -class_Rings_Osemiring(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),hAPP(c_Int_Onumber__class_Onumber__of(A),D)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),D))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),hAPP(c_Int_Onumber__class_Onumber__of(A),D))) # label(fact_left__distrib__number__of) # label(axiom). [clausify(3883)]. 131.71/131.55 Derived: -class_Int_Onumber(tc_Code__Numeral_Ocode__numeral) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),B)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),C))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Code__Numeral_Ocode__numeral),C))). [resolve(6155,b,6153,a)]. 131.71/131.55 Derived: -class_Int_Onumber(tc_Nat_Onat) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),C))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Nat_Onat),C))). [resolve(6155,b,6154,a)]. 131.71/131.55 6156 -class_Rings_Osemiring(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),D)),E) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D)),E)) # label(fact_combine__common__factor) # label(axiom). [clausify(4166)]. 131.71/131.55 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B)),C)),D) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),C)),D)). [resolve(6156,a,6152,a)]. 131.71/131.55 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_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),B)),C)),D) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),B),C)),D)). [resolve(6156,a,6153,a)]. 139.14/138.98 Derived: 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),A),B)),C)),D) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),C)),D)). [resolve(6156,a,6154,a)]. 139.14/138.98 6157 -class_Rings_Olinordered__semiring(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),B),D)) # label(fact_mult__right__less__imp__less) # label(axiom). [clausify(2712)]. 139.14/138.98 6158 class_Rings_Olinordered__semiring(tc_Int_Oint) # label(arity_Int__Oint__Rings_Olinordered__semiring) # label(axiom). [assumption]. 139.14/138.98 6159 class_Rings_Olinordered__semiring(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Rings_Olinordered__semiring) # label(axiom). [assumption]. 139.14/138.98 6160 class_Rings_Olinordered__semiring(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Olinordered__semiring) # label(axiom). [assumption]. 139.14/138.98 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(6160,a,6157,a)]. 139.14/138.98 6161 -class_Rings_Olinordered__semiring(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),B)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),C),D)) # label(fact_mult__left__less__imp__less) # label(axiom). [clausify(4587)]. 139.14/138.98 6162 class_Rings_Oring(tc_Int_Oint) # label(arity_Int__Oint__Rings_Oring) # label(axiom). [assumption]. 139.14/138.98 6163 -class_Rings_Oring(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__ring) # label(axiom). [clausify(628)]. 139.14/138.98 6164 -class_Rings_Oring(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),C) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(c_Groups_Ouminus__class_Ouminus(A),C)) # label(fact_minus__mult__commute) # label(axiom). [clausify(656)]. 139.14/138.98 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B)). [resolve(6162,a,6164,a)]. 139.14/138.98 6165 -class_Rings_Oring(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),hAPP(c_Groups_Ouminus__class_Ouminus(A),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C) # label(fact_minus__mult__minus) # label(axiom). [clausify(1991)]. 139.14/138.98 Derived: hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A)),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B). [resolve(6165,a,6162,a)]. 139.41/139.26 6166 -class_Rings_Oring(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) | 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) = D # label(fact_eq__add__iff2) # label(axiom). [clausify(2375)]. 139.41/139.26 Derived: 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) | 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) = C. [resolve(6166,a,6162,a)]. 139.41/139.26 6167 -class_Rings_Oring(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) | 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) != D # label(fact_eq__add__iff2) # label(axiom). [clausify(2375)]. 139.41/139.26 Derived: 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) | 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) != C. [resolve(6167,a,6162,a)]. 139.41/139.26 6168 -class_Int_Onumber(A) | -class_Rings_Oring(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),C),D)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Int_Onumber__class_Onumber__of(A),B)),D)) # label(fact_right__diff__distrib__number__of) # label(axiom). [clausify(2746)]. 139.41/139.26 Derived: -class_Int_Onumber(tc_Int_Oint) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),B),C)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),A)),C)). [resolve(6168,b,6162,a)]. 139.41/139.26 6169 -class_Int_Onumber(A) | -class_Rings_Oring(A) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),C)),hAPP(c_Int_Onumber__class_Onumber__of(A),D)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(c_Int_Onumber__class_Onumber__of(A),D))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),hAPP(c_Int_Onumber__class_Onumber__of(A),D))) # label(fact_left__diff__distrib__number__of) # label(axiom). [clausify(3098)]. 139.41/139.26 Derived: -class_Int_Onumber(tc_Int_Oint) | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),B)),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),C)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),C))),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),hAPP(c_Int_Onumber__class_Onumber__of(tc_Int_Oint),C))). [resolve(6169,b,6162,a)]. 139.41/139.26 6170 -class_Rings_Oring(A) | hAPP(c_Groups_Ouminus__class_Ouminus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),hAPP(c_Groups_Ouminus__class_Ouminus(A),C)) # label(fact_minus__mult__right) # label(axiom). [clausify(3193)]. 147.98/147.81 Derived: hAPP(c_Groups_Ouminus__class_Ouminus(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),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B)). [resolve(6170,a,6162,a)]. 147.98/147.81 6171 -class_Rings_Oring(A) | hAPP(c_Groups_Ouminus__class_Ouminus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),B)),C) # label(fact_minus__mult__left) # label(axiom). [clausify(4181)]. 147.98/147.81 Derived: hAPP(c_Groups_Ouminus__class_Ouminus(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_Ouminus__class_Ouminus(tc_Int_Oint),A)),B). [resolve(6171,a,6162,a)]. 147.98/147.81 6172 -class_Rings_Oring(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 | 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_eq__add__iff1) # label(axiom). [clausify(4598)]. 147.98/147.81 Derived: 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 | 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(6172,a,6162,a)]. 147.98/147.81 6173 -class_Rings_Oring(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 | 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_eq__add__iff1) # label(axiom). [clausify(4598)]. 147.98/147.81 Derived: 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 | 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(6173,a,6162,a)]. 147.98/147.81 6174 class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(tc_Int_Oint) # label(arity_Int__Oint__Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct) # label(axiom). [assumption]. 147.98/147.81 6175 -class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(A) | B = C | D = E | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),E)) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),E)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D)) # label(fact_crossproduct__noteq) # label(axiom). [clausify(803)]. 147.98/147.81 6176 -class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(A) | B != C | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),E)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),E)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D)) # label(fact_crossproduct__noteq) # label(axiom). [clausify(803)]. 147.98/147.81 6177 -class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(A) | B != C | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),E),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),E),B)) # label(fact_crossproduct__noteq) # label(axiom). [clausify(803)]. 148.10/147.99 6178 -class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(A) | B != C | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),E)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),E)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D)) # label(fact_crossproduct__eq) # label(axiom). [clausify(1168)]. 148.10/147.99 6179 -class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(A) | B != C | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),E),C)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),E),B)) # label(fact_crossproduct__eq) # label(axiom). [clausify(1168)]. 148.10/147.99 6180 -class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(A) | B = C | D = E | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),E)) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),E)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D)) # label(fact_crossproduct__eq) # label(axiom). [clausify(1168)]. 148.10/147.99 6181 -class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(A) | c_Groups_Ozero__class_Ozero(A) != B | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),B) = C # label(fact_add__0__iff) # label(axiom). [clausify(2491)]. 148.10/147.99 6182 -class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(A) | c_Groups_Ozero__class_Ozero(A) = B | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),B) != C # label(fact_add__0__iff) # label(axiom). [clausify(2491)]. 148.10/147.99 Derived: A = B | C = D | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),D)) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),C)). [resolve(6174,a,6175,a)]. 148.10/147.99 Derived: A != B | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),D)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),C)). [resolve(6174,a,6176,a)]. 148.10/147.99 Derived: A != B | hAPP(hAPP(c_Groups_Oplus__class_Oplus(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),D),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(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),D),A)). [resolve(6174,a,6177,a)]. 148.10/147.99 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),A) = B. [resolve(6174,a,6181,a)]. 148.10/147.99 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),A) != B. [resolve(6174,a,6182,a)]. 148.10/147.99 6183 -class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(A) | c_Groups_Ozero__class_Ozero(A) = B | C != D | E = F | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),E)) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),D),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),F)) # label(fact_add__scale__eq__noteq) # label(axiom). [clausify(4601)]. 148.10/147.99 Derived: c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A | B != C | D = E | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),D)) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),E)). [resolve(6183,a,6174,a)]. 148.10/147.99 6184 class_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct(tc_Nat_Onat) # label(arity_Nat__Onat__Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct) # label(axiom). [assumption]. 150.51/150.38 Derived: A = B | C = D | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),D)) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),C)). [resolve(6184,a,6175,a)]. 150.51/150.38 Derived: A != B | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),D)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),B),C)). [resolve(6184,a,6176,a)]. 150.51/150.38 Derived: A != B | hAPP(hAPP(c_Groups_Oplus__class_Oplus(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),D),B)) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),C),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),D),A)). [resolve(6184,a,6177,a)]. 150.51/150.38 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) != A | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),A) = B. [resolve(6184,a,6181,a)]. 150.51/150.38 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),A) != B. [resolve(6184,a,6182,a)]. 150.51/150.38 Derived: c_Groups_Ozero__class_Ozero(tc_Nat_Onat) = A | B != C | D = E | hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),D)) != hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),C),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Nat_Onat),A),E)). [resolve(6184,a,6183,a)]. 150.51/150.38 6185 class_Rings_Oring__1(tc_Int_Oint) # label(arity_Int__Oint__Rings_Oring__1) # label(axiom). [assumption]. 150.51/150.38 6186 -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(815)]. 150.51/150.38 6187 -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(1225)]. 150.51/150.38 6188 -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(1356)]. 150.51/150.38 6189 -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(1429)]. 150.51/150.38 6190 -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(1638)]. 150.51/150.38 6191 -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(1666)]. 150.51/150.38 6192 -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(1666)]. 150.51/150.38 6193 -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(1758)]. 150.51/150.38 6194 -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(1793)]. 150.51/150.38 6195 -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(1842)]. 150.51/150.38 6196 -class_Rings_Oring__1(A) | -class_Rings_Oinverse(A) | c_Nitpick_Oof__frac(B,A,C) = hAPP(hAPP(c_Rings_Oinverse__class_Odivide(A),hAPP(c_Int_Oring__1__class_Oof__int(A),hAPP(c_Nitpick_Onum(B),C))),hAPP(c_Int_Oring__1__class_Oof__int(A),hAPP(c_Nitpick_Odenom(B),C))) # label(fact_Nitpick_Oof__frac__def) # label(axiom). [clausify(2068)]. 150.51/150.38 6197 -class_Rings_Oring__1(A) | 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))),B)),hAPP(hAPP(c_Power_Opower__class_Opower(A),C),B)) = hAPP(hAPP(c_Power_Opower__class_Opower(A),hAPP(c_Groups_Ouminus__class_Ouminus(A),C)),B) # label(fact_power__minus) # label(axiom). [clausify(2143)]. 150.51/150.38 6198 -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(2397)]. 150.51/150.38 6199 -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(2419)]. 150.51/150.38 6200 -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(2474)]. 150.51/150.38 6201 -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(2765)]. 150.51/150.38 6202 -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(2785)]. 150.51/150.38 6203 -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(3445)]. 150.51/150.38 6204 -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(3609)]. 150.51/150.38 6205 -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(3612)]. 150.51/150.38 6206 -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(3975)]. 150.51/150.38 6207 -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(3990)]. 150.51/150.38 6208 -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(4117)]. 150.51/150.38 6209 -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(4117)]. 150.51/150.38 6210 -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(4117)]. 150.51/150.38 6211 -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(4229)]. 150.51/150.38 6212 -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(4251)]. 150.51/150.39 6213 -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(4265)]. 150.51/150.39 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(6185,a,6186,a)]. 150.51/150.39 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(6185,a,6187,a)]. 150.51/150.39 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(6185,a,6188,a)]. 150.51/150.39 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(6185,a,6189,a)]. 150.51/150.39 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Nat_Onat),A),B)) | hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Nat_Onat),B),A)) = hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),B)),hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),A)). [resolve(6185,a,6190,a)]. 150.51/150.39 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(6185,a,6191,a)]. 150.51/150.39 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(6185,a,6192,a)]. 150.51/150.39 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(6185,a,6193,a)]. 150.51/150.39 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(6185,a,6194,a)]. 150.51/150.39 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(6185,a,6195,a)]. 150.51/150.39 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(6185,a,6196,a)]. 150.51/150.39 Derived: 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))),A)),hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),B),A)) = hAPP(hAPP(c_Power_Opower__class_Opower(tc_Int_Oint),hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),B)),A). [resolve(6185,a,6197,a)]. 150.51/150.39 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(6185,a,6198,b)]. 150.51/150.39 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(6185,a,6199,a)]. 150.51/150.39 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(6185,a,6200,a)]. 150.51/150.39 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(6185,a,6201,a)]. 150.51/150.39 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(6185,a,6202,a)]. 150.51/150.39 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(6185,a,6203,a)]. 150.51/150.39 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(6185,a,6204,a)]. 150.51/150.39 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(6185,a,6205,a)]. 150.51/150.39 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(6185,a,6206,a)]. 150.51/150.39 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(6185,a,6207,a)]. 150.51/150.39 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(6185,a,6208,a)]. 150.51/150.39 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(6185,a,6209,a)]. 150.63/150.49 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(6185,a,6210,a)]. 150.63/150.49 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(6185,a,6211,a)]. 150.63/150.49 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(6185,a,6212,a)]. 150.63/150.49 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(6185,a,6213,a)]. 150.63/150.49 6214 -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(4440)]. 150.63/150.49 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(6214,a,6185,a)]. 150.63/150.49 6215 -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(4442)]. 150.63/150.49 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(6215,a,6185,a)]. 150.63/150.49 6216 -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(4560)]. 153.45/153.28 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(6216,a,6185,a)]. 153.45/153.28 6217 -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(4737)]. 153.45/153.28 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(6217,a,6185,a)]. 153.45/153.28 6218 -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(4822)]. 153.45/153.28 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(6218,a,6185,a)]. 153.45/153.28 6219 -class_Groups_Oab__semigroup__add(A) | c_List_Ofoldr(A,A,c_Groups_Oplus__class_Oplus(A),B,C) = hAPP(c_List_Ofoldl(A,A,c_Groups_Oplus__class_Oplus(A),C),B) # label(fact_foldr__conv__foldl) # label(axiom). [clausify(4427)]. 153.45/153.28 6220 class_Groups_Oab__semigroup__add(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Oab__semigroup__add) # label(axiom). [assumption]. 153.45/153.28 6221 class_Groups_Oab__semigroup__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Oab__semigroup__add) # label(axiom). [assumption]. 153.45/153.28 6222 class_Groups_Oab__semigroup__add(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Groups_Oab__semigroup__add) # label(axiom). [assumption]. 153.45/153.28 Derived: c_List_Ofoldr(tc_Nat_Onat,tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A,B) = hAPP(c_List_Ofoldl(tc_Nat_Onat,tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),A). [resolve(6219,a,6220,a)]. 153.45/153.28 Derived: c_List_Ofoldr(tc_Int_Oint,tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint),A,B) = hAPP(c_List_Ofoldl(tc_Int_Oint,tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),A). [resolve(6219,a,6221,a)]. 153.45/153.28 Derived: c_List_Ofoldr(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A,B) = hAPP(c_List_Ofoldl(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),B),A). [resolve(6219,a,6222,a)]. 153.45/153.28 6223 -class_Groups_Oab__semigroup__add(A) | hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),D) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),C),D)) # label(fact_ab__semigroup__add__class_Oadd__ac_I1_J) # label(axiom). [clausify(4565)]. 153.45/153.28 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B)),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),C)). [resolve(6223,a,6221,a)]. 153.45/153.28 Derived: hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),B)),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),B),C)). [resolve(6223,a,6222,a)]. 160.91/160.75 6224 -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(4370)]. 160.91/160.75 6225 class_Nat_Osemiring__char__0(tc_Int_Oint) # label(arity_Int__Oint__Nat_Osemiring__char__0) # label(axiom). [assumption]. 160.91/160.75 6226 class_Nat_Osemiring__char__0(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Nat_Osemiring__char__0) # label(axiom). [assumption]. 160.91/160.75 Derived: hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),A) != hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),B) | A = B. [resolve(6224,a,6225,a)]. 160.91/160.75 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(6224,a,6226,a)]. 160.91/160.75 6227 -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(4370)]. 160.91/160.75 Derived: hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),A) = hAPP(c_Nat_Osemiring__1__class_Oof__nat(tc_Int_Oint),B) | A != B. [resolve(6227,a,6225,a)]. 160.91/160.75 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(6227,a,6226,a)]. 160.91/160.75 6228 -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(4659)]. 160.91/160.75 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(6228,a,6225,a)]. 160.91/160.75 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(6228,a,6226,a)]. 160.91/160.75 6229 class_Nat_Osemiring__char__0(tc_Nat_Onat) # label(arity_Nat__Onat__Nat_Osemiring__char__0) # label(axiom). [assumption]. 160.91/160.75 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(6229,a,6224,a)]. 160.91/160.75 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(6229,a,6227,a)]. 160.91/160.75 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(6229,a,6228,a)]. 160.91/160.75 6230 -c_Equiv__Relations_Oequivp(A,B) | c_Predicate_Otransp(A,B) # label(fact_equivpE) # label(axiom). [clausify(2038)]. 160.91/160.75 6231 -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(1018)]. 160.91/160.75 6232 -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(1020)]. 160.91/160.75 6233 -c_Equiv__Relations_Oequivp(A,B) | c_Predicate_Otransp(A,B) # label(fact_equivp__reflp__symp__transp) # label(axiom). [clausify(2196)]. 160.91/160.75 6234 c_Equiv__Relations_Oequivp(A,B) | -c_Predicate_Oreflp(A,B) | -c_Predicate_Otransp(A,B) | -c_Predicate_Osymp(A,B) # label(fact_equivp__reflp__symp__transp) # label(axiom). [clausify(2196)]. 160.91/160.75 6235 -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(3562)]. 160.91/160.75 Derived: -c_Relation_Otrans(A,hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))) | -hBOOL(hAPP(hAPP(B,C),D)) | -hBOOL(hAPP(hAPP(B,D),E)) | hBOOL(hAPP(hAPP(B,C),E)). [resolve(6235,b,6231,a)]. 160.91/160.75 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_Predicate_Oreflp(A,B) | -c_Predicate_Osymp(A,B) | c_Equiv__Relations_Oequivp(A,B). [resolve(6235,b,6232,c)]. 162.34/162.17 6236 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(3562)]. 162.34/162.17 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(6236,b,6230,b)]. 162.34/162.17 6237 -hBOOL(hAPP(hAPP(A,B),B)) | -c_Predicate_Osymp(C,A) | -c_Predicate_Otransp(C,A) | hBOOL(hAPP(hAPP(A,f67(A,C)),f67(A,C))). [resolve(6123,a,6124,a)]. 162.34/162.17 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | -c_Predicate_Osymp(C,A) | hBOOL(hAPP(hAPP(A,f67(A,C)),f67(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(6237,c,6235,b)]. 162.34/162.17 6238 -hBOOL(hAPP(hAPP(A,B),C)) | -hBOOL(hAPP(hAPP(A,C),D)) | hBOOL(hAPP(hAPP(A,B),D)) | -hBOOL(hAPP(hAPP(A,E),E)) | -c_Predicate_Osymp(F,A) | -c_Predicate_Otransp(F,A). [resolve(6128,a,6123,a)]. 162.34/162.17 6239 hBOOL(hAPP(hAPP(A,f312(A,B)),f312(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -c_Predicate_Osymp(B,A) | -c_Predicate_Otransp(B,A). [resolve(6129,a,6123,a)]. 162.34/162.17 Derived: hBOOL(hAPP(hAPP(A,f312(A,B)),f312(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -c_Predicate_Osymp(B,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(6239,d,6235,b)]. 162.34/162.17 6240 -hBOOL(hAPP(hAPP(A,B),C)) | hBOOL(hAPP(hAPP(A,C),B)) | -hBOOL(hAPP(hAPP(A,D),D)) | -c_Predicate_Osymp(E,A) | -c_Predicate_Otransp(E,A). [resolve(6132,a,6123,a)]. 162.34/162.17 Derived: -hBOOL(hAPP(hAPP(A,B),C)) | hBOOL(hAPP(hAPP(A,C),B)) | -hBOOL(hAPP(hAPP(A,D),D)) | -c_Predicate_Osymp(E,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(6240,e,6235,b)]. 162.34/162.17 6241 hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f529(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_Predicate_Osymp(A,B) | -c_Predicate_Otransp(A,B). [resolve(6133,a,6123,a)]. 162.34/162.17 Derived: hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f529(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_Predicate_Osymp(A,B) | -c_Relation_Otrans(A,hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))). [resolve(6241,d,6235,b)]. 162.34/162.17 6242 -c_Equiv__Relations_Oequivp(A,B) | c_Predicate_Otransp(A,B). [resolve(6134,b,6126,a)]. 162.34/162.17 Derived: -c_Equiv__Relations_Oequivp(A,B) | -hBOOL(hAPP(hAPP(B,C),C)) | -c_Predicate_Osymp(A,B) | hBOOL(hAPP(hAPP(B,f67(B,A)),f67(B,A))). [resolve(6242,b,6237,c)]. 162.34/162.17 6243 -hBOOL(hAPP(hAPP(A,B),B)) | hAPP(A,B) != hAPP(A,C) | -hBOOL(hAPP(hAPP(A,C),C)) | hBOOL(hAPP(hAPP(A,C),B)) | -hBOOL(hAPP(hAPP(A,D),D)) | -c_Predicate_Osymp(E,A) | -c_Predicate_Otransp(E,A). [resolve(6135,a,6123,a)]. 162.36/162.19 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hAPP(A,B) != hAPP(A,C) | -hBOOL(hAPP(hAPP(A,C),C)) | hBOOL(hAPP(hAPP(A,C),B)) | -hBOOL(hAPP(hAPP(A,D),D)) | -c_Predicate_Osymp(E,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(6243,g,6235,b)]. 162.36/162.19 6244 hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,C),B)) | -hBOOL(hAPP(hAPP(A,D),D)) | -c_Predicate_Osymp(E,A) | -c_Predicate_Otransp(E,A). [resolve(6136,a,6123,a)]. 162.36/162.19 Derived: hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,C),B)) | -hBOOL(hAPP(hAPP(A,D),D)) | -c_Predicate_Osymp(E,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(6244,e,6235,b)]. 162.36/162.19 6245 hAPP(A,B) = hAPP(A,C) | -hBOOL(hAPP(hAPP(A,C),B)) | -hBOOL(hAPP(hAPP(A,D),D)) | -c_Predicate_Osymp(E,A) | -c_Predicate_Otransp(E,A). [resolve(6137,a,6123,a)]. 162.36/162.19 Derived: hAPP(A,B) = hAPP(A,C) | -hBOOL(hAPP(hAPP(A,C),B)) | -hBOOL(hAPP(hAPP(A,D),D)) | -c_Predicate_Osymp(E,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(6245,e,6235,b)]. 162.36/162.19 6246 hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,B),C)) | -hBOOL(hAPP(hAPP(A,D),D)) | -c_Predicate_Osymp(E,A) | -c_Predicate_Otransp(E,A). [resolve(6138,a,6123,a)]. 162.36/162.19 Derived: hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,B),C)) | -hBOOL(hAPP(hAPP(A,D),D)) | -c_Predicate_Osymp(E,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(6246,e,6235,b)]. 162.36/162.19 6247 hBOOL(hAPP(hAPP(A,f534(A,B)),f534(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -c_Predicate_Osymp(B,A) | -c_Predicate_Otransp(B,A). [resolve(6139,a,6123,a)]. 162.36/162.19 Derived: hBOOL(hAPP(hAPP(A,f534(A,B)),f534(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -c_Predicate_Osymp(B,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(6247,d,6235,b)]. 162.36/162.19 6248 hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | c_Predicate_Otransp(B,A). [resolve(6140,a,6126,a)]. 162.36/162.19 Derived: hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -c_Predicate_Oreflp(B,A) | -c_Predicate_Osymp(B,A) | c_Equiv__Relations_Oequivp(B,A). [resolve(6248,d,6232,c)]. 162.36/162.19 Derived: hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | 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(6248,d,6236,b)]. 162.36/162.19 Derived: hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),D)) | -c_Predicate_Osymp(B,A) | hBOOL(hAPP(hAPP(A,f67(A,B)),f67(A,B))). [resolve(6248,d,6237,c)]. 162.36/162.19 6249 hAPP(A,f536(A,B)) = hAPP(A,f535(A,B)) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | c_Predicate_Otransp(B,A). [resolve(6141,a,6126,a)]. 162.36/162.19 Derived: hAPP(A,f536(A,B)) = hAPP(A,f535(A,B)) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -c_Predicate_Oreflp(B,A) | -c_Predicate_Osymp(B,A) | c_Equiv__Relations_Oequivp(B,A). [resolve(6249,d,6232,c)]. 162.36/162.19 Derived: hAPP(A,f536(A,B)) = hAPP(A,f535(A,B)) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | 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(6249,d,6236,b)]. 162.36/162.19 Derived: hAPP(A,f536(A,B)) = hAPP(A,f535(A,B)) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),D)) | -c_Predicate_Osymp(B,A) | hBOOL(hAPP(hAPP(A,f67(A,B)),f67(A,B))). [resolve(6249,d,6237,c)]. 162.36/162.19 6250 hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | c_Predicate_Otransp(B,A). [resolve(6142,a,6126,a)]. 163.86/163.71 Derived: hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -c_Predicate_Oreflp(B,A) | -c_Predicate_Osymp(B,A) | c_Equiv__Relations_Oequivp(B,A). [resolve(6250,d,6232,c)]. 163.86/163.71 Derived: hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | 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(6250,d,6236,b)]. 163.86/163.71 Derived: hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),D)) | -c_Predicate_Osymp(B,A) | hBOOL(hAPP(hAPP(A,f67(A,B)),f67(A,B))). [resolve(6250,d,6237,c)]. 163.86/163.71 6251 -hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hAPP(A,f536(A,B)) != hAPP(A,f535(A,B)) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | c_Predicate_Otransp(B,A). [resolve(6143,a,6126,a)]. 163.86/163.71 Derived: -hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hAPP(A,f536(A,B)) != hAPP(A,f535(A,B)) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -c_Predicate_Oreflp(B,A) | -c_Predicate_Osymp(B,A) | c_Equiv__Relations_Oequivp(B,A). [resolve(6251,f,6232,c)]. 163.86/163.71 Derived: -hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hAPP(A,f536(A,B)) != hAPP(A,f535(A,B)) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | 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(6251,f,6236,b)]. 163.86/163.71 Derived: -hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hAPP(A,f536(A,B)) != hAPP(A,f535(A,B)) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),D)) | -c_Predicate_Osymp(B,A) | hBOOL(hAPP(hAPP(A,f67(A,B)),f67(A,B))). [resolve(6251,f,6237,c)]. 163.86/163.71 6252 -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(2426)]. 163.86/163.71 6253 class_Rings_Oordered__comm__semiring(tc_Nat_Onat) # label(arity_Nat__Onat__Rings_Oordered__comm__semiring) # label(axiom). [assumption]. 163.86/163.71 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),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(6252,a,6253,a)]. 163.86/163.71 6254 class_Rings_Oordered__comm__semiring(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Rings_Oordered__comm__semiring) # label(axiom). [assumption]. 163.86/163.71 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),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(6254,a,6252,a)]. 163.86/163.71 6255 class_Rings_Oordered__comm__semiring(tc_Int_Oint) # label(arity_Int__Oint__Rings_Oordered__comm__semiring) # label(axiom). [assumption]. 163.86/163.71 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),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(6255,a,6252,a)]. 168.50/168.39 6256 -class_Groups_Osemigroup__add(A) | hAPP(c_List_Ofoldl(A,A,c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),C)),D) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),B),hAPP(c_List_Ofoldl(A,A,c_Groups_Oplus__class_Oplus(A),C),D)) # label(fact_foldl__assoc) # label(axiom). [clausify(2166)]. 168.50/168.39 6257 class_Groups_Osemigroup__add(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Groups_Osemigroup__add) # label(axiom). [assumption]. 168.50/168.39 6258 class_Groups_Osemigroup__add(tc_Int_Oint) # label(arity_Int__Oint__Groups_Osemigroup__add) # label(axiom). [assumption]. 168.50/168.39 Derived: hAPP(c_List_Ofoldl(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),B)),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),A),hAPP(c_List_Ofoldl(tc_Code__Numeral_Ocode__numeral,tc_Code__Numeral_Ocode__numeral,c_Groups_Oplus__class_Oplus(tc_Code__Numeral_Ocode__numeral),B),C)). [resolve(6256,a,6257,a)]. 168.50/168.39 Derived: hAPP(c_List_Ofoldl(tc_Int_Oint,tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),B)),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),A),hAPP(c_List_Ofoldl(tc_Int_Oint,tc_Int_Oint,c_Groups_Oplus__class_Oplus(tc_Int_Oint),B),C)). [resolve(6256,a,6258,a)]. 168.50/168.39 6259 class_Groups_Osemigroup__add(tc_Nat_Onat) # label(arity_Nat__Onat__Groups_Osemigroup__add) # label(axiom). [assumption]. 168.50/168.39 Derived: hAPP(c_List_Ofoldl(tc_Nat_Onat,tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),B)),C) = hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Nat_Onat),A),hAPP(c_List_Ofoldl(tc_Nat_Onat,tc_Nat_Onat,c_Groups_Oplus__class_Oplus(tc_Nat_Onat),B),C)). [resolve(6259,a,6256,a)]. 168.50/168.39 6260 class_Rings_Oordered__ring(tc_Int_Oint) # label(arity_Int__Oint__Rings_Oordered__ring) # label(axiom). [assumption]. 168.50/168.39 6261 -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),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),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),B),D)),E)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D)),F))) # label(fact_less__add__iff1) # label(axiom). [clausify(1165)]. 168.50/168.39 6262 -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),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),B),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),B),D)),E)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),D)),F))) # label(fact_less__add__iff1) # label(axiom). [clausify(1165)]. 168.50/168.39 6263 -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(1693)]. 168.50/168.39 6264 -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(1693)]. 168.86/168.68 6265 -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(1886)]. 168.86/168.68 6266 -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(1886)]. 168.86/168.68 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),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),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),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(6260,a,6261,a)]. 168.86/168.68 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),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),A),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),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(6260,a,6262,a)]. 168.86/168.68 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(6260,a,6263,a)]. 168.86/168.68 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(6260,a,6264,a)]. 168.86/168.68 6267 -class_Rings_Oordered__ring(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),D)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),E),C)),F))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),E),B)),C)),F))) # label(fact_less__add__iff2) # label(axiom). [clausify(3435)]. 168.86/168.68 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),D),B)),E))) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),C),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),D),A)),B)),E))). [resolve(6267,a,6260,a)]. 169.05/168.95 6268 -class_Rings_Oordered__ring(A) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),D)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),E),C)),F))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(A),D),hAPP(hAPP(c_Groups_Oplus__class_Oplus(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),hAPP(hAPP(c_Groups_Ominus__class_Ominus(A),E),B)),C)),F))) # label(fact_less__add__iff2) # label(axiom). [clausify(3435)]. 169.05/168.95 Derived: hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),C)),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),D),B)),E))) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless(tc_Int_Oint),C),hAPP(hAPP(c_Groups_Oplus__class_Oplus(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),hAPP(hAPP(c_Groups_Ominus__class_Ominus(tc_Int_Oint),D),A)),B)),E))). [resolve(6268,a,6260,a)]. 169.05/168.95 6269 -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(3767)]. 169.05/168.95 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(6269,a,6260,a)]. 169.05/168.95 6270 -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(3767)]. 169.05/168.95 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(6270,a,6260,a)]. 169.05/168.95 6271 -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(4083)]. 171.32/171.15 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(6271,a,6260,a)]. 171.32/171.15 6272 -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(4328)]. 171.32/171.15 6273 -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(4856)]. 171.32/171.15 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(6273,a,6260,a)]. 171.32/171.15 6274 -class_Rings_Oidom(A) | hAPP(c_Groups_Ouminus__class_Ouminus(A),B) != C | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),C) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),B) # label(fact_square__eq__iff) # label(axiom). [clausify(2535)]. 171.32/171.15 6275 class_Rings_Oidom(tc_Int_Oint) # label(arity_Int__Oint__Rings_Oidom) # label(axiom). [assumption]. 171.32/171.15 Derived: hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A) != B | 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(6274,a,6275,a)]. 171.32/171.15 6276 -class_Rings_Oidom(A) | B != C | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),B) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),C) # label(fact_square__eq__iff) # label(axiom). [clausify(2535)]. 171.32/171.15 Derived: A != B | hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),A) = hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),B),B). [resolve(6276,a,6275,a)]. 171.32/171.15 6277 -class_Rings_Oidom(A) | hAPP(c_Groups_Ouminus__class_Ouminus(A),B) = C | C = B | hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),C) != hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),B) # label(fact_square__eq__iff) # label(axiom). [clausify(2535)]. 171.32/171.15 Derived: hAPP(c_Groups_Ouminus__class_Ouminus(tc_Int_Oint),A) = B | B = A | 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(6277,a,6275,a)]. 171.32/171.15 6278 -class_Rings_Oidom(A) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D))) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),C),D)) | c_Groups_Ozero__class_Ozero(A) = B # label(fact_dvd__mult__cancel__left) # label(axiom). [clausify(2782)]. 171.32/171.15 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C))) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),B),C)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = A. [resolve(6278,a,6275,a)]. 171.32/171.15 6279 -class_Rings_Oidom(A) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D))) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),C),D)) # label(fact_dvd__mult__cancel__left) # label(axiom). [clausify(2782)]. 176.08/175.91 Derived: hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C))) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),B),C)). [resolve(6279,a,6275,a)]. 176.08/175.91 6280 -class_Rings_Oidom(A) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D))) | c_Groups_Ozero__class_Ozero(A) != B # label(fact_dvd__mult__cancel__left) # label(axiom). [clausify(2782)]. 176.08/175.91 Derived: hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),C))) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != A. [resolve(6280,a,6275,a)]. 176.08/175.91 6281 -class_Rings_Oidom(A) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C))) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),D)) | c_Groups_Ozero__class_Ozero(A) = C # label(fact_dvd__mult__cancel__right) # label(axiom). [clausify(3243)]. 176.08/175.91 Derived: -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),B))) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),C)) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) = B. [resolve(6281,a,6275,a)]. 176.08/175.91 6282 -class_Rings_Oidom(A) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C))) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),B),D)) # label(fact_dvd__mult__cancel__right) # label(axiom). [clausify(3243)]. 176.08/175.91 Derived: hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),B))) | -hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),A),C)). [resolve(6282,a,6275,a)]. 176.08/175.91 6283 -class_Rings_Oidom(A) | hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C))) | c_Groups_Ozero__class_Ozero(A) != C # label(fact_dvd__mult__cancel__right) # label(axiom). [clausify(3243)]. 176.08/175.91 Derived: hBOOL(hAPP(hAPP(c_Rings_Odvd__class_Odvd(tc_Int_Oint),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),A),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Int_Oint),C),B))) | c_Groups_Ozero__class_Ozero(tc_Int_Oint) != B. [resolve(6283,a,6275,a)]. 176.08/175.91 6284 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_Predicate_Osymp(A,B) # label(fact_symp__def) # label(axiom). [clausify(1369)]. 176.08/175.91 6285 -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_Predicate_Osymp(A,B) # label(fact_symp__def) # label(axiom). [clausify(1369)]. 176.08/175.91 6286 -c_Equiv__Relations_Oequivp(A,B) | c_Predicate_Osymp(A,B) # label(fact_equivpE) # label(axiom). [clausify(2038)]. 176.08/175.91 Derived: -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(6286,b,6284,b)]. 176.08/175.91 6287 -c_Equiv__Relations_Oequivp(A,B) | c_Predicate_Osymp(A,B) # label(fact_equivp__reflp__symp__transp) # label(axiom). [clausify(2196)]. 176.08/175.91 6288 -c_Predicate_Osymp(A,B) | -hBOOL(hAPP(hAPP(B,C),D)) | hBOOL(hAPP(hAPP(B,D),C)) # label(fact_sympE) # label(axiom). [clausify(2872)]. 176.08/175.91 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(6288,a,6285,b)]. 176.08/175.91 6289 -c_Equiv__Relations_Oequivp(A,B) | c_Predicate_Osymp(A,B). [resolve(6134,b,6125,a)]. 176.08/175.91 6290 hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | c_Predicate_Osymp(B,A). [resolve(6140,a,6125,a)]. 176.15/176.00 Derived: hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(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))). [resolve(6290,d,6284,b)]. 176.15/176.00 6291 hAPP(A,f536(A,B)) = hAPP(A,f535(A,B)) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | c_Predicate_Osymp(B,A). [resolve(6141,a,6125,a)]. 176.15/176.00 Derived: hAPP(A,f536(A,B)) = hAPP(A,f535(A,B)) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(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))). [resolve(6291,d,6284,b)]. 176.15/176.00 6292 hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | c_Predicate_Osymp(B,A). [resolve(6142,a,6125,a)]. 176.15/176.00 Derived: hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(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))). [resolve(6292,d,6284,b)]. 176.15/176.00 6293 -hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hAPP(A,f536(A,B)) != hAPP(A,f535(A,B)) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | c_Predicate_Osymp(B,A). [resolve(6143,a,6125,a)]. 176.15/176.00 Derived: -hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hAPP(A,f536(A,B)) != hAPP(A,f535(A,B)) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f536(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))). [resolve(6293,f,6284,b)]. 176.15/176.00 6294 -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_Oreflp(A,B) | -c_Predicate_Osymp(A,B) | c_Equiv__Relations_Oequivp(A,B). [resolve(6235,b,6232,c)]. 176.15/176.00 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_Predicate_Oreflp(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(6294,c,6285,b)]. 176.15/176.00 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_Predicate_Oreflp(A,B) | c_Equiv__Relations_Oequivp(A,B) | hBOOL(hAPP(hAPP(B,f536(B,A)),f536(B,A))) | hBOOL(hAPP(hAPP(B,f535(B,A)),f536(B,A))) | -hBOOL(hAPP(hAPP(B,C),C)). [resolve(6294,c,6290,d)]. 176.15/176.00 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_Predicate_Oreflp(A,B) | c_Equiv__Relations_Oequivp(A,B) | hAPP(B,f536(B,A)) = hAPP(B,f535(B,A)) | hBOOL(hAPP(hAPP(B,f535(B,A)),f536(B,A))) | -hBOOL(hAPP(hAPP(B,C),C)). [resolve(6294,c,6291,d)]. 176.15/176.00 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_Predicate_Oreflp(A,B) | c_Equiv__Relations_Oequivp(A,B) | hBOOL(hAPP(hAPP(B,f535(B,A)),f535(B,A))) | hBOOL(hAPP(hAPP(B,f535(B,A)),f536(B,A))) | -hBOOL(hAPP(hAPP(B,C),C)). [resolve(6294,c,6292,d)]. 176.15/176.00 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_Predicate_Oreflp(A,B) | c_Equiv__Relations_Oequivp(A,B) | -hBOOL(hAPP(hAPP(B,f536(B,A)),f536(B,A))) | hAPP(B,f536(B,A)) != hAPP(B,f535(B,A)) | -hBOOL(hAPP(hAPP(B,f535(B,A)),f535(B,A))) | -hBOOL(hAPP(hAPP(B,f535(B,A)),f536(B,A))) | -hBOOL(hAPP(hAPP(B,C),C)). [resolve(6294,c,6293,f)]. 176.15/176.00 6295 -hBOOL(hAPP(hAPP(A,B),B)) | -c_Predicate_Osymp(C,A) | hBOOL(hAPP(hAPP(A,f67(A,C)),f67(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(6237,c,6235,b)]. 176.15/176.00 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f67(A,C)),f67(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))) | -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(6295,b,6285,b)]. 176.15/176.00 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f67(A,C)),f67(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))) | -c_Equiv__Relations_Oequivp(C,A). [resolve(6295,b,6286,b)]. 176.15/176.00 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f67(A,C)),f67(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))) | hBOOL(hAPP(hAPP(A,f536(A,C)),f536(A,C))) | hBOOL(hAPP(hAPP(A,f535(A,C)),f536(A,C))) | -hBOOL(hAPP(hAPP(A,D),D)). [resolve(6295,b,6290,d)]. 176.15/176.00 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f67(A,C)),f67(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))) | hAPP(A,f536(A,C)) = hAPP(A,f535(A,C)) | hBOOL(hAPP(hAPP(A,f535(A,C)),f536(A,C))) | -hBOOL(hAPP(hAPP(A,D),D)). [resolve(6295,b,6291,d)]. 176.15/176.00 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f67(A,C)),f67(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))) | hBOOL(hAPP(hAPP(A,f535(A,C)),f535(A,C))) | hBOOL(hAPP(hAPP(A,f535(A,C)),f536(A,C))) | -hBOOL(hAPP(hAPP(A,D),D)). [resolve(6295,b,6292,d)]. 176.15/176.00 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hBOOL(hAPP(hAPP(A,f67(A,C)),f67(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))) | -hBOOL(hAPP(hAPP(A,f536(A,C)),f536(A,C))) | hAPP(A,f536(A,C)) != hAPP(A,f535(A,C)) | -hBOOL(hAPP(hAPP(A,f535(A,C)),f535(A,C))) | -hBOOL(hAPP(hAPP(A,f535(A,C)),f536(A,C))) | -hBOOL(hAPP(hAPP(A,D),D)). [resolve(6295,b,6293,f)]. 176.15/176.00 6296 hBOOL(hAPP(hAPP(A,f312(A,B)),f312(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -c_Predicate_Osymp(B,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(6239,d,6235,b)]. 176.15/176.00 Derived: hBOOL(hAPP(hAPP(A,f312(A,B)),f312(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -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))) | -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(6296,c,6285,b)]. 176.15/176.00 Derived: hBOOL(hAPP(hAPP(A,f312(A,B)),f312(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -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))) | -c_Equiv__Relations_Oequivp(B,A). [resolve(6296,c,6286,b)]. 176.15/176.00 Derived: hBOOL(hAPP(hAPP(A,f312(A,B)),f312(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -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))) | hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,D),D)). [resolve(6296,c,6290,d)]. 176.15/176.00 Derived: hBOOL(hAPP(hAPP(A,f312(A,B)),f312(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -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))) | hAPP(A,f536(A,B)) = hAPP(A,f535(A,B)) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,D),D)). [resolve(6296,c,6291,d)]. 176.15/176.00 Derived: hBOOL(hAPP(hAPP(A,f312(A,B)),f312(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -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))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,D),D)). [resolve(6296,c,6292,d)]. 176.15/176.00 Derived: hBOOL(hAPP(hAPP(A,f312(A,B)),f312(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -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))) | -hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hAPP(A,f536(A,B)) != hAPP(A,f535(A,B)) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,D),D)). [resolve(6296,c,6293,f)]. 176.15/176.00 6297 -hBOOL(hAPP(hAPP(A,B),C)) | hBOOL(hAPP(hAPP(A,C),B)) | -hBOOL(hAPP(hAPP(A,D),D)) | -c_Predicate_Osymp(E,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(6240,e,6235,b)]. 176.15/176.00 Derived: -hBOOL(hAPP(hAPP(A,B),C)) | hBOOL(hAPP(hAPP(A,C),B)) | -hBOOL(hAPP(hAPP(A,D),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))) | -c_Equiv__Relations_Oequivp(E,A). [resolve(6297,d,6286,b)]. 176.15/176.00 Derived: -hBOOL(hAPP(hAPP(A,B),C)) | hBOOL(hAPP(hAPP(A,C),B)) | -hBOOL(hAPP(hAPP(A,D),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))) | hBOOL(hAPP(hAPP(A,f536(A,E)),f536(A,E))) | hBOOL(hAPP(hAPP(A,f535(A,E)),f536(A,E))) | -hBOOL(hAPP(hAPP(A,F),F)). [resolve(6297,d,6290,d)]. 176.15/176.00 Derived: -hBOOL(hAPP(hAPP(A,B),C)) | hBOOL(hAPP(hAPP(A,C),B)) | -hBOOL(hAPP(hAPP(A,D),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))) | hAPP(A,f536(A,E)) = hAPP(A,f535(A,E)) | hBOOL(hAPP(hAPP(A,f535(A,E)),f536(A,E))) | -hBOOL(hAPP(hAPP(A,F),F)). [resolve(6297,d,6291,d)]. 176.15/176.00 Derived: -hBOOL(hAPP(hAPP(A,B),C)) | hBOOL(hAPP(hAPP(A,C),B)) | -hBOOL(hAPP(hAPP(A,D),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))) | hBOOL(hAPP(hAPP(A,f535(A,E)),f535(A,E))) | hBOOL(hAPP(hAPP(A,f535(A,E)),f536(A,E))) | -hBOOL(hAPP(hAPP(A,F),F)). [resolve(6297,d,6292,d)]. 176.15/176.00 Derived: -hBOOL(hAPP(hAPP(A,B),C)) | hBOOL(hAPP(hAPP(A,C),B)) | -hBOOL(hAPP(hAPP(A,D),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))) | -hBOOL(hAPP(hAPP(A,f536(A,E)),f536(A,E))) | hAPP(A,f536(A,E)) != hAPP(A,f535(A,E)) | -hBOOL(hAPP(hAPP(A,f535(A,E)),f535(A,E))) | -hBOOL(hAPP(hAPP(A,f535(A,E)),f536(A,E))) | -hBOOL(hAPP(hAPP(A,F),F)). [resolve(6297,d,6293,f)]. 176.15/176.00 6298 hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f529(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_Predicate_Osymp(A,B) | -c_Relation_Otrans(A,hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))). [resolve(6241,d,6235,b)]. 176.15/176.00 Derived: hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f529(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_Otrans(A,hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))) | -c_Relation_Osym(A,hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))). [resolve(6298,c,6285,b)]. 176.15/176.00 Derived: hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f529(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_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(6298,c,6286,b)]. 176.15/176.00 Derived: hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f529(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_Otrans(A,hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))) | hBOOL(hAPP(hAPP(B,f536(B,A)),f536(B,A))) | hBOOL(hAPP(hAPP(B,f535(B,A)),f536(B,A))) | -hBOOL(hAPP(hAPP(B,D),D)). [resolve(6298,c,6290,d)]. 176.15/176.00 Derived: hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f529(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_Otrans(A,hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))) | hAPP(B,f536(B,A)) = hAPP(B,f535(B,A)) | hBOOL(hAPP(hAPP(B,f535(B,A)),f536(B,A))) | -hBOOL(hAPP(hAPP(B,D),D)). [resolve(6298,c,6291,d)]. 176.15/176.00 Derived: hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f529(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_Otrans(A,hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))) | hBOOL(hAPP(hAPP(B,f535(B,A)),f535(B,A))) | hBOOL(hAPP(hAPP(B,f535(B,A)),f536(B,A))) | -hBOOL(hAPP(hAPP(B,D),D)). [resolve(6298,c,6292,d)]. 176.15/176.00 Derived: hBOOL(hAPP(hAPP(c_member(tc_fun(A,tc_HOL_Obool)),f529(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_Otrans(A,hAPP(c_Set_OCollect(tc_prod(A,A)),hAPP(c_Product__Type_Oprod_Oprod__case(A,A,tc_HOL_Obool),B))) | -hBOOL(hAPP(hAPP(B,f536(B,A)),f536(B,A))) | hAPP(B,f536(B,A)) != hAPP(B,f535(B,A)) | -hBOOL(hAPP(hAPP(B,f535(B,A)),f535(B,A))) | -hBOOL(hAPP(hAPP(B,f535(B,A)),f536(B,A))) | -hBOOL(hAPP(hAPP(B,D),D)). [resolve(6298,c,6293,f)]. 176.15/176.00 6299 -c_Equiv__Relations_Oequivp(A,B) | -hBOOL(hAPP(hAPP(B,C),C)) | -c_Predicate_Osymp(A,B) | hBOOL(hAPP(hAPP(B,f67(B,A)),f67(B,A))). [resolve(6242,b,6237,c)]. 176.15/176.00 Derived: -c_Equiv__Relations_Oequivp(A,B) | -hBOOL(hAPP(hAPP(B,C),C)) | hBOOL(hAPP(hAPP(B,f67(B,A)),f67(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(6299,c,6285,b)]. 176.15/176.00 Derived: -c_Equiv__Relations_Oequivp(A,B) | -hBOOL(hAPP(hAPP(B,C),C)) | hBOOL(hAPP(hAPP(B,f67(B,A)),f67(B,A))) | -c_Equiv__Relations_Oequivp(A,B). [resolve(6299,c,6286,b)]. 176.15/176.00 6300 -hBOOL(hAPP(hAPP(A,B),B)) | hAPP(A,B) != hAPP(A,C) | -hBOOL(hAPP(hAPP(A,C),C)) | hBOOL(hAPP(hAPP(A,C),B)) | -hBOOL(hAPP(hAPP(A,D),D)) | -c_Predicate_Osymp(E,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(6243,g,6235,b)]. 176.15/176.00 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hAPP(A,B) != hAPP(A,C) | -hBOOL(hAPP(hAPP(A,C),C)) | hBOOL(hAPP(hAPP(A,C),B)) | -hBOOL(hAPP(hAPP(A,D),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))) | -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))). [resolve(6300,f,6285,b)]. 176.15/176.00 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hAPP(A,B) != hAPP(A,C) | -hBOOL(hAPP(hAPP(A,C),C)) | hBOOL(hAPP(hAPP(A,C),B)) | -hBOOL(hAPP(hAPP(A,D),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))) | -c_Equiv__Relations_Oequivp(E,A). [resolve(6300,f,6286,b)]. 176.15/176.00 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hAPP(A,B) != hAPP(A,C) | -hBOOL(hAPP(hAPP(A,C),C)) | hBOOL(hAPP(hAPP(A,C),B)) | -hBOOL(hAPP(hAPP(A,D),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))) | hBOOL(hAPP(hAPP(A,f536(A,E)),f536(A,E))) | hBOOL(hAPP(hAPP(A,f535(A,E)),f536(A,E))) | -hBOOL(hAPP(hAPP(A,F),F)). [resolve(6300,f,6290,d)]. 176.15/176.00 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hAPP(A,B) != hAPP(A,C) | -hBOOL(hAPP(hAPP(A,C),C)) | hBOOL(hAPP(hAPP(A,C),B)) | -hBOOL(hAPP(hAPP(A,D),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))) | hAPP(A,f536(A,E)) = hAPP(A,f535(A,E)) | hBOOL(hAPP(hAPP(A,f535(A,E)),f536(A,E))) | -hBOOL(hAPP(hAPP(A,F),F)). [resolve(6300,f,6291,d)]. 176.15/176.00 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hAPP(A,B) != hAPP(A,C) | -hBOOL(hAPP(hAPP(A,C),C)) | hBOOL(hAPP(hAPP(A,C),B)) | -hBOOL(hAPP(hAPP(A,D),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))) | hBOOL(hAPP(hAPP(A,f535(A,E)),f535(A,E))) | hBOOL(hAPP(hAPP(A,f535(A,E)),f536(A,E))) | -hBOOL(hAPP(hAPP(A,F),F)). [resolve(6300,f,6292,d)]. 176.15/176.00 Derived: -hBOOL(hAPP(hAPP(A,B),B)) | hAPP(A,B) != hAPP(A,C) | -hBOOL(hAPP(hAPP(A,C),C)) | hBOOL(hAPP(hAPP(A,C),B)) | -hBOOL(hAPP(hAPP(A,D),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))) | -hBOOL(hAPP(hAPP(A,f536(A,E)),f536(A,E))) | hAPP(A,f536(A,E)) != hAPP(A,f535(A,E)) | -hBOOL(hAPP(hAPP(A,f535(A,E)),f535(A,E))) | -hBOOL(hAPP(hAPP(A,f535(A,E)),f536(A,E))) | -hBOOL(hAPP(hAPP(A,F),F)). [resolve(6300,f,6293,f)]. 176.15/176.01 6301 hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,C),B)) | -hBOOL(hAPP(hAPP(A,D),D)) | -c_Predicate_Osymp(E,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(6244,e,6235,b)]. 176.15/176.01 Derived: hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,C),B)) | -hBOOL(hAPP(hAPP(A,D),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))) | -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))). [resolve(6301,d,6285,b)]. 176.15/176.01 Derived: hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,C),B)) | -hBOOL(hAPP(hAPP(A,D),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))) | -c_Equiv__Relations_Oequivp(E,A). [resolve(6301,d,6286,b)]. 176.15/176.01 Derived: hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,C),B)) | -hBOOL(hAPP(hAPP(A,D),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))) | hBOOL(hAPP(hAPP(A,f536(A,E)),f536(A,E))) | hBOOL(hAPP(hAPP(A,f535(A,E)),f536(A,E))) | -hBOOL(hAPP(hAPP(A,F),F)). [resolve(6301,d,6290,d)]. 176.15/176.01 Derived: hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,C),B)) | -hBOOL(hAPP(hAPP(A,D),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))) | hAPP(A,f536(A,E)) = hAPP(A,f535(A,E)) | hBOOL(hAPP(hAPP(A,f535(A,E)),f536(A,E))) | -hBOOL(hAPP(hAPP(A,F),F)). [resolve(6301,d,6291,d)]. 176.15/176.01 Derived: hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,C),B)) | -hBOOL(hAPP(hAPP(A,D),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))) | hBOOL(hAPP(hAPP(A,f535(A,E)),f535(A,E))) | hBOOL(hAPP(hAPP(A,f535(A,E)),f536(A,E))) | -hBOOL(hAPP(hAPP(A,F),F)). [resolve(6301,d,6292,d)]. 176.15/176.01 Derived: hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,C),B)) | -hBOOL(hAPP(hAPP(A,D),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))) | -hBOOL(hAPP(hAPP(A,f536(A,E)),f536(A,E))) | hAPP(A,f536(A,E)) != hAPP(A,f535(A,E)) | -hBOOL(hAPP(hAPP(A,f535(A,E)),f535(A,E))) | -hBOOL(hAPP(hAPP(A,f535(A,E)),f536(A,E))) | -hBOOL(hAPP(hAPP(A,F),F)). [resolve(6301,d,6293,f)]. 176.15/176.01 6302 hAPP(A,B) = hAPP(A,C) | -hBOOL(hAPP(hAPP(A,C),B)) | -hBOOL(hAPP(hAPP(A,D),D)) | -c_Predicate_Osymp(E,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(6245,e,6235,b)]. 176.15/176.01 Derived: hAPP(A,B) = hAPP(A,C) | -hBOOL(hAPP(hAPP(A,C),B)) | -hBOOL(hAPP(hAPP(A,D),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))) | -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))). [resolve(6302,d,6285,b)]. 176.15/176.01 Derived: hAPP(A,B) = hAPP(A,C) | -hBOOL(hAPP(hAPP(A,C),B)) | -hBOOL(hAPP(hAPP(A,D),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))) | -c_Equiv__Relations_Oequivp(E,A). [resolve(6302,d,6286,b)]. 176.15/176.01 Derived: hAPP(A,B) = hAPP(A,C) | -hBOOL(hAPP(hAPP(A,C),B)) | -hBOOL(hAPP(hAPP(A,D),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))) | hBOOL(hAPP(hAPP(A,f536(A,E)),f536(A,E))) | hBOOL(hAPP(hAPP(A,f535(A,E)),f536(A,E))) | -hBOOL(hAPP(hAPP(A,F),F)). [resolve(6302,d,6290,d)]. 176.15/176.01 Derived: hAPP(A,B) = hAPP(A,C) | -hBOOL(hAPP(hAPP(A,C),B)) | -hBOOL(hAPP(hAPP(A,D),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))) | hAPP(A,f536(A,E)) = hAPP(A,f535(A,E)) | hBOOL(hAPP(hAPP(A,f535(A,E)),f536(A,E))) | -hBOOL(hAPP(hAPP(A,F),F)). [resolve(6302,d,6291,d)]. 176.15/176.01 Derived: hAPP(A,B) = hAPP(A,C) | -hBOOL(hAPP(hAPP(A,C),B)) | -hBOOL(hAPP(hAPP(A,D),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))) | hBOOL(hAPP(hAPP(A,f535(A,E)),f535(A,E))) | hBOOL(hAPP(hAPP(A,f535(A,E)),f536(A,E))) | -hBOOL(hAPP(hAPP(A,F),F)). [resolve(6302,d,6292,d)]. 176.15/176.01 Derived: hAPP(A,B) = hAPP(A,C) | -hBOOL(hAPP(hAPP(A,C),B)) | -hBOOL(hAPP(hAPP(A,D),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))) | -hBOOL(hAPP(hAPP(A,f536(A,E)),f536(A,E))) | hAPP(A,f536(A,E)) != hAPP(A,f535(A,E)) | -hBOOL(hAPP(hAPP(A,f535(A,E)),f535(A,E))) | -hBOOL(hAPP(hAPP(A,f535(A,E)),f536(A,E))) | -hBOOL(hAPP(hAPP(A,F),F)). [resolve(6302,d,6293,f)]. 176.15/176.01 6303 hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,B),C)) | -hBOOL(hAPP(hAPP(A,D),D)) | -c_Predicate_Osymp(E,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(6246,e,6235,b)]. 176.15/176.01 Derived: hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,B),C)) | -hBOOL(hAPP(hAPP(A,D),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))) | -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))). [resolve(6303,d,6285,b)]. 176.15/176.01 Derived: hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,B),C)) | -hBOOL(hAPP(hAPP(A,D),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))) | -c_Equiv__Relations_Oequivp(E,A). [resolve(6303,d,6286,b)]. 176.15/176.01 Derived: hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,B),C)) | -hBOOL(hAPP(hAPP(A,D),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))) | hBOOL(hAPP(hAPP(A,f536(A,E)),f536(A,E))) | hBOOL(hAPP(hAPP(A,f535(A,E)),f536(A,E))) | -hBOOL(hAPP(hAPP(A,F),F)). [resolve(6303,d,6290,d)]. 176.15/176.01 Derived: hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,B),C)) | -hBOOL(hAPP(hAPP(A,D),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))) | hAPP(A,f536(A,E)) = hAPP(A,f535(A,E)) | hBOOL(hAPP(hAPP(A,f535(A,E)),f536(A,E))) | -hBOOL(hAPP(hAPP(A,F),F)). [resolve(6303,d,6291,d)]. 176.15/176.01 Derived: hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,B),C)) | -hBOOL(hAPP(hAPP(A,D),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))) | hBOOL(hAPP(hAPP(A,f535(A,E)),f535(A,E))) | hBOOL(hAPP(hAPP(A,f535(A,E)),f536(A,E))) | -hBOOL(hAPP(hAPP(A,F),F)). [resolve(6303,d,6292,d)]. 176.15/176.01 Derived: hBOOL(hAPP(hAPP(A,B),B)) | -hBOOL(hAPP(hAPP(A,B),C)) | -hBOOL(hAPP(hAPP(A,D),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))) | -hBOOL(hAPP(hAPP(A,f536(A,E)),f536(A,E))) | hAPP(A,f536(A,E)) != hAPP(A,f535(A,E)) | -hBOOL(hAPP(hAPP(A,f535(A,E)),f535(A,E))) | -hBOOL(hAPP(hAPP(A,f535(A,E)),f536(A,E))) | -hBOOL(hAPP(hAPP(A,F),F)). [resolve(6303,d,6293,f)]. 176.15/176.01 6304 hBOOL(hAPP(hAPP(A,f534(A,B)),f534(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -c_Predicate_Osymp(B,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(6247,d,6235,b)]. 176.15/176.01 Derived: hBOOL(hAPP(hAPP(A,f534(A,B)),f534(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -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))) | -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(6304,c,6285,b)]. 176.15/176.01 Derived: hBOOL(hAPP(hAPP(A,f534(A,B)),f534(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -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))) | -c_Equiv__Relations_Oequivp(B,A). [resolve(6304,c,6286,b)]. 176.15/176.01 Derived: hBOOL(hAPP(hAPP(A,f534(A,B)),f534(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -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))) | hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,D),D)). [resolve(6304,c,6290,d)]. 176.15/176.01 Derived: hBOOL(hAPP(hAPP(A,f534(A,B)),f534(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -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))) | hAPP(A,f536(A,B)) = hAPP(A,f535(A,B)) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,D),D)). [resolve(6304,c,6291,d)]. 176.15/176.01 Derived: hBOOL(hAPP(hAPP(A,f534(A,B)),f534(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -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))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,D),D)). [resolve(6304,c,6292,d)]. 176.15/176.01 Derived: hBOOL(hAPP(hAPP(A,f534(A,B)),f534(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -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))) | -hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hAPP(A,f536(A,B)) != hAPP(A,f535(A,B)) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,D),D)). [resolve(6304,c,6293,f)]. 176.15/176.01 6305 hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -c_Predicate_Oreflp(B,A) | -c_Predicate_Osymp(B,A) | c_Equiv__Relations_Oequivp(B,A). [resolve(6248,d,6232,c)]. 176.15/176.01 Derived: hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -c_Predicate_Oreflp(B,A) | c_Equiv__Relations_Oequivp(B,A) | -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(6305,e,6285,b)]. 176.15/176.01 Derived: hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -c_Predicate_Oreflp(B,A) | c_Equiv__Relations_Oequivp(B,A) | hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,D),D)). [resolve(6305,e,6290,d)]. 176.15/176.01 6306 hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),D)) | -c_Predicate_Osymp(B,A) | hBOOL(hAPP(hAPP(A,f67(A,B)),f67(A,B))). [resolve(6248,d,6237,c)]. 176.15/176.01 Derived: hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),D)) | hBOOL(hAPP(hAPP(A,f67(A,B)),f67(A,B))) | -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(6306,e,6285,b)]. 176.15/176.01 Derived: hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),D)) | hBOOL(hAPP(hAPP(A,f67(A,B)),f67(A,B))) | hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,E),E)). [resolve(6306,e,6290,d)]. 176.15/176.01 6307 hAPP(A,f536(A,B)) = hAPP(A,f535(A,B)) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -c_Predicate_Oreflp(B,A) | -c_Predicate_Osymp(B,A) | c_Equiv__Relations_Oequivp(B,A). [resolve(6249,d,6232,c)]. 176.15/176.01 Derived: hAPP(A,f536(A,B)) = hAPP(A,f535(A,B)) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -c_Predicate_Oreflp(B,A) | c_Equiv__Relations_Oequivp(B,A) | -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(6307,e,6285,b)]. 176.15/176.01 Derived: hAPP(A,f536(A,B)) = hAPP(A,f535(A,B)) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -c_Predicate_Oreflp(B,A) | c_Equiv__Relations_Oequivp(B,A) | hAPP(A,f536(A,B)) = hAPP(A,f535(A,B)) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,D),D)). [resolve(6307,e,6291,d)]. 176.15/176.02 6308 hAPP(A,f536(A,B)) = hAPP(A,f535(A,B)) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),D)) | -c_Predicate_Osymp(B,A) | hBOOL(hAPP(hAPP(A,f67(A,B)),f67(A,B))). [resolve(6249,d,6237,c)]. 176.15/176.02 Derived: hAPP(A,f536(A,B)) = hAPP(A,f535(A,B)) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),D)) | hBOOL(hAPP(hAPP(A,f67(A,B)),f67(A,B))) | -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(6308,e,6285,b)]. 176.15/176.02 Derived: hAPP(A,f536(A,B)) = hAPP(A,f535(A,B)) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),D)) | hBOOL(hAPP(hAPP(A,f67(A,B)),f67(A,B))) | hAPP(A,f536(A,B)) = hAPP(A,f535(A,B)) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,E),E)). [resolve(6308,e,6291,d)]. 176.15/176.02 6309 hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -c_Predicate_Oreflp(B,A) | -c_Predicate_Osymp(B,A) | c_Equiv__Relations_Oequivp(B,A). [resolve(6250,d,6232,c)]. 176.15/176.02 Derived: hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -c_Predicate_Oreflp(B,A) | c_Equiv__Relations_Oequivp(B,A) | -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(6309,e,6285,b)]. 176.15/176.02 Derived: hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -c_Predicate_Oreflp(B,A) | c_Equiv__Relations_Oequivp(B,A) | hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,D),D)). [resolve(6309,e,6292,d)]. 176.15/176.02 6310 hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),D)) | -c_Predicate_Osymp(B,A) | hBOOL(hAPP(hAPP(A,f67(A,B)),f67(A,B))). [resolve(6250,d,6237,c)]. 176.15/176.02 Derived: hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),D)) | hBOOL(hAPP(hAPP(A,f67(A,B)),f67(A,B))) | -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(6310,e,6285,b)]. 176.15/176.02 Derived: hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),D)) | hBOOL(hAPP(hAPP(A,f67(A,B)),f67(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,E),E)). [resolve(6310,e,6292,d)]. 176.15/176.02 6311 -hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hAPP(A,f536(A,B)) != hAPP(A,f535(A,B)) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -c_Predicate_Oreflp(B,A) | -c_Predicate_Osymp(B,A) | c_Equiv__Relations_Oequivp(B,A). [resolve(6251,f,6232,c)]. 176.15/176.02 Derived: -hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hAPP(A,f536(A,B)) != hAPP(A,f535(A,B)) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -c_Predicate_Oreflp(B,A) | c_Equiv__Relations_Oequivp(B,A) | -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(6311,g,6285,b)]. 176.15/176.02 Derived: -hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hAPP(A,f536(A,B)) != hAPP(A,f535(A,B)) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -c_Predicate_Oreflp(B,A) | c_Equiv__Relations_Oequivp(B,A) | -hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hAPP(A,f536(A,B)) != hAPP(A,f535(A,B)) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,D),D)). [resolve(6311,g,6293,f)]. 176.15/176.02 6312 -hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hAPP(A,f536(A,B)) != hAPP(A,f535(A,B)) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),D)) | -c_Predicate_Osymp(B,A) | hBOOL(hAPP(hAPP(A,f67(A,B)),f67(A,B))). [resolve(6251,f,6237,c)]. 179.48/179.37 Derived: -hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hAPP(A,f536(A,B)) != hAPP(A,f535(A,B)) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),D)) | hBOOL(hAPP(hAPP(A,f67(A,B)),f67(A,B))) | -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(6312,g,6285,b)]. 179.48/179.37 Derived: -hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hAPP(A,f536(A,B)) != hAPP(A,f535(A,B)) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,C),C)) | -hBOOL(hAPP(hAPP(A,D),D)) | hBOOL(hAPP(hAPP(A,f67(A,B)),f67(A,B))) | -hBOOL(hAPP(hAPP(A,f536(A,B)),f536(A,B))) | hAPP(A,f536(A,B)) != hAPP(A,f535(A,B)) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f535(A,B))) | -hBOOL(hAPP(hAPP(A,f535(A,B)),f536(A,B))) | -hBOOL(hAPP(hAPP(A,E),E)). [resolve(6312,g,6293,f)]. 179.48/179.37 6313 class_Rings_Oordered__semiring(tc_Code__Numeral_Ocode__numeral) # label(arity_Code____Numeral__Ocode____numeral__Rings_Oordered__semiring) # label(axiom). [assumption]. 179.48/179.37 6314 -class_Rings_Oordered__semiring(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),D),E)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),E))) # label(fact_mult__mono_H) # label(axiom). [clausify(1459)]. 179.48/179.37 6315 -class_Rings_Oordered__semiring(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),B)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),D),C))) # label(fact_mult__left__mono) # label(axiom). [clausify(1615)]. 179.48/179.37 6316 -class_Rings_Oordered__semiring(A) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),B),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),D),E)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),C)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),c_Groups_Ozero__class_Ozero(A)),D)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(A),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),B),D)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(A),C),E))) # label(fact_mult__mono) # label(axiom). [clausify(2945)]. 179.48/179.37 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),C),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),A)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),c_Groups_Ozero__class_Ozero(tc_Code__Numeral_Ocode__numeral)),C)) | hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),A),C)),hAPP(hAPP(c_Groups_Otimes__class_Otimes(tc_Code__Numeral_Ocode__numeral),B),D))). [resolve(6313,a,6314,a)]. 179.48/179.37 Derived: -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),A),B)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),C),D)) | -hBOOL(hAPP(hAPP(c_Orderings_Oord__class_Oless__eq(tc_Code__Numeral_Ocode__numeral),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(cCputime limit exceeded (core dumped) 180.04/179.87 EOF