0.06/0.12 % Problem : theBenchmark.p : TPTP v0.0.0. Released v0.0.0. 0.06/0.12 % Command : tptp2X_and_run_prover9 %d %s 0.13/0.33 % Computer : n011.cluster.edu 0.13/0.33 % Model : x86_64 x86_64 0.13/0.33 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz 0.13/0.33 % Memory : 8042.1875MB 0.13/0.33 % OS : Linux 3.10.0-693.el7.x86_64 0.13/0.33 % CPULimit : 1200 0.13/0.33 % DateTime : Wed Jul 14 16:44:40 EDT 2021 0.13/0.34 % CPUTime : 1.00/1.28 ============================== Prover9 =============================== 1.00/1.28 Prover9 (32) version 2009-11A, November 2009. 1.00/1.28 Process 20486 was started by sandbox on n011.cluster.edu, 1.00/1.28 Wed Jul 14 16:44:40 2021 1.00/1.28 The command was "/export/starexec/sandbox/solver/bin/prover9 -t 1200 -f /tmp/Prover9_20293_n011.cluster.edu". 1.00/1.28 ============================== end of head =========================== 1.00/1.28 1.00/1.28 ============================== INPUT ================================= 1.00/1.28 1.00/1.28 % Reading from file /tmp/Prover9_20293_n011.cluster.edu 1.00/1.28 1.00/1.28 set(prolog_style_variables). 1.00/1.28 set(auto2). 1.00/1.28 % set(auto2) -> set(auto). 1.00/1.28 % set(auto) -> set(auto_inference). 1.00/1.28 % set(auto) -> set(auto_setup). 1.00/1.28 % set(auto_setup) -> set(predicate_elim). 1.00/1.28 % set(auto_setup) -> assign(eq_defs, unfold). 1.00/1.28 % set(auto) -> set(auto_limits). 1.00/1.28 % set(auto_limits) -> assign(max_weight, "100.000"). 1.00/1.28 % set(auto_limits) -> assign(sos_limit, 20000). 1.00/1.28 % set(auto) -> set(auto_denials). 1.00/1.28 % set(auto) -> set(auto_process). 1.00/1.28 % set(auto2) -> assign(new_constants, 1). 1.00/1.28 % set(auto2) -> assign(fold_denial_max, 3). 1.00/1.28 % set(auto2) -> assign(max_weight, "200.000"). 1.00/1.28 % set(auto2) -> assign(max_hours, 1). 1.00/1.28 % assign(max_hours, 1) -> assign(max_seconds, 3600). 1.00/1.28 % set(auto2) -> assign(max_seconds, 0). 1.00/1.28 % set(auto2) -> assign(max_minutes, 5). 1.00/1.28 % assign(max_minutes, 5) -> assign(max_seconds, 300). 1.00/1.28 % set(auto2) -> set(sort_initial_sos). 1.00/1.28 % set(auto2) -> assign(sos_limit, -1). 1.00/1.28 % set(auto2) -> assign(lrs_ticks, 3000). 1.00/1.28 % set(auto2) -> assign(max_megs, 400). 1.00/1.28 % set(auto2) -> assign(stats, some). 1.00/1.28 % set(auto2) -> clear(echo_input). 1.00/1.28 % set(auto2) -> set(quiet). 1.00/1.28 % set(auto2) -> clear(print_initial_clauses). 1.00/1.28 % set(auto2) -> clear(print_given). 1.00/1.28 assign(lrs_ticks,-1). 1.00/1.28 assign(sos_limit,10000). 1.00/1.28 assign(order,kbo). 1.00/1.28 set(lex_order_vars). 1.00/1.28 clear(print_given). 1.00/1.28 1.00/1.28 % formulas(sos). % not echoed (449 formulas) 1.00/1.28 1.00/1.28 ============================== end of input ========================== 1.00/1.28 1.00/1.28 % From the command line: assign(max_seconds, 1200). 1.00/1.28 1.00/1.28 ============================== PROCESS NON-CLAUSAL FORMULAS ========== 1.00/1.28 1.00/1.28 % Formulas that are not ordinary clauses: 1.00/1.28 1 (all F all A (hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f2009550088ol_nat(finite1306199131a_bool,image_112932426a_bool(F,A))),hAPP_f921600141ol_nat(finite_card_pname,A))))) # label(fact_72_card__image__le) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 2 (all M_2 all N_1 (N_1 = M_2 -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_1)))) # label(fact_125_eq__imp__le) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 3 (all Pa (is_fun949378684l_bool(Pa) -> collect_fun_a_bool(Pa) = Pa)) # label(fact_258_Collect__def) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 4 (all F all A (hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_fun_a_bool_nat(finite_card_a,image_876012084bool_a(F,A))),hAPP_f55526627ol_nat(finite1340463720e_bool,A))))) # label(fact_66_card__image__le) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 5 (all P all Q all R hAPP_pname_bool(hAPP_a93125764e_bool(cOMBC_pname_a_bool(P),Q),R) = hAPP_a_bool(hAPP_p1534023578a_bool(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__Com__Opname_000t__a_000tc__HOL__Obool_U) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 6 (all A all B (is_fun_pname_bool(A) & is_fun_pname_bool(B) -> (hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B)) -> (hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f921600141ol_nat(finite_card_pname,B)),hAPP_f921600141ol_nat(finite_card_pname,A))) -> A = B))))) # label(fact_89_card__seteq) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 7 (all B_1_1 all B_2 is_fun_pname_bool(image_990671762_pname(B_1_1,B_2))) # label(gsy_c_Set_Oimage_000tc__fun_Itc__fun_It__a_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J_0_001) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 8 (all B all X_2 all A (hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A)) -> (hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B)) -> hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),B))))) # label(fact_242_set__rev__mp) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 9 (all Y_1 all A all X_2 (hBOOL(hAPP_nat_bool(A,X_2)) | X_2 = Y_1 <-> hBOOL(hAPP_nat_bool(insert_nat(Y_1,A),X_2)))) # label(fact_213_insert__code) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 10 (all B_1 all A_1 all B (hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_1),B)) -> hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_1),insert_a(B_1,B))))) # label(fact_221_insertI2) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 11 (all X_2 all A all B (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,insert_nat(X_2,A)),B)) <-> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)) & hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),B)))) # label(fact_272_insert__subset) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 12 (all A_1 all A (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),A)) -> A = insert_nat(A_1,A))) # label(fact_222_insert__absorb) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 13 (all A_1 all A (hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A)) -> hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,insert1325755072e_bool(A_1,A))))) # label(fact_52_finite_OinsertI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 14 (all P all Q all R hAPP_f389811538l_bool(hAPP_f1759205631l_bool(P,R),Q) = hAPP_f389811538l_bool(hAPP_f1759205631l_bool(cOMBC_336095980l_bool(P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__fun_Itc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obo) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 15 (all B_1_1 all B_2 (is_a(B_2) -> is_fun_pname_bool(hAPP_a93125764e_bool(B_1_1,B_2)))) # label(gsy_c_hAPP_000t__a_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 16 (all M_3 all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,Na)),M_3)) <-> -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_3),Na)))) # label(fact_158_not__less__eq__eq) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 17 (all A all B (is_fun_pname_bool(B) & is_fun_pname_bool(A) -> (hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B)) -> (hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B),A)) -> A = B)))) # label(fact_178_equalityI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 18 (all B all F all A (hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A)) -> (hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),image_pname_a(F,A))) -> hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B))))) # label(fact_169_finite__surj) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 19 (all B all A_1 hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),insert_a(A_1,B)))) # label(fact_271_subset__insertI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 20 (all H all F_1 (hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_1)) -> hBOOL(hAPP_f937997336l_bool(finite1701474069l_bool,image_1154884483l_bool(H,F_1))))) # label(fact_32_finite__imageI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 21 (all C_1 all A all B (hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B)) -> (hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,C_1),A)) -> hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,C_1),B))))) # label(fact_181_subsetD) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 22 (all A all B (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B)) -> (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)) -> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))))) # label(fact_151_rev__finite__subset) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 23 (all X all Y (is_fun_a_bool(X) & is_fun_a_bool(Y) -> Y = X | -hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(fequal_fun_a_bool,X),Y)))) # label(help_fequal_1_1_fequal_000tc__fun_It__a_Mtc__HOL__Obool_J_T) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 24 (all A all B (hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B)) -> (hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B)) -> hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))))) # label(fact_143_finite__subset) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 25 (all F all X_2 all A (hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A)) -> insert_a(hAPP_pname_a(F,X_2),image_pname_a(F,A)) = image_pname_a(F,A))) # label(fact_285_insert__image) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 26 (all A all B (B = A -> hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B),A)))) # label(fact_235_equalityD2) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 27 (all A_1 all A (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) <-> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,insert_nat(A_1,A))))) # label(fact_135_finite__insert) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 28 (all A all B (hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B)) -> (hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B)) -> hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A))))) # label(fact_149_rev__finite__subset) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 29 (all B_1_1 all B_2 is_bool(hAPP_f1637334154l_bool(B_1_1,B_2))) # label(gsy_c_hAPP_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__HOL__Obool_) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 30 (all B_1_1 all B_2 (is_fun1661590463l_bool(B_2) -> is_fun1661590463l_bool(hAPP_f559147733l_bool(B_1_1,B_2)))) # label(gsy_c_hAPP_000tc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_Mtc__HOL__Oboo_005) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 31 (all F all X_2 all A (hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A)) -> hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,hAPP_pname_a(F,X_2)),image_pname_a(F,A))))) # label(fact_261_imageI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 32 (all X_2 all A all B (hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B)) -> (hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A)) -> hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),B))))) # label(fact_245_set__mp) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 33 (all X all Y (hAPP_nat_nat(suc,Y) = hAPP_nat_nat(suc,X) -> X = Y)) # label(fact_119_Suc__inject) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 34 (all A all B (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),A)) & hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)) <-> B = A)) # label(fact_228_set__eq__subset) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 35 (all B_1_1 all B_2 (is_pname(B_2) -> is_fun_a_bool(hAPP_p1534023578a_bool(B_1_1,B_2)))) # label(gsy_c_hAPP_000tc__Com__Opname_000tc__fun_It__a_Mtc__HOL__Obool_J) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 36 (all F all A (hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_fun_a_bool_nat(finite_card_a,image_fun_nat_bool_a(F,A))),hAPP_f696928925ol_nat(finite346522414t_bool,A))))) # label(fact_67_card__image__le) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 37 (all B_1_1 all B_2 (is_fun_pname_bool(B_2) -> is_fun_pname_bool(cOMBB_647938656_pname(B_1_1,B_2)))) # label(gsy_c_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Com__Opname) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 38 (all P all Q all R hAPP_bool_bool(hAPP_n1006566506l_bool(P,R),hAPP_nat_bool(Q,R)) = hAPP_nat_bool(cOMBS_nat_bool_bool(P,Q),R)) # label(help_COMBS_1_1_COMBS_000tc__Nat__Onat_000tc__HOL__Obool_000tc__HOL__Obool_U) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 39 (all Na all K all M_3 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),M_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),Na)) -> (hAPP_nat_nat(minus_minus_nat(M_3),K) = hAPP_nat_nat(minus_minus_nat(Na),K) <-> M_3 = Na)))) # label(fact_164_eq__diff__iff) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 40 (all B_1 all F all X_2 all A (hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A)) -> (hAPP_pname_a(F,X_2) = B_1 -> hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,B_1),image_pname_a(F,A)))))) # label(fact_262_rev__image__eqI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 41 (all X_2 all A (hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A)) -> (-hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_2),A)) -> hAPP_fun_a_bool_nat(finite_card_a,insert_a(X_2,A)) = hAPP_nat_nat(suc,hAPP_fun_a_bool_nat(finite_card_a,A))) & (hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_2),A)) -> hAPP_fun_a_bool_nat(finite_card_a,insert_a(X_2,A)) = hAPP_fun_a_bool_nat(finite_card_a,A)))) # label(fact_103_card__insert__if) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 42 (all A_1 all B collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fdisj,hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(fequal_nat),A_1)),hAPP_f800510211t_bool(cOMBC_226598744l_bool(member_nat),B))) = insert_nat(A_1,B)) # label(fact_192_insert__compr) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 43 (all H all F_1 (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1)) -> hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,image_26036933t_bool(H,F_1))))) # label(fact_30_finite__imageI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 44 (all M_2 all N_1 all K_1 hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(suc,M_2)),N_1)),hAPP_nat_nat(suc,K_1)) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(minus_minus_nat(M_2),N_1)),K_1)) # label(fact_160_Suc__diff__diff) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 45 (all A all B (hBOOL(hAPP_f1935102916l_bool(hAPP_f510955609l_bool(ord_le675606854l_bool,A),B)) -> (hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,B)) -> hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A))))) # label(fact_142_finite__subset) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 46 (all X_2 all A (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) -> (-hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A)) -> hAPP_nat_nat(suc,hAPP_f22106695ol_nat(finite_card_nat,A)) = hAPP_f22106695ol_nat(finite_card_nat,insert_nat(X_2,A))))) # label(fact_107_card__insert__disjoint) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 47 (all N_1 all M_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M_2)) -> hAPP_nat_nat(suc,hAPP_nat_nat(minus_minus_nat(M_2),N_1)) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(suc,M_2)),N_1))) # label(fact_116_Suc__diff__le) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 48 (all A all B (is_fun1661590463l_bool(B) & is_fun1661590463l_bool(A) -> (hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,B)) -> (hBOOL(hAPP_f1935102916l_bool(hAPP_f510955609l_bool(ord_le675606854l_bool,A),B)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f55526627ol_nat(finite1340463720e_bool,B)),hAPP_f55526627ol_nat(finite1340463720e_bool,A))) -> A = B))))) # label(fact_87_card__seteq) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 49 (all B_1_1 (is_fun_a_bool(B_1_1) -> is_fun949378684l_bool(p(B_1_1)))) # label(gsy_v_P) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 50 (all A_1 all B insert_fun_a_bool(A_1,B) = collect_fun_a_bool(cOMBS_1035972772l_bool(cOMBB_338059395a_bool(fdisj,hAPP_f1631501043l_bool(cOMBC_1732670874l_bool(fequal_fun_a_bool),A_1)),hAPP_f2117159681l_bool(cOMBC_1880041174l_bool(member_fun_a_bool),B)))) # label(fact_197_insert__compr) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 51 (all P all Q all R hAPP_f1476298914l_bool(cOMBB_2095475776e_bool(P,Q),R) = hAPP_b589554111l_bool(P,hAPP_f1664156314l_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_010) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 52 (all B_1_1 all B_2 is_fun_a_bool(image_573985017bool_a(B_1_1,B_2))) # label(gsy_c_Set_Oimage_000tc__fun_Itc__fun_It__a_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J_0) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 53 (all P all Q all R hAPP_fun_a_bool_bool(cOMBS_1035972772l_bool(P,Q),R) = hAPP_bool_bool(hAPP_f198738859l_bool(P,R),hAPP_fun_a_bool_bool(Q,R))) # label(help_COMBS_1_1_COMBS_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__HOL__Obool_000tc_) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 54 (all A_1 all A (is_fun_a_bool(A) -> (hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_1),A)) -> insert_a(A_1,A) = A))) # label(fact_224_insert__absorb) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 55 (all A all B (A = B -> hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B)))) # label(fact_233_equalityD1) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 56 (all B_1_1 all B_2 (is_fun_a_bool(B_2) -> is_fun_pname_bool(image_a_pname(B_1_1,B_2)))) # label(gsy_c_Set_Oimage_000t__a_000tc__Com__Opname) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 57 (all X_1 all Xa collect_fun_a_bool(cOMBS_1035972772l_bool(cOMBB_338059395a_bool(fdisj,hAPP_f1631501043l_bool(cOMBC_1732670874l_bool(fequal_fun_a_bool),X_1)),hAPP_f2117159681l_bool(cOMBC_1880041174l_bool(member_fun_a_bool),Xa))) = insert_fun_a_bool(X_1,Xa)) # label(fact_268_insert__compr__raw) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 58 (all Pa all Q_1 (hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,collect_fun_a_bool(Q_1))) & hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,collect_fun_a_bool(Pa))) <-> hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,collect_fun_a_bool(cOMBS_1035972772l_bool(cOMBB_338059395a_bool(fdisj,Pa),Q_1)))))) # label(fact_132_finite__Collect__disjI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 59 (all H all F_1 (hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,F_1)) -> hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,image_1921560913_pname(H,F_1))))) # label(fact_23_finite__imageI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 60 (all F all A (hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(finite_card_nat,image_pname_nat(F,A))),hAPP_f921600141ol_nat(finite_card_pname,A))))) # label(fact_54_card__image__le) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 61 (all X_2 all A all B (hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,insert_pname(X_2,A)),B)) <-> hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),B)) & hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B)))) # label(fact_273_insert__subset) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 62 (all F all A (hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_fun_a_bool_nat(finite_card_a,image_fun_a_bool_a(F,A))),hAPP_f2009550088ol_nat(finite1306199131a_bool,A))))) # label(fact_65_card__image__le) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 63 (all X_2 all A all B (hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B)) -> (hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_2),A)) -> hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_2),B))))) # label(fact_238_in__mono) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 64 (all X_2 all A insert_a(X_2,insert_a(X_2,A)) = insert_a(X_2,A)) # label(fact_206_insert__absorb2) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 65 (all B_1_1 all B_2 is_fun1661590463l_bool(image_1655916159e_bool(B_1_1,B_2))) # label(gsy_c_Set_Oimage_000tc__Nat__Onat_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 66 (all B_1_1 all B_2 (is_fun_a_bool(B_2) -> is_fun_a_bool(hAPP_f2050579477a_bool(B_1_1,B_2)))) # label(gsy_c_hAPP_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__fun_It__a_Mtc__HOL__Obool_J) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 67 (all H all F_1 (hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,F_1)) -> hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,image_876012084bool_a(H,F_1))))) # label(fact_41_finite__imageI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 68 (all X_1 all Xa collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fdisj,hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(fequal_nat),X_1)),hAPP_f800510211t_bool(cOMBC_226598744l_bool(member_nat),Xa))) = insert_nat(X_1,Xa)) # label(fact_263_insert__compr__raw) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 69 (all F all A (hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f921600141ol_nat(finite_card_pname,image_1283814551_pname(F,A))),hAPP_f55526627ol_nat(finite1340463720e_bool,A))))) # label(fact_78_card__image__le) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 70 (all A_1 all A (hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A)) -> hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,insert_fun_nat_bool(A_1,A))))) # label(fact_53_finite_OinsertI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 71 (all A all B (A = B -> -(hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B)) -> -hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B),A))))) # label(fact_250_equalityE) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 72 (all H all F_1 (hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_1)) -> hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,image_2129980159t_bool(H,F_1))))) # label(fact_37_finite__imageI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 73 (all Pa all Q_1 (hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,collect_a(cOMBS_a_bool_bool(cOMBB_1972296269bool_a(fdisj,Pa),Q_1)))) <-> hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,collect_a(Pa))) & hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,collect_a(Q_1))))) # label(fact_134_finite__Collect__disjI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 74 (all B_1_1 all B_2 (is_fun_pname_bool(B_2) -> is_fun_pname_bool(hAPP_f759274231e_bool(B_1_1,B_2)))) # label(gsy_c_hAPP_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__fun_Itc__Com__Op) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 75 (all A hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),A))) # label(fact_226_subset__refl) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 76 (all P all Q all R hAPP_f54304608l_bool(cOMBS_1187019125l_bool(P,Q),R) = hAPP_bool_bool(hAPP_f1748468828l_bool(P,R),hAPP_f54304608l_bool(Q,R))) # label(help_COMBS_1_1_COMBS_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__HOL__Obo) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 77 (all A_1 all C all D (hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,C),D)) -> hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,insert_pname(A_1,C)),insert_pname(A_1,D))))) # label(fact_282_insert__mono) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 78 (all B_1_1 all B_2 (is_a(B_2) -> is_fun_a_bool(hAPP_a_fun_a_bool(B_1_1,B_2)))) # label(gsy_c_hAPP_000t__a_000tc__fun_It__a_Mtc__HOL__Obool_J) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 79 (all A_1 all B hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),insert_pname(A_1,B)))) # label(fact_190_insertI1) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 80 (all A_1 all B collect_pname(cOMBS_568398431l_bool(cOMBB_675860798_pname(fdisj,hAPP_p61793385e_bool(cOMBC_1149511130e_bool(fequal_pname),A_1)),hAPP_f759274231e_bool(cOMBC_1058051404l_bool(member_pname),B))) = insert_pname(A_1,B)) # label(fact_193_insert__compr) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 81 (all P all Q all R hAPP_p393069232l_bool(cOMBB_675860798_pname(P,Q),R) = hAPP_b589554111l_bool(P,hAPP_pname_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_007) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 82 (all C all A all B (hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B)) -> (hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B),C)) -> hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),C))))) # label(fact_247_subset__trans) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 83 (all A_1 all C all D (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,C),D)) -> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,insert_nat(A_1,C)),insert_nat(A_1,D))))) # label(fact_281_insert__mono) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 84 (all B all X_2 all A (hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_2),A)) -> (hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B)) -> hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_2),B))))) # label(fact_241_set__rev__mp) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 85 (all H all F_1 (hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_1)) -> hBOOL(hAPP_f292226953l_bool(finite1381704300l_bool,image_1420695166l_bool(H,F_1))))) # label(fact_34_finite__imageI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 86 (all X all Y (hBOOL(hAPP_a_bool(hAPP_a_fun_a_bool(fequal_a,X),Y)) | X != Y)) # label(help_fequal_2_1_fequal_000t__a_T) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 87 (all Pa Pa = collect_nat(Pa)) # label(fact_259_Collect__def) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 88 (all P all Q all R hAPP_f1664156314l_bool(cOMBS_350070575l_bool(P,Q),R) = hAPP_bool_bool(hAPP_f1476298914l_bool(P,R),hAPP_f1664156314l_bool(Q,R))) # label(help_COMBS_1_1_COMBS_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__HOL__O) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 89 (all P all Q all R hAPP_f1748468828l_bool(cOMBB_444170502t_bool(P,Q),R) = hAPP_b589554111l_bool(P,hAPP_f54304608l_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_009) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 90 (all B_1 all A_1 all B ((-hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),B)) -> B_1 = A_1) -> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),insert_nat(B_1,B))))) # label(fact_183_insertCI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 91 (all B_1_1 all B_2 is_fun_pname_bool(image_1604018183_pname(B_1_1,B_2))) # label(gsy_c_Set_Oimage_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__HOL___003) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 92 (all P all Q (-hBOOL(Q) | hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fdisj,P),Q)))) # label(help_fdisj_2_1_U) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 93 (all B_1_1 all B_2 (is_fun1661590463l_bool(B_2) -> is_fun_pname_bool(image_1283814551_pname(B_1_1,B_2)))) # label(gsy_c_Set_Oimage_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__Com__Opnam) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 94 (all Ts all G (hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,Ts),G)) -> hBOOL(hAPP_fun_a_bool_bool(p(G),Ts)))) # label(fact_0_assms_I1_J) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 95 (all P (hBOOL(hAPP_bool_bool(fNot,P)) | hBOOL(P))) # label(help_fNot_2_1_U) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 96 (all A (hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A)) -> hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,collect_fun_a_bool(hAPP_f1631501043l_bool(cOMBC_1732670874l_bool(ord_le1311769555a_bool),A)))))) # label(fact_3_finite__Collect__subsets) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 97 (all X_2 hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,X_2),X_2))) # label(fact_296_order__refl) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 98 (all H all F_1 (hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_1)) -> hBOOL(hAPP_f389811538l_bool(finite786885583l_bool,image_1642285373l_bool(H,F_1))))) # label(fact_33_finite__imageI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 99 (all A_1 all B hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_1),insert_a(A_1,B)))) # label(fact_191_insertI1) # label(axiom) # label(non_clause). [assumption]. 1.00/1.28 100 (all A all B (hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B)) -> (hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B)) -> hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))))) # label(fact_146_finite__subset) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 101 (all Na hAPP_nat_nat(suc,Na) = hAPP_f22106695ol_nat(finite_card_nat,collect_nat(hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(ord_less_eq_nat),Na)))) # label(fact_118_card__Collect__le__nat) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 102 (all X_2 all A (hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f921600141ol_nat(finite_card_pname,A)),hAPP_f921600141ol_nat(finite_card_pname,insert_pname(X_2,A)))))) # label(fact_95_card__insert__le) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 103 (all F all A (hBOOL(hAPP_f937997336l_bool(finite1701474069l_bool,A)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_fun_a_bool_nat(finite_card_a,image_526090948bool_a(F,A))),hAPP_f1690079119ol_nat(finite1352710292l_bool,A))))) # label(fact_62_card__image__le) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 104 (all Pa all Q_1 (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fdisj,Pa),Q_1)))) <-> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(Q_1))) & hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(Pa))))) # label(fact_133_finite__Collect__disjI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 105 (all B all F all A (is_fun_a_bool(B) -> ((exists AA (hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,AA),A)) & image_pname_a(F,AA) = B & is_fun_pname_bool(AA))) <-> hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),image_pname_a(F,A)))))) # label(fact_286_subset__image__iff) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 106 (all X all Y (hBOOL(hAPP_pname_bool(hAPP_p61793385e_bool(fequal_pname,X),Y)) | X != Y)) # label(help_fequal_2_1_fequal_000tc__Com__Opname_T) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 107 (all B_1_1 all B_2 is_fun_a_bool(image_nat_a(B_1_1,B_2))) # label(gsy_c_Set_Oimage_000tc__Nat__Onat_000t__a) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 108 (all X_3 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_3),X_3))) # label(fact_297_order__refl) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 109 (all B_1_1 all B_2 is_bool(hAPP_f937997336l_bool(B_1_1,B_2))) # label(gsy_c_hAPP_000tc__fun_Itc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__HO) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 110 (all X_2 all A (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A)) <-> hBOOL(hAPP_nat_bool(A,X_2)))) # label(fact_252_mem__def) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 111 (all F all A (hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(finite_card_nat,image_496248727ol_nat(F,A))),hAPP_f696928925ol_nat(finite346522414t_bool,A))))) # label(fact_56_card__image__le) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 112 (all B_1 all A all B (hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B)) -> hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),insert_a(B_1,B))))) # label(fact_280_subset__insertI2) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 113 (all F all A (hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f921600141ol_nat(finite_card_pname,image_a_pname(F,A))),hAPP_fun_a_bool_nat(finite_card_a,A))))) # label(fact_59_card__image__le) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 114 (all Q_1 all Pa (hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,collect_a(Q_1))) | hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,collect_a(Pa))) -> hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,collect_a(cOMBS_a_bool_bool(cOMBB_1972296269bool_a(fconj,Pa),Q_1)))))) # label(fact_113_finite__Collect__conjI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 115 (all A_1 all B_1 all A (is_a(B_1) & is_a(A_1) -> (hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_1),insert_a(B_1,A))) -> (B_1 != A_1 -> hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_1),A)))))) # label(fact_188_insertE) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 116 (all X_1 all Xa collec1974731493e_bool(cOMBS_350070575l_bool(cOMBB_2095475776e_bool(fdisj,hAPP_f434788991l_bool(cOMBC_1284144636l_bool(fequal533582459e_bool),X_1)),hAPP_f559147733l_bool(cOMBC_1988546018l_bool(member799430823e_bool),Xa))) = insert1325755072e_bool(X_1,Xa)) # label(fact_267_insert__compr__raw) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 117 (all B_1_1 all B_2 (is_fun_a_bool(B_2) -> is_fun949378684l_bool(hAPP_f1631501043l_bool(B_1_1,B_2)))) # label(gsy_c_hAPP_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_It__a_Mtc__HOL) # label(hypothesis) # label(non_clause). [assumption]. 1.00/1.29 118 (all X_2 all Y_1 all A insert_pname(X_2,insert_pname(Y_1,A)) = insert_pname(Y_1,insert_pname(X_2,A))) # label(fact_208_insert__commute) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 119 (all F all B all A ((all X_1 (is_pname(X_1) -> (hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A)) -> hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,hAPP_pname_a(F,X_1)),B))))) -> hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,image_pname_a(F,A)),B)))) # label(fact_294_image__subsetI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 120 (all B_1_1 all B_2 (is_pname(B_1_1) & is_fun_pname_bool(B_2) -> is_fun_pname_bool(insert_pname(B_1_1,B_2)))) # label(gsy_c_Set_Oinsert_000tc__Com__Opname) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 121 (all A all B (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B)) -> (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(finite_card_nat,B)),hAPP_f22106695ol_nat(finite_card_nat,A))) -> B = A)))) # label(fact_91_card__seteq) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 122 (all P all Q all R hAPP_n1006566506l_bool(cOMBB_1015721476ol_nat(P,Q),R) = hAPP_b589554111l_bool(P,hAPP_nat_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_006) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 123 (all P all Q all R hAPP_nat_bool(hAPP_n1699378549t_bool(P,R),Q) = hAPP_nat_bool(hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__Nat__Onat_000tc__HOL__Obool_U) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 124 (all P all Q all R hAPP_f1637334154l_bool(hAPP_f1951378235l_bool(P,R),Q) = hAPP_f54304608l_bool(hAPP_f1246832597l_bool(cOMBC_1245412066l_bool(P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc__011) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 125 (all F all A (-hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A)) -> (hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,image_pname_a(F,A))) -> (exists X_1 (is_pname(X_1) & -hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,collect_pname(cOMBS_568398431l_bool(cOMBB_675860798_pname(fconj,hAPP_f759274231e_bool(cOMBC_1058051404l_bool(member_pname),A)),hAPP_a93125764e_bool(cOMBC_pname_a_bool(cOMBB_1897541054_pname(fequal_a,F)),hAPP_pname_a(F,X_1)))))) & hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A))))))) # label(fact_175_pigeonhole__infinite) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 126 (all X_2 all Y_1 all A insert_a(Y_1,insert_a(X_2,A)) = insert_a(X_2,insert_a(Y_1,A))) # label(fact_209_insert__commute) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 127 (all X_2 all A (hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A)) -> (-hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A)) -> hAPP_f921600141ol_nat(finite_card_pname,insert_pname(X_2,A)) = hAPP_nat_nat(suc,hAPP_f921600141ol_nat(finite_card_pname,A))))) # label(fact_108_card__insert__disjoint) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 128 (all B all A ((all X_1 (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),A)) -> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),B)))) -> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)))) # label(fact_289_subsetI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 129 (all Pa all Q_1 (hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,collec1974731493e_bool(cOMBS_350070575l_bool(cOMBB_2095475776e_bool(fdisj,Pa),Q_1)))) <-> hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,collec1974731493e_bool(Q_1))) & hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,collec1974731493e_bool(Pa))))) # label(fact_131_finite__Collect__disjI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 130 (all A all B (is_fun949378684l_bool(B) & is_fun949378684l_bool(A) -> (hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,B)) -> (hBOOL(hAPP_f621171935l_bool(hAPP_f1434722111l_bool(ord_le1375614389l_bool,A),B)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f2009550088ol_nat(finite1306199131a_bool,B)),hAPP_f2009550088ol_nat(finite1306199131a_bool,A))) -> A = B))))) # label(fact_88_card__seteq) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 131 (all X all Y (is_a(X) & is_a(Y) -> -hBOOL(hAPP_a_bool(hAPP_a_fun_a_bool(fequal_a,X),Y)) | X = Y)) # label(help_fequal_1_1_fequal_000t__a_T) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 132 (all B_1_1 all B_2 (is_a(B_2) & is_fun_a_bool(B_1_1) -> is_bool(hAPP_a_bool(B_1_1,B_2)))) # label(gsy_c_hAPP_000t__a_000tc__HOL__Obool) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 133 (all C all A all B (hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B)) -> (hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),C)) -> hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),C))))) # label(fact_248_subset__trans) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 134 (all A all B (hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,B)) -> (hBOOL(hAPP_f621171935l_bool(hAPP_f1434722111l_bool(ord_le1375614389l_bool,A),B)) -> hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A))))) # label(fact_150_rev__finite__subset) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 135 (all I_1 all J all K_1 hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(minus_minus_nat(I_1),K_1)),J) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(minus_minus_nat(I_1),J)),K_1)) # label(fact_128_diff__commute) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 136 (all A (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) -> hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,collect_fun_nat_bool(hAPP_f103356543l_bool(cOMBC_1693257480l_bool(ord_le1568362934t_bool),A)))))) # label(fact_1_finite__Collect__subsets) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 137 (all P all Q (-hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q)) | hBOOL(P))) # label(help_fconj_2_1_U) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 138 (all P all Q all R hAPP_fun_a_bool_bool(hAPP_f2117159681l_bool(cOMBC_1880041174l_bool(P),Q),R) = hAPP_f621171935l_bool(hAPP_f285962445l_bool(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_It__) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 139 (all Na all M_3 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,Na)),hAPP_nat_nat(suc,M_3))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),M_3)))) # label(fact_156_Suc__le__mono) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 140 (all A_1 all B collect_fun_nat_bool(cOMBS_1187019125l_bool(cOMBB_444170502t_bool(fdisj,hAPP_f103356543l_bool(cOMBC_1693257480l_bool(fequal_fun_nat_bool),A_1)),hAPP_f1246832597l_bool(cOMBC_1245412066l_bool(member_fun_nat_bool),B))) = insert_fun_nat_bool(A_1,B)) # label(fact_195_insert__compr) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 141 (all X_2 all A insert_pname(X_2,insert_pname(X_2,A)) = insert_pname(X_2,A)) # label(fact_205_insert__absorb2) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 142 (all X_2 all A (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) -> (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A)) -> hAPP_f22106695ol_nat(finite_card_nat,A) = hAPP_f22106695ol_nat(finite_card_nat,insert_nat(X_2,A))) & (-hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A)) -> hAPP_nat_nat(suc,hAPP_f22106695ol_nat(finite_card_nat,A)) = hAPP_f22106695ol_nat(finite_card_nat,insert_nat(X_2,A))))) # label(fact_101_card__insert__if) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 143 (all F all A (hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(finite_card_nat,image_fun_a_bool_nat(F,A))),hAPP_f2009550088ol_nat(finite1306199131a_bool,A))))) # label(fact_58_card__image__le) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 144 (all P all Q all R hAPP_bool_bool(hAPP_a_fun_bool_bool(P,R),hAPP_a_bool(Q,R)) = hAPP_a_bool(cOMBS_a_bool_bool(P,Q),R)) # label(help_COMBS_1_1_COMBS_000t__a_000tc__HOL__Obool_000tc__HOL__Obool_U) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 145 (all Q all P (-hBOOL(P) | hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fdisj,P),Q)))) # label(help_fdisj_1_1_U) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 146 (all B_1_1 all B_2 (is_fun_a_bool(B_2) -> is_fun_a_bool(cOMBS_a_bool_bool(B_1_1,B_2)))) # label(gsy_c_COMBS_000t__a_000tc__HOL__Obool_000tc__HOL__Obool) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 147 (all B_1_1 all B_2 is_fun_pname_bool(image_1705983821_pname(B_1_1,B_2))) # label(gsy_c_Set_Oimage_000tc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_Mtc__HOL_002) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 148 (all H all F_1 (hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_1)) -> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,image_pname_nat(H,F_1))))) # label(fact_43_finite__imageI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 149 (all A all B (is_fun_a_bool(A) & is_fun_a_bool(B) -> (hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B)) -> (hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_fun_a_bool_nat(finite_card_a,B)),hAPP_fun_a_bool_nat(finite_card_a,A))) -> B = A))))) # label(fact_90_card__seteq) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 150 (all A_1 all B collec1974731493e_bool(cOMBS_350070575l_bool(cOMBB_2095475776e_bool(fdisj,hAPP_f434788991l_bool(cOMBC_1284144636l_bool(fequal533582459e_bool),A_1)),hAPP_f559147733l_bool(cOMBC_1988546018l_bool(member799430823e_bool),B))) = insert1325755072e_bool(A_1,B)) # label(fact_196_insert__compr) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 151 (all X all Y (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(fequal_nat,X),Y)) | Y = X)) # label(help_fequal_1_1_fequal_000tc__Nat__Onat_T) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 152 (all K_1 all I_1 all J (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),J)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,J),K_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),K_1))))) # label(fact_124_le__trans) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 153 (all H all F_1 (hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,F_1)) -> hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,image_a_a(H,F_1))))) # label(fact_39_finite__imageI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 154 (all X_2 all A (hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A)) -> (-hBOOL(hAPP_f1637334154l_bool(hAPP_f1951378235l_bool(member_fun_nat_bool,X_2),A)) -> hAPP_f696928925ol_nat(finite346522414t_bool,insert_fun_nat_bool(X_2,A)) = hAPP_nat_nat(suc,hAPP_f696928925ol_nat(finite346522414t_bool,A))))) # label(fact_104_card__insert__disjoint) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 155 (all B_1_1 all B_2 is_fun_pname_bool(image_nat_pname(B_1_1,B_2))) # label(gsy_c_Set_Oimage_000tc__Nat__Onat_000tc__Com__Opname) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 156 (all B_1_1 all B_2 (is_fun1661590463l_bool(B_2) -> is_fun1661590463l_bool(cOMBB_307249310e_bool(B_1_1,B_2)))) # label(gsy_c_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__fun_Itc__Com__Opname_Mtc_) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 157 (all H all F_1 (hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,F_1)) -> hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,image_fun_a_bool_a(H,F_1))))) # label(fact_42_finite__imageI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 158 (all H all F_1 (hBOOL(hAPP_f389811538l_bool(finite786885583l_bool,F_1)) -> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,image_1079571347ol_nat(H,F_1))))) # label(fact_12_finite__imageI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 159 (all F all A (hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(finite_card_nat,image_1551609309ol_nat(F,A))),hAPP_f55526627ol_nat(finite1340463720e_bool,A))))) # label(fact_57_card__image__le) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 160 (all B all X_2 all A (-hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A)) -> (hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B)) <-> hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),insert_pname(X_2,B)))))) # label(fact_276_subset__insert) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 161 (all B_1_1 all B_2 is_bool(hAPP_f54304608l_bool(B_1_1,B_2))) # label(gsy_c_hAPP_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__HOL__Obool) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 162 (all N_1 all K_1 all M_2 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),M_2)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K_1),N_1)) -> hAPP_nat_nat(minus_minus_nat(M_2),N_1) = hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(minus_minus_nat(M_2),K_1)),hAPP_nat_nat(minus_minus_nat(N_1),K_1))))) # label(fact_163_Nat_Odiff__diff__eq) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 163 (all P all Q all R hAPP_f1664156314l_bool(hAPP_f434788991l_bool(P,R),Q) = hAPP_f1664156314l_bool(hAPP_f434788991l_bool(cOMBC_1284144636l_bool(P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__fun_It) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 164 (all A (hBOOL(hAPP_f292226953l_bool(finite1381704300l_bool,A)) -> hBOOL(hAPP_f1363661463l_bool(finite1343359508l_bool,collec1635217238l_bool(hAPP_f1050622307l_bool(cOMBC_636888218l_bool(ord_le967226251l_bool),A)))))) # label(fact_6_finite__Collect__subsets) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 165 (all H all F_1 (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1)) -> hBOOL(hAPP_f937997336l_bool(finite1701474069l_bool,image_1607900221l_bool(H,F_1))))) # label(fact_25_finite__imageI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 166 (all P all Q (hBOOL(P) | hBOOL(Q) | -hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fdisj,P),Q)))) # label(help_fdisj_3_1_U) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 167 (all B_1_1 all B_2 (is_fun1661590463l_bool(B_2) -> is_fun_a_bool(image_876012084bool_a(B_1_1,B_2)))) # label(gsy_c_Set_Oimage_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000t__a) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 168 (all Na all N_3 all F ((all N_2 hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_n1699378549t_bool(F,N_2)),hAPP_n1699378549t_bool(F,hAPP_nat_nat(suc,N_2))))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),N_3)) -> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,hAPP_n1699378549t_bool(F,Na)),hAPP_n1699378549t_bool(F,N_3)))))) # label(fact_171_lift__Suc__mono__le) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 169 (all B_1_1 all B_2 (is_fun1661590463l_bool(B_2) -> is_fun1661590463l_bool(cOMBS_350070575l_bool(B_1_1,B_2)))) # label(gsy_c_COMBS_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__HOL__Obool_000t) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 170 (all B_1_1 all B_2 (is_fun1661590463l_bool(B_2) & is_fun_pname_bool(B_1_1) -> is_fun1661590463l_bool(insert1325755072e_bool(B_1_1,B_2)))) # label(gsy_c_Set_Oinsert_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 171 (all A_1 all B hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),insert_nat(A_1,B)))) # label(fact_189_insertI1) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 172 (all M_2 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_1)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M_2)) -> M_2 = N_1))) # label(fact_123_le__antisym) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 173 (all B_1 all F all A (is_a(B_1) -> (hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,B_1),image_pname_a(F,A))) -> -(all X_1 (is_pname(X_1) -> (hAPP_pname_a(F,X_1) = B_1 -> -hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A)))))))) # label(fact_288_imageE) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 174 (all H all F_1 (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1)) -> hBOOL(hAPP_f389811538l_bool(finite786885583l_bool,image_1874789623l_bool(H,F_1))))) # label(fact_26_finite__imageI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 175 (all B all X_2 all A (-hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A)) -> (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),insert_nat(X_2,B))) <-> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B))))) # label(fact_275_subset__insert) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 176 (all P (-hBOOL(hAPP_bool_bool(fNot,P)) | -hBOOL(P))) # label(help_fNot_1_1_U) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 177 (all B_1 all A all B (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)) -> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),insert_nat(B_1,B))))) # label(fact_278_subset__insertI2) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 178 (all B_1_1 all B_2 (is_fun949378684l_bool(B_2) & is_fun_a_bool(B_1_1) -> is_fun949378684l_bool(insert_fun_a_bool(B_1_1,B_2)))) # label(gsy_c_Set_Oinsert_000tc__fun_It__a_Mtc__HOL__Obool_J) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 179 (all F all A (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f921600141ol_nat(finite_card_pname,image_nat_pname(F,A))),hAPP_f22106695ol_nat(finite_card_nat,A))))) # label(fact_60_card__image__le) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 180 (all B_1 all A_1 all B ((-hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),B)) -> B_1 = A_1) -> hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),insert_pname(B_1,B))))) # label(fact_184_insertCI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 181 (all B_1_1 all B_2 (is_fun949378684l_bool(B_2) -> is_fun949378684l_bool(hAPP_f2117159681l_bool(B_1_1,B_2)))) # label(gsy_c_hAPP_000tc__fun_Itc__fun_It__a_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J_000tc___004) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 182 (all H all F_1 (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1)) -> hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,image_nat_pname(H,F_1))))) # label(fact_44_finite__imageI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 183 (all P all Q all R hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(P,R),Q) = hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(cOMBC_1732670874l_bool(P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__fun_It__a_Mtc__HO) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 184 (all P all Q all R hAPP_a_bool(hAPP_a_fun_a_bool(P,R),Q) = hAPP_a_bool(hAPP_a_fun_a_bool(cOMBC_a_a_bool(P),Q),R)) # label(help_COMBC_1_1_COMBC_000t__a_000t__a_000tc__HOL__Obool_U) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 185 (all A_1 all A (hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,insert1325755072e_bool(A_1,A))) <-> hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A)))) # label(fact_139_finite__insert) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 186 (all P all Q all R hAPP_f621171935l_bool(hAPP_f1434722111l_bool(P,R),Q) = hAPP_f621171935l_bool(hAPP_f1434722111l_bool(cOMBC_331553030l_bool(P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__fun_Itc__fun_It__a_Mtc__HOL__Obool_J_Mtc__HOL__Obool) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 187 (all Y_1 all A all X_2 (is_a(Y_1) & is_a(X_2) -> (X_2 = Y_1 | hBOOL(hAPP_a_bool(A,X_2)) <-> hBOOL(hAPP_a_bool(insert_a(Y_1,A),X_2))))) # label(fact_215_insert__code) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 188 (all A all B (is_fun_a_bool(A) & is_fun_a_bool(B) -> (hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),A)) & hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B)) <-> B = A))) # label(fact_230_set__eq__subset) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 189 (all B_1_1 all B_2 is_fun_pname_bool(image_1921560913_pname(B_1_1,B_2))) # label(gsy_c_Set_Oimage_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__Com__Opname) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 190 (all X all Y (Y = X | -hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(fequal_fun_nat_bool,X),Y)))) # label(help_fequal_1_1_fequal_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_T) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 191 (all B_1_1 all B_2 (is_fun949378684l_bool(B_2) -> is_fun_a_bool(image_fun_a_bool_a(B_1_1,B_2)))) # label(gsy_c_Set_Oimage_000tc__fun_It__a_Mtc__HOL__Obool_J_000t__a) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 192 (all X_2 all A (hBOOL(hAPP_a_bool(A,X_2)) <-> hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_2),A)))) # label(fact_253_mem__def) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 193 (all H all F_1 (hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,F_1)) -> hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,image_1283814551_pname(H,F_1))))) # label(fact_22_finite__imageI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 194 (all A_1 all B_1 all A (B_1 = A_1 | hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),A)) <-> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),insert_nat(B_1,A))))) # label(fact_210_insert__iff) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 195 (all X_2 hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,X_2),X_2))) # label(fact_295_order__refl) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 196 (all H all F_1 (hBOOL(hAPP_f292226953l_bool(finite1381704300l_bool,F_1)) -> hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,image_990671762_pname(H,F_1))))) # label(fact_20_finite__imageI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 197 (all B_1_1 all B_2 (is_pname(B_2) -> is_fun1661590463l_bool(hAPP_p338031245l_bool(B_1_1,B_2)))) # label(gsy_c_hAPP_000tc__Com__Opname_000tc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obo) # label(hypothesis) # label(non_clause). [assumption]. 1.00/1.29 198 (all A all B (hBOOL(hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(ord_le1454342156l_bool,A),B)) -> (hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,B)) -> hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A))))) # label(fact_141_finite__subset) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 199 (all X all Y (Y != X | hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(fequal_fun_a_bool,X),Y)))) # label(help_fequal_2_1_fequal_000tc__fun_It__a_Mtc__HOL__Obool_J_T) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 200 (all F all A (hBOOL(hAPP_f389811538l_bool(finite786885583l_bool,A)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_fun_a_bool_nat(finite_card_a,image_349102846bool_a(F,A))),hAPP_f98387925ol_nat(finite269641166l_bool,A))))) # label(fact_63_card__image__le) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 201 (all B_1_1 all B_2 (is_fun949378684l_bool(B_2) -> is_fun_pname_bool(image_1854862208_pname(B_1_1,B_2)))) # label(gsy_c_Set_Oimage_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__Com__Opname) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 202 (all F all A (hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_fun_a_bool_nat(finite_card_a,image_a_a(F,A))),hAPP_fun_a_bool_nat(finite_card_a,A))))) # label(fact_68_card__image__le) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 203 (all B_1 all A all B (hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B)) -> hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),insert_pname(B_1,B))))) # label(fact_279_subset__insertI2) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 204 (all X_2 all Y_1 all A insert_nat(X_2,insert_nat(Y_1,A)) = insert_nat(Y_1,insert_nat(X_2,A))) # label(fact_207_insert__commute) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 205 (all M_2 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),hAPP_nat_nat(suc,N_1))))) # label(fact_155_le__SucI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 206 (all A_1 all B_1 all A (is_pname(A_1) & is_pname(B_1) -> (hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),A)) | B_1 = A_1 <-> hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),insert_pname(B_1,A)))))) # label(fact_211_insert__iff) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 207 (all H all F_1 (hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_1)) -> hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,image_pname_pname(H,F_1))))) # label(fact_38_finite__imageI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 208 (all B_1_1 all B_2 (is_fun_pname_bool(B_2) -> is_fun_pname_bool(cOMBS_568398431l_bool(B_1_1,B_2)))) # label(gsy_c_COMBS_000tc__Com__Opname_000tc__HOL__Obool_000tc__HOL__Obool) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 209 (all A_1 all A (hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A)) -> hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,insert_fun_a_bool(A_1,A))))) # label(fact_51_finite_OinsertI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 210 (all F all A all B (is_fun_a_bool(B) -> (hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B)) -> (hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),image_pname_a(F,A))) -> (exists C_2 (hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,C_2),A)) & image_pname_a(F,C_2) = B & hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,C_2)) & is_fun_pname_bool(C_2))))))) # label(fact_170_finite__subset__image) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 211 (all A all B (hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,B)) -> (hBOOL(hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(ord_le1454342156l_bool,A),B)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f696928925ol_nat(finite346522414t_bool,A)),hAPP_f696928925ol_nat(finite346522414t_bool,B)))))) # label(fact_80_card__mono) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 212 (all A (hBOOL(hAPP_f937997336l_bool(finite1701474069l_bool,A)) -> hBOOL(hAPP_f1295398978l_bool(finite719726885l_bool,collec1874991203l_bool(hAPP_f760187903l_bool(cOMBC_1269652216l_bool(ord_le65145710l_bool),A)))))) # label(fact_4_finite__Collect__subsets) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 213 (all P all Q all R hAPP_p1534023578a_bool(cOMBB_1897541054_pname(P,Q),R) = hAPP_a_fun_a_bool(P,hAPP_pname_a(Q,R))) # label(help_COMBB_1_1_COMBB_000t__a_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__Com__Opna) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 214 (all L all M_2 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(minus_minus_nat(L),N_1)),hAPP_nat_nat(minus_minus_nat(L),M_2))))) # label(fact_167_diff__le__mono2) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 215 (all L all M_2 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(minus_minus_nat(M_2),L)),hAPP_nat_nat(minus_minus_nat(N_1),L))))) # label(fact_166_diff__le__mono) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 216 (all B_1 all A_1 all B (hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),B)) -> hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),insert_pname(B_1,B))))) # label(fact_220_insertI2) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 217 (all A (hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A)) -> hBOOL(hAPP_f937997336l_bool(finite1701474069l_bool,collec1015864663l_bool(hAPP_f1772781669l_bool(cOMBC_595898202l_bool(ord_le1454342156l_bool),A)))))) # label(fact_9_finite__Collect__subsets) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 218 (all Pa (is_fun_pname_bool(Pa) -> collect_pname(Pa) = Pa)) # label(fact_255_Collect__def) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 219 (all B_1_1 all B_2 is_fun_a_bool(hAPP_nat_fun_a_bool(B_1_1,B_2))) # label(gsy_c_hAPP_000tc__Nat__Onat_000tc__fun_It__a_Mtc__HOL__Obool_J) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 220 (all A (hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A)) -> hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,collec1974731493e_bool(hAPP_f434788991l_bool(cOMBC_1284144636l_bool(ord_le313189616e_bool),A)))))) # label(fact_2_finite__Collect__subsets) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 221 (all A all B (hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,B)) -> (hBOOL(hAPP_f1935102916l_bool(hAPP_f510955609l_bool(ord_le675606854l_bool,A),B)) -> hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A))))) # label(fact_148_rev__finite__subset) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 222 (all P all Q all R hAPP_bool_bool(P,hAPP_pname_bool(Q,R)) = hAPP_pname_bool(cOMBB_647938656_pname(P,Q),R)) # label(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Com__Opname_U) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 223 (all A all B (hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,B)) -> (hBOOL(hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(ord_le1454342156l_bool,A),B)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f696928925ol_nat(finite346522414t_bool,B)),hAPP_f696928925ol_nat(finite346522414t_bool,A))) -> A = B)))) # label(fact_86_card__seteq) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 224 (all B_1_1 all B_2 is_bool(hAPP_f1363661463l_bool(B_1_1,B_2))) # label(gsy_c_hAPP_000tc__fun_Itc__fun_Itc__fun_Itc__fun_It__a_Mtc__HOL__Obool_J_Mtc__HO) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 225 (all A_1 all Pa collect_a(cOMBS_a_bool_bool(cOMBB_1972296269bool_a(fimplies,cOMBB_bool_bool_a(fNot,hAPP_a_fun_a_bool(cOMBC_a_a_bool(fequal_a),A_1))),Pa)) = insert_a(A_1,collect_a(Pa))) # label(fact_200_insert__Collect) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 226 (all B_1_1 all B_2 is_bool(hAPP_nat_bool(B_1_1,B_2))) # label(gsy_c_hAPP_000tc__Nat__Onat_000tc__HOL__Obool) # label(hypothesis) # label(non_clause). [assumption]. 1.00/1.29 227 (all P all Q all R hAPP_f1935102916l_bool(hAPP_f510955609l_bool(cOMBC_7971162l_bool(P),Q),R) = hAPP_f1935102916l_bool(hAPP_f510955609l_bool(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_Mtc_) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 228 (all X_2 all A (hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A)) <-> hBOOL(hAPP_pname_bool(A,X_2)))) # label(fact_254_mem__def) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 229 (all H all F_1 (hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,F_1)) -> hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,image_1854862208_pname(H,F_1))))) # label(fact_21_finite__imageI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 230 (all H all F_1 (hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,F_1)) -> hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,image_fun_nat_bool_a(H,F_1))))) # label(fact_40_finite__imageI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.29 231 (all X all Y (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(fequal_fun_nat_bool,X),Y)) | Y != X)) # label(help_fequal_2_1_fequal_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_T) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 232 (all X_2 all A (hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A)) -> (-hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_2),A)) -> hAPP_fun_a_bool_nat(finite_card_a,insert_a(X_2,A)) = hAPP_nat_nat(suc,hAPP_fun_a_bool_nat(finite_card_a,A))))) # label(fact_109_card__insert__disjoint) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 233 (all X_2 all A (hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A)) -> (-hBOOL(hAPP_f621171935l_bool(hAPP_f285962445l_bool(member_fun_a_bool,X_2),A)) -> hAPP_f2009550088ol_nat(finite1306199131a_bool,insert_fun_a_bool(X_2,A)) = hAPP_nat_nat(suc,hAPP_f2009550088ol_nat(finite1306199131a_bool,A))) & (hBOOL(hAPP_f621171935l_bool(hAPP_f285962445l_bool(member_fun_a_bool,X_2),A)) -> hAPP_f2009550088ol_nat(finite1306199131a_bool,insert_fun_a_bool(X_2,A)) = hAPP_f2009550088ol_nat(finite1306199131a_bool,A)))) # label(fact_100_card__insert__if) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 234 (all A all B (hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,B)) -> (hBOOL(hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(ord_le1454342156l_bool,A),B)) -> hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A))))) # label(fact_147_rev__finite__subset) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 235 (all X_2 all A (hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A)) -> (hBOOL(hAPP_f1637334154l_bool(hAPP_f1951378235l_bool(member_fun_nat_bool,X_2),A)) -> hAPP_f696928925ol_nat(finite346522414t_bool,A) = hAPP_f696928925ol_nat(finite346522414t_bool,insert_fun_nat_bool(X_2,A))) & (-hBOOL(hAPP_f1637334154l_bool(hAPP_f1951378235l_bool(member_fun_nat_bool,X_2),A)) -> hAPP_f696928925ol_nat(finite346522414t_bool,insert_fun_nat_bool(X_2,A)) = hAPP_nat_nat(suc,hAPP_f696928925ol_nat(finite346522414t_bool,A))))) # label(fact_98_card__insert__if) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 236 (all P all Q all R hAPP_pname_bool(cOMBS_568398431l_bool(P,Q),R) = hAPP_bool_bool(hAPP_p393069232l_bool(P,R),hAPP_pname_bool(Q,R))) # label(help_COMBS_1_1_COMBS_000tc__Com__Opname_000tc__HOL__Obool_000tc__HOL__Obool_U) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 237 (all M_3 all Na (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_3),hAPP_nat_nat(suc,Na))) <-> hAPP_nat_nat(suc,Na) = M_3 | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_3),Na)))) # label(fact_157_le__Suc__eq) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 238 (all X_2 all A all B (hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,insert_a(X_2,A)),B)) <-> hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B)) & hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_2),B)))) # label(fact_274_insert__subset) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 239 (all B_1_1 (is_fun949378684l_bool(B_1_1) -> is_fun949378684l_bool(collect_fun_a_bool(B_1_1)))) # label(gsy_c_Set_OCollect_000tc__fun_It__a_Mtc__HOL__Obool_J) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 240 (all B_1_1 all B_2 (is_a(B_2) -> is_fun949378684l_bool(hAPP_a85458249l_bool(B_1_1,B_2)))) # label(gsy_c_hAPP_000t__a_000tc__fun_Itc__fun_It__a_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J) # label(hypothesis) # label(non_clause). [assumption]. 1.00/1.30 241 (all F all A (hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f55526627ol_nat(finite1340463720e_bool,image_47868345e_bool(F,A))),hAPP_f921600141ol_nat(finite_card_pname,A))))) # label(fact_71_card__image__le) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 242 (all A all B (hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,B)) -> (hBOOL(hAPP_f1935102916l_bool(hAPP_f510955609l_bool(ord_le675606854l_bool,A),B)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f55526627ol_nat(finite1340463720e_bool,A)),hAPP_f55526627ol_nat(finite1340463720e_bool,B)))))) # label(fact_81_card__mono) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 243 (all C_1 all A all B (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)) -> (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_1),A)) -> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,C_1),B))))) # label(fact_180_subsetD) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 244 (all A_1 all B_1 all A (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),insert_nat(B_1,A))) -> (B_1 != A_1 -> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),A))))) # label(fact_186_insertE) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 245 (all A_1 all Pa collec1974731493e_bool(cOMBS_350070575l_bool(cOMBB_2095475776e_bool(fimplies,cOMBB_307249310e_bool(fNot,hAPP_f434788991l_bool(cOMBC_1284144636l_bool(fequal533582459e_bool),A_1))),Pa)) = insert1325755072e_bool(A_1,collec1974731493e_bool(Pa))) # label(fact_202_insert__Collect) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 246 (all F all A (hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f921600141ol_nat(finite_card_pname,image_1921560913_pname(F,A))),hAPP_f696928925ol_nat(finite346522414t_bool,A))))) # label(fact_77_card__image__le) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 247 (all A all B (hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,B)) -> (hBOOL(hAPP_f621171935l_bool(hAPP_f1434722111l_bool(ord_le1375614389l_bool,A),B)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f2009550088ol_nat(finite1306199131a_bool,A)),hAPP_f2009550088ol_nat(finite1306199131a_bool,B)))))) # label(fact_82_card__mono) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 248 (all B_1_1 all B_2 (is_fun_pname_bool(B_2) -> is_fun1661590463l_bool(image_47868345e_bool(B_1_1,B_2)))) # label(gsy_c_Set_Oimage_000tc__Com__Opname_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 249 (all Pa (is_fun1661590463l_bool(Pa) -> collec1974731493e_bool(Pa) = Pa)) # label(fact_257_Collect__def) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 250 (all Pa all Q_1 (hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,collect_fun_nat_bool(Pa))) & hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,collect_fun_nat_bool(Q_1))) <-> hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,collect_fun_nat_bool(cOMBS_1187019125l_bool(cOMBB_444170502t_bool(fdisj,Pa),Q_1)))))) # label(fact_130_finite__Collect__disjI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 251 (all Pa collect_fun_nat_bool(Pa) = Pa) # label(fact_256_Collect__def) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 252 (all B_1_1 (is_fun_a_bool(B_1_1) -> is_fun_a_bool(collect_a(B_1_1)))) # label(gsy_c_Set_OCollect_000t__a) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 253 (all P all Q all R hAPP_f54304608l_bool(cOMBB_238756964t_bool(P,Q),R) = hAPP_bool_bool(P,hAPP_f54304608l_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__fun_Itc__Nat__On) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 254 (all X_1 all Xa insert_fun_nat_bool(X_1,Xa) = collect_fun_nat_bool(cOMBS_1187019125l_bool(cOMBB_444170502t_bool(fdisj,hAPP_f103356543l_bool(cOMBC_1693257480l_bool(fequal_fun_nat_bool),X_1)),hAPP_f1246832597l_bool(cOMBC_1245412066l_bool(member_fun_nat_bool),Xa)))) # label(fact_266_insert__compr__raw) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 255 (all B all X_2 all A (is_fun_pname_bool(B) & is_fun_pname_bool(A) -> (-hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A)) -> (-hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),B)) -> (insert_pname(X_2,A) = insert_pname(X_2,B) <-> A = B))))) # label(fact_217_insert__ident) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 256 (all P all Q all R hAPP_f292226953l_bool(hAPP_f1050622307l_bool(P,R),Q) = hAPP_f292226953l_bool(hAPP_f1050622307l_bool(cOMBC_636888218l_bool(P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__fun_Itc__fun_Itc__fun_It__a_Mtc__HOL__Obool_J_Mtc__H) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 257 (all Na all N_3 all F ((all N_2 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(F,N_2)),hAPP_nat_nat(F,hAPP_nat_nat(suc,N_2))))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),N_3)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(F,Na)),hAPP_nat_nat(F,N_3)))))) # label(fact_173_lift__Suc__mono__le) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 258 (all N_1 all M_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N_1)),M_1)) -> (exists M hAPP_nat_nat(suc,M) = M_1))) # label(fact_293_Suc__le__D) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 259 (all B_1_1 all B_2 is_fun_a_bool(image_349102846bool_a(B_1_1,B_2))) # label(gsy_c_Set_Oimage_000tc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_Mtc__HOL) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 260 (all A all B (hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B)) -> (hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B)) -> hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A))))) # label(fact_152_rev__finite__subset) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 261 (all P all Q (hBOOL(Q) | -hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q)))) # label(help_fconj_3_1_U) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 262 (all A_1 all Pa insert_fun_nat_bool(A_1,collect_fun_nat_bool(Pa)) = collect_fun_nat_bool(cOMBS_1187019125l_bool(cOMBB_444170502t_bool(fimplies,cOMBB_238756964t_bool(fNot,hAPP_f103356543l_bool(cOMBC_1693257480l_bool(fequal_fun_nat_bool),A_1))),Pa))) # label(fact_201_insert__Collect) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 263 (all A_1 all A (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) -> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,insert_nat(A_1,A))))) # label(fact_46_finite_OinsertI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 264 (all X all Y (is_pname(X) & is_pname(Y) -> -hBOOL(hAPP_pname_bool(hAPP_p61793385e_bool(fequal_pname,X),Y)) | X = Y)) # label(help_fequal_1_1_fequal_000tc__Com__Opname_T) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 265 (all H all F_1 (hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_1)) -> hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,image_pname_a(H,F_1))))) # label(fact_10_finite__imageI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 266 (all P all Q all R hAPP_pname_bool(hAPP_p61793385e_bool(cOMBC_1149511130e_bool(P),Q),R) = hAPP_pname_bool(hAPP_p61793385e_bool(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__Com__Opname_000tc__Com__Opname_000tc__HOL__Obool_U) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 267 (all B_1_1 all B_2 (is_fun_a_bool(B_2) -> is_fun_a_bool(image_a_a(B_1_1,B_2)))) # label(gsy_c_Set_Oimage_000t__a_000t__a) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 268 (all A_1 all A (hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A)) -> hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,insert_a(A_1,A))))) # label(fact_45_finite_OinsertI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 269 (all N_1 -hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,N_1)),N_1))) # label(fact_159_Suc__n__not__le__n) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 270 (all X_2 all A all B (hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B)) -> (hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_2),A)) -> hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_2),B))))) # label(fact_244_set__mp) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 271 (all A_1 all Pa insert_nat(A_1,collect_nat(Pa)) = collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fimplies,cOMBB_bool_bool_nat(fNot,hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(fequal_nat),A_1))),Pa))) # label(fact_198_insert__Collect) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 272 (all Q_1 all Pa (hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,collect_fun_nat_bool(Pa))) | hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,collect_fun_nat_bool(Q_1))) -> hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,collect_fun_nat_bool(cOMBS_1187019125l_bool(cOMBB_444170502t_bool(fconj,Pa),Q_1)))))) # label(fact_110_finite__Collect__conjI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 273 (all A all B_1 all F all X_2 (B_1 = hAPP_pname_a(F,X_2) -> (hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A)) -> hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,B_1),image_pname_a(F,A)))))) # label(fact_176_image__eqI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 274 (all X_2 all A (hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f55526627ol_nat(finite1340463720e_bool,A)),hAPP_f55526627ol_nat(finite1340463720e_bool,insert1325755072e_bool(X_2,A)))))) # label(fact_93_card__insert__le) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 275 (all N (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,N)) <-> (exists M all X_1 (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_1),N)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,X_1),M)))))) # label(fact_299_finite__nat__set__iff__bounded__le) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 276 (all N_1 hAPP_nat_nat(suc,N_1) != N_1) # label(fact_121_Suc__n__not__n) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 277 (all Pa all Q_1 (hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,collect_pname(Pa))) & hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,collect_pname(Q_1))) <-> hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,collect_pname(cOMBS_568398431l_bool(cOMBB_675860798_pname(fdisj,Pa),Q_1)))))) # label(fact_129_finite__Collect__disjI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 278 (all H all F_1 (hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,F_1)) -> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,image_496248727ol_nat(H,F_1))))) # label(fact_16_finite__imageI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 279 (all B all A_1 hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B),insert_pname(A_1,B)))) # label(fact_270_subset__insertI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 280 (all H all F_1 (hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,F_1)) -> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,image_a_nat(H,F_1))))) # label(fact_17_finite__imageI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 281 (all A_1 all A (is_fun_pname_bool(A) -> (hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),A)) -> insert_pname(A_1,A) = A))) # label(fact_223_insert__absorb) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 282 (all A all B (hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,B)) -> (hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f921600141ol_nat(finite_card_pname,A)),hAPP_f921600141ol_nat(finite_card_pname,B)))))) # label(fact_83_card__mono) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 283 (all H all F_1 (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1)) -> hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,image_1655916159e_bool(H,F_1))))) # label(fact_29_finite__imageI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 284 (all Q_1 all Pa (hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,collec1974731493e_bool(Pa))) | hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,collec1974731493e_bool(Q_1))) -> hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,collec1974731493e_bool(cOMBS_350070575l_bool(cOMBB_2095475776e_bool(fconj,Pa),Q_1)))))) # label(fact_111_finite__Collect__conjI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 285 (all Y_1 all A all X_2 (is_pname(Y_1) & is_pname(X_2) -> (hBOOL(hAPP_pname_bool(insert_pname(Y_1,A),X_2)) <-> hBOOL(hAPP_pname_bool(A,X_2)) | Y_1 = X_2))) # label(fact_214_insert__code) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 286 (all X all Y (is_fun_pname_bool(Y) & is_fun_pname_bool(X) -> Y = X | -hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(fequal533582459e_bool,X),Y)))) # label(help_fequal_1_1_fequal_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_T) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 287 (all F all A (hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f921600141ol_nat(finite_card_pname,image_pname_pname(F,A))),hAPP_f921600141ol_nat(finite_card_pname,A))))) # label(fact_69_card__image__le) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 288 (all B_1_1 all B_2 is_fun_pname_bool(hAPP_n1025906991e_bool(B_1_1,B_2))) # label(gsy_c_hAPP_000tc__Nat__Onat_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 289 (all B_1_1 all B_2 (is_pname(B_2) -> is_fun_pname_bool(hAPP_p61793385e_bool(B_1_1,B_2)))) # label(gsy_c_hAPP_000tc__Com__Opname_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 290 (all A_1 all A (hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A)) -> hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,insert_pname(A_1,A))))) # label(fact_47_finite_OinsertI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 291 (all Q_1 all Pa (hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,collect_pname(Pa))) | hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,collect_pname(Q_1))) -> hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,collect_pname(cOMBS_568398431l_bool(cOMBB_675860798_pname(fconj,Pa),Q_1)))))) # label(fact_114_finite__Collect__conjI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 292 (all M_2 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),M_2)) | hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_1)))) # label(fact_126_nat__le__linear) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 293 (all B_1_1 all B_2 is_bool(hAPP_f595608956l_bool(B_1_1,B_2))) # label(gsy_c_hAPP_000tc__fun_Itc__fun_Itc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Oboo) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 294 (all B all A ((all X_1 (is_a(X_1) -> (hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),A)) -> hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_1),B))))) -> hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B)))) # label(fact_290_subsetI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 295 (all M_2 all N_1 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(minus_minus_nat(M_2),N_1)),M_2))) # label(fact_168_diff__le__self) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 296 (all B_1_1 all B_2 is_bool(hAPP_f1295398978l_bool(B_1_1,B_2))) # label(gsy_c_hAPP_000tc__fun_Itc__fun_Itc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 297 (all C_1 all A all B (hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B)) -> (hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_1),A)) -> hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,C_1),B))))) # label(fact_182_subsetD) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 298 (all B_1 all A_1 all B (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),B)) -> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,A_1),insert_nat(B_1,B))))) # label(fact_219_insertI2) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 299 (all A_1 all A (hBOOL(hAPP_f389811538l_bool(finite786885583l_bool,A)) -> hBOOL(hAPP_f389811538l_bool(finite786885583l_bool,insert1117693814l_bool(A_1,A))))) # label(fact_49_finite_OinsertI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 300 (all A all B (B = A -> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),A)))) # label(fact_234_equalityD2) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 301 (all A_1 all A (hBOOL(hAPP_f937997336l_bool(finite1701474069l_bool,A)) -> hBOOL(hAPP_f937997336l_bool(finite1701474069l_bool,insert2003652156l_bool(A_1,A))))) # label(fact_48_finite_OinsertI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 302 (all F all A (hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_fun_a_bool_nat(finite_card_a,image_pname_a(F,A))),hAPP_f921600141ol_nat(finite_card_pname,A))))) # label(fact_61_card__image__le) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 303 (all B_1_1 all B_2 (is_fun1661590463l_bool(B_2) -> is_bool(hAPP_f1935102916l_bool(B_1_1,B_2)))) # label(gsy_c_hAPP_000tc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_Mtc__HOL__Oboo) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 304 (all Na all K all M_3 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),M_3)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,K),Na)) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(minus_minus_nat(M_3),K)),hAPP_nat_nat(minus_minus_nat(Na),K))) <-> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_3),Na)))))) # label(fact_162_le__diff__iff) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 305 (all P all Q all R hAPP_f54304608l_bool(hAPP_f103356543l_bool(P,R),Q) = hAPP_f54304608l_bool(hAPP_f103356543l_bool(cOMBC_1693257480l_bool(P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000tc__fun_Itc_) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 306 (all B_1_1 all B_2 (is_fun_pname_bool(B_1_1) & is_pname(B_2) -> is_bool(hAPP_pname_bool(B_1_1,B_2)))) # label(gsy_c_hAPP_000tc__Com__Opname_000tc__HOL__Obool) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 307 (all A_1 all A (hBOOL(hAPP_f292226953l_bool(finite1381704300l_bool,A)) -> hBOOL(hAPP_f292226953l_bool(finite1381704300l_bool,insert1457093509l_bool(A_1,A))))) # label(fact_50_finite_OinsertI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 308 (all A all B (A = B -> hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),A)))) # label(fact_236_equalityD2) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 309 (all F all A (hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f696928925ol_nat(finite346522414t_bool,image_2129980159t_bool(F,A))),hAPP_f921600141ol_nat(finite_card_pname,A))))) # label(fact_70_card__image__le) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 310 (all B_1_1 all B_2 (is_fun949378684l_bool(B_2) -> is_bool(hAPP_f621171935l_bool(B_1_1,B_2)))) # label(gsy_c_hAPP_000tc__fun_Itc__fun_It__a_Mtc__HOL__Obool_J_Mtc__HOL__Obool_J_000tc__) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 311 (all X_2 all A (hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A)) -> (hBOOL(hAPP_f1935102916l_bool(hAPP_f556039215l_bool(member799430823e_bool,X_2),A)) -> hAPP_f55526627ol_nat(finite1340463720e_bool,insert1325755072e_bool(X_2,A)) = hAPP_f55526627ol_nat(finite1340463720e_bool,A)) & (-hBOOL(hAPP_f1935102916l_bool(hAPP_f556039215l_bool(member799430823e_bool,X_2),A)) -> hAPP_nat_nat(suc,hAPP_f55526627ol_nat(finite1340463720e_bool,A)) = hAPP_f55526627ol_nat(finite1340463720e_bool,insert1325755072e_bool(X_2,A))))) # label(fact_99_card__insert__if) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 312 (all H all F_1 (hBOOL(hAPP_f937997336l_bool(finite1701474069l_bool,F_1)) -> hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,image_1604018183_pname(H,F_1))))) # label(fact_18_finite__imageI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 313 (all X_2 all A insert_nat(X_2,A) = insert_nat(X_2,insert_nat(X_2,A))) # label(fact_204_insert__absorb2) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 314 (all Nat_1 all Nat (Nat = Nat_1 <-> hAPP_nat_nat(suc,Nat) = hAPP_nat_nat(suc,Nat_1))) # label(fact_120_nat_Oinject) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 315 (all A (hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A)) -> hBOOL(hAPP_f292226953l_bool(finite1381704300l_bool,collec707592106l_bool(hAPP_f1434722111l_bool(cOMBC_331553030l_bool(ord_le1375614389l_bool),A)))))) # label(fact_7_finite__Collect__subsets) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 316 (all P all Q all R hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(P,R),Q) = hAPP_a_bool(hAPP_f2050579477a_bool(cOMBC_1355376034l_bool(P),Q),R)) # label(help_COMBC_1_1_COMBC_000t__a_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__HOL__Oboo) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 317 (all P all Q all R hAPP_bool_bool(P,hAPP_a_bool(Q,R)) = hAPP_a_bool(cOMBB_bool_bool_a(P,Q),R)) # label(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000t__a_U) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 318 (all A all B (B = A -> -(hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B)) -> -hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),A))))) # label(fact_251_equalityE) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 319 (all B_1_1 (is_fun1661590463l_bool(B_1_1) -> is_fun1661590463l_bool(collec1974731493e_bool(B_1_1)))) # label(gsy_c_Set_OCollect_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 320 (all B all X_2 all A (is_fun_a_bool(A) & is_fun_a_bool(B) -> (-hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_2),A)) -> (-hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_2),B)) -> (B = A <-> insert_a(X_2,B) = insert_a(X_2,A)))))) # label(fact_218_insert__ident) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 321 (all B_1_1 all B_2 is_bool(hAPP_f292226953l_bool(B_1_1,B_2))) # label(gsy_c_hAPP_000tc__fun_Itc__fun_Itc__fun_It__a_Mtc__HOL__Obool_J_Mtc__HOL__Obool_) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 322 (all H all F_1 (hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,F_1)) -> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,image_1551609309ol_nat(H,F_1))))) # label(fact_15_finite__imageI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 323 (all B_1_1 all B_2 (is_fun_pname_bool(B_2) & is_fun1661590463l_bool(B_1_1) -> is_bool(hAPP_f1664156314l_bool(B_1_1,B_2)))) # label(gsy_c_hAPP_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__HOL__Obool) # label(hypothesis) # label(non_clause). [assumption]. 1.00/1.30 324 (all B_1_1 all B_2 (is_pname(B_2) -> is_a(hAPP_pname_a(B_1_1,B_2)))) # label(gsy_c_hAPP_000tc__Com__Opname_000t__a) # label(hypothesis) # label(non_clause). [assumption]. 1.00/1.30 325 (all H all F_1 (hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,F_1)) -> hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,image_a_pname(H,F_1))))) # label(fact_24_finite__imageI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 326 (all F all A (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f55526627ol_nat(finite1340463720e_bool,image_1655916159e_bool(F,A))),hAPP_f22106695ol_nat(finite_card_nat,A))))) # label(fact_75_card__image__le) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 327 (all H all F_1 (hBOOL(hAPP_f937997336l_bool(finite1701474069l_bool,F_1)) -> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,image_2089570637ol_nat(H,F_1))))) # label(fact_11_finite__imageI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 328 (all P all Q (hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fimplies,P),Q)) | -hBOOL(Q))) # label(help_fimplies_2_1_U) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 329 (all Na all N_3 all F ((all N_2 hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,hAPP_n1025906991e_bool(F,N_2)),hAPP_n1025906991e_bool(F,hAPP_nat_nat(suc,N_2))))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),N_3)) -> hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,hAPP_n1025906991e_bool(F,Na)),hAPP_n1025906991e_bool(F,N_3)))))) # label(fact_172_lift__Suc__mono__le) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 330 (all X all Y (hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(fequal533582459e_bool,X),Y)) | Y != X)) # label(help_fequal_2_1_fequal_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_T) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 331 (all X_2 all A (hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_fun_a_bool_nat(finite_card_a,A)),hAPP_fun_a_bool_nat(finite_card_a,insert_a(X_2,A)))))) # label(fact_97_card__insert__le) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 332 (all A_1 all A (hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,insert_pname(A_1,A))) <-> hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A)))) # label(fact_136_finite__insert) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 333 (all P all Q all R hAPP_a_fun_bool_bool(cOMBB_1972296269bool_a(P,Q),R) = hAPP_b589554111l_bool(P,hAPP_a_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 334 (all Q_1 all Pa (hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,collect_fun_a_bool(Q_1))) | hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,collect_fun_a_bool(Pa))) -> hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,collect_fun_a_bool(cOMBS_1035972772l_bool(cOMBB_338059395a_bool(fconj,Pa),Q_1)))))) # label(fact_112_finite__Collect__conjI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 335 (all B_1_1 all B_2 (is_fun_a_bool(B_2) & is_a(B_1_1) -> is_fun_a_bool(insert_a(B_1_1,B_2)))) # label(gsy_c_Set_Oinsert_000t__a) # label(hypothesis) # label(non_clause). [assumption]. 1.00/1.30 336 (all I all Pa all K (hBOOL(hAPP_nat_bool(Pa,K)) -> ((all N_2 (hBOOL(hAPP_nat_bool(Pa,hAPP_nat_nat(suc,N_2))) -> hBOOL(hAPP_nat_bool(Pa,N_2)))) -> hBOOL(hAPP_nat_bool(Pa,hAPP_nat_nat(minus_minus_nat(K),I)))))) # label(fact_292_zero__induct__lemma) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 337 (all P all Q all R hAPP_nat_bool(cOMBB_bool_bool_nat(P,Q),R) = hAPP_bool_bool(P,hAPP_nat_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__Nat__Onat_U) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 338 (all P all Q all R hAPP_f1664156314l_bool(hAPP_p338031245l_bool(P,R),Q) = hAPP_pname_bool(hAPP_f759274231e_bool(cOMBC_1058051404l_bool(P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__Com__Opname_000tc__fun_Itc__Com__Opname_Mtc__HOL__Ob) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 339 (all A_1 all Pa collect_pname(cOMBS_568398431l_bool(cOMBB_675860798_pname(fimplies,cOMBB_647938656_pname(fNot,hAPP_p61793385e_bool(cOMBC_1149511130e_bool(fequal_pname),A_1))),Pa)) = insert_pname(A_1,collect_pname(Pa))) # label(fact_199_insert__Collect) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 340 (all A hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),A))) # label(fact_227_subset__refl) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 341 (all P all Q all R hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(P,R),Q) = hAPP_f1637334154l_bool(hAPP_f1772781669l_bool(cOMBC_595898202l_bool(P),Q),R)) # label(help_COMBC_1_1_COMBC_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__H) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 342 (all A all B (hBOOL(hAPP_f621171935l_bool(hAPP_f1434722111l_bool(ord_le1375614389l_bool,A),B)) -> (hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,B)) -> hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A))))) # label(fact_144_finite__subset) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 343 (all P all Q all R hAPP_nat_bool(hAPP_f800510211t_bool(cOMBC_226598744l_bool(P),Q),R) = hAPP_f54304608l_bool(hAPP_n215258509l_bool(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__Nat__Onat_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 344 (all Na all N_3 all F ((all N_2 hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,hAPP_nat_fun_a_bool(F,N_2)),hAPP_nat_fun_a_bool(F,hAPP_nat_nat(suc,N_2))))) -> (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,Na),N_3)) -> hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,hAPP_nat_fun_a_bool(F,Na)),hAPP_nat_fun_a_bool(F,N_3)))))) # label(fact_174_lift__Suc__mono__le) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 345 (all M_2 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_nat_nat(suc,M_2)),N_1)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_1)))) # label(fact_153_Suc__leD) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 346 (all B all A ((all X_1 (is_pname(X_1) -> (hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A)) -> hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),B))))) -> hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B)))) # label(fact_291_subsetI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 347 (all A_1 all B_1 all A (is_a(A_1) & is_a(B_1) -> (B_1 = A_1 | hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_1),A)) <-> hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_1),insert_a(B_1,A)))))) # label(fact_212_insert__iff) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 348 (all N_1 N_1 != hAPP_nat_nat(suc,N_1)) # label(fact_122_n__not__Suc__n) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 349 (all P all Q all R hAPP_f937997336l_bool(hAPP_f760187903l_bool(cOMBC_1269652216l_bool(P),Q),R) = hAPP_f937997336l_bool(hAPP_f760187903l_bool(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__fun_Itc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 350 (all M_2 all N_1 hAPP_nat_nat(minus_minus_nat(hAPP_nat_nat(suc,M_2)),hAPP_nat_nat(suc,N_1)) = hAPP_nat_nat(minus_minus_nat(M_2),N_1)) # label(fact_161_diff__Suc__Suc) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 351 (all X_2 all A all B (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)) -> (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A)) -> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),B))))) # label(fact_243_set__mp) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 352 (all H all F_1 (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1)) -> hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,image_nat_a(H,F_1))))) # label(fact_31_finite__imageI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 353 (all F all A (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_fun_a_bool_nat(finite_card_a,image_nat_a(F,A))),hAPP_f22106695ol_nat(finite_card_nat,A))))) # label(fact_73_card__image__le) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 354 (all X_2 all A (hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,A)) -> (hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A)) -> hAPP_f921600141ol_nat(finite_card_pname,A) = hAPP_f921600141ol_nat(finite_card_pname,insert_pname(X_2,A))) & (-hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A)) -> hAPP_nat_nat(suc,hAPP_f921600141ol_nat(finite_card_pname,A)) = hAPP_f921600141ol_nat(finite_card_pname,insert_pname(X_2,A))))) # label(fact_102_card__insert__if) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 355 (all B_1_1 all B_2 (is_fun949378684l_bool(B_2) -> is_fun949378684l_bool(cOMBB_2140588453a_bool(B_1_1,B_2)))) # label(gsy_c_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__fun_It__a_Mtc__HOL__Obool) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 356 (all B_1_1 all B_2 is_bool(hAPP_f389811538l_bool(B_1_1,B_2))) # label(gsy_c_hAPP_000tc__fun_Itc__fun_Itc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_Mtc__) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 357 (all X_2 all A (hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A)) -> (-hBOOL(hAPP_f621171935l_bool(hAPP_f285962445l_bool(member_fun_a_bool,X_2),A)) -> hAPP_nat_nat(suc,hAPP_f2009550088ol_nat(finite1306199131a_bool,A)) = hAPP_f2009550088ol_nat(finite1306199131a_bool,insert_fun_a_bool(X_2,A))))) # label(fact_106_card__insert__disjoint) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 358 (all B_1_1 all B_2 (is_fun_pname_bool(B_2) -> is_fun_pname_bool(image_pname_pname(B_1_1,B_2)))) # label(gsy_c_Set_Oimage_000tc__Com__Opname_000tc__Com__Opname) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 359 (all B_1_1 all B_2 is_fun_a_bool(image_526090948bool_a(B_1_1,B_2))) # label(gsy_c_Set_Oimage_000tc__fun_Itc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_Mtc__HOL__) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 360 (all A_1 all C all D (hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,C),D)) -> hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,insert_a(A_1,C)),insert_a(A_1,D))))) # label(fact_283_insert__mono) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 361 (all A all B (B = A -> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)))) # label(fact_231_equalityD1) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 362 (all Q all P (hBOOL(P) | hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fimplies,P),Q)))) # label(help_fimplies_1_1_U) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 363 (all K hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(hAPP_n1699378549t_bool(cOMBC_nat_nat_bool(ord_less_eq_nat),K))))) # label(fact_117_finite__Collect__le__nat) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 364 (all P all Q all R hAPP_f1664156314l_bool(cOMBB_307249310e_bool(P,Q),R) = hAPP_bool_bool(P,hAPP_f1664156314l_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__fun_Itc__Com__Op) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 365 (all B_1_1 all B_2 (is_fun949378684l_bool(B_2) -> is_fun949378684l_bool(cOMBS_1035972772l_bool(B_1_1,B_2)))) # label(gsy_c_COMBS_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__HOL__Obool_000tc__HOL__Obo) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 366 (all Z all F all A (is_a(Z) -> (hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,Z),image_pname_a(F,A))) <-> (exists X_1 (is_pname(X_1) & hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_1),A)) & hAPP_pname_a(F,X_1) = Z))))) # label(fact_260_image__iff) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 367 (all P all Q (-hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fimplies,P),Q)) | -hBOOL(P) | hBOOL(Q))) # label(help_fimplies_3_1_U) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 368 (all B_1_1 (is_fun_pname_bool(B_1_1) -> is_fun_pname_bool(collect_pname(B_1_1)))) # label(gsy_c_Set_OCollect_000tc__Com__Opname) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 369 (all P all Q all R hAPP_f1664156314l_bool(hAPP_f559147733l_bool(cOMBC_1988546018l_bool(P),Q),R) = hAPP_f1935102916l_bool(hAPP_f556039215l_bool(P,R),Q)) # label(help_COMBC_1_1_COMBC_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__fun_It_012) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 370 (all A_1 all B insert_a(A_1,B) = collect_a(cOMBS_a_bool_bool(cOMBB_1972296269bool_a(fdisj,hAPP_a_fun_a_bool(cOMBC_a_a_bool(fequal_a),A_1)),hAPP_f2050579477a_bool(cOMBC_1355376034l_bool(member_a),B)))) # label(fact_194_insert__compr) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 371 (all H all F_1 (hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_1)) -> hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,image_47868345e_bool(H,F_1))))) # label(fact_36_finite__imageI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 372 (all A all B (is_fun_pname_bool(B) & is_fun_pname_bool(A) -> (B = A <-> hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B)) & hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,B),A))))) # label(fact_229_set__eq__subset) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 373 (all A all B (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)) -> (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),A)) -> B = A))) # label(fact_177_equalityI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 374 (all B all X_2 all A (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A)) -> (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)) -> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),B))))) # label(fact_240_set__rev__mp) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 375 (all F all A (hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(finite_card_nat,image_a_nat(F,A))),hAPP_fun_a_bool_nat(finite_card_a,A))))) # label(fact_55_card__image__le) # label(axiom) # label(non_clause). [assumption]. 1.00/1.30 376 (all X_2 all A (hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f696928925ol_nat(finite346522414t_bool,A)),hAPP_f696928925ol_nat(finite346522414t_bool,insert_fun_nat_bool(X_2,A)))))) # label(fact_92_card__insert__le) # label(axiom) # label(non_clause). [assumption]. 1.00/1.31 377 (all H all F_1 (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1)) -> hBOOL(hAPP_f292226953l_bool(finite1381704300l_bool,image_1208015684l_bool(H,F_1))))) # label(fact_27_finite__imageI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.31 378 (all B all A_1 hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),insert_nat(A_1,B)))) # label(fact_269_subset__insertI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.31 379 (all A all B (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)) -> (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B)) -> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A))))) # label(fact_145_finite__subset) # label(axiom) # label(non_clause). [assumption]. 1.00/1.31 380 (all F all A (hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f921600141ol_nat(finite_card_pname,image_1854862208_pname(F,A))),hAPP_f2009550088ol_nat(finite1306199131a_bool,A))))) # label(fact_79_card__image__le) # label(axiom) # label(non_clause). [assumption]. 1.00/1.31 381 (all F all A (hBOOL(hAPP_f292226953l_bool(finite1381704300l_bool,A)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_fun_a_bool_nat(finite_card_a,image_573985017bool_a(F,A))),hAPP_f1253658590ol_nat(finite1659325229l_bool,A))))) # label(fact_64_card__image__le) # label(axiom) # label(non_clause). [assumption]. 1.00/1.31 382 (all B_1_1 all B_2 (is_fun_a_bool(B_2) & is_fun949378684l_bool(B_1_1) -> is_bool(hAPP_fun_a_bool_bool(B_1_1,B_2)))) # label(gsy_c_hAPP_000tc__fun_It__a_Mtc__HOL__Obool_J_000tc__HOL__Obool) # label(hypothesis) # label(non_clause). [assumption]. 1.00/1.31 383 (all H all F_1 (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,F_1)) -> hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,image_nat_fun_a_bool(H,F_1))))) # label(fact_28_finite__imageI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.31 384 (all B all X_2 all A (-hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_2),A)) -> (hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),insert_a(X_2,B))) <-> hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B))))) # label(fact_277_subset__insert) # label(axiom) # label(non_clause). [assumption]. 1.00/1.31 385 (all X_2 all A (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(finite_card_nat,A)),hAPP_f22106695ol_nat(finite_card_nat,insert_nat(X_2,A)))))) # label(fact_96_card__insert__le) # label(axiom) # label(non_clause). [assumption]. 1.00/1.31 386 (all F all A_1 all B image_pname_a(F,insert_pname(A_1,B)) = insert_a(hAPP_pname_a(F,A_1),image_pname_a(F,B))) # label(fact_284_image__insert) # label(axiom) # label(non_clause). [assumption]. 1.00/1.31 387 (all I_1 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,I_1),N_1)) -> hAPP_nat_nat(minus_minus_nat(N_1),hAPP_nat_nat(minus_minus_nat(N_1),I_1)) = I_1)) # label(fact_165_diff__diff__cancel) # label(axiom) # label(non_clause). [assumption]. 1.00/1.31 388 (all A all B (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,B)) -> (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f22106695ol_nat(finite_card_nat,A)),hAPP_f22106695ol_nat(finite_card_nat,B)))))) # label(fact_85_card__mono) # label(axiom) # label(non_clause). [assumption]. 1.00/1.31 389 (all A hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),A))) # label(fact_225_subset__refl) # label(axiom) # label(non_clause). [assumption]. 1.00/1.31 390 (all B_1 all A_1 all B ((-hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_1),B)) -> B_1 = A_1) -> hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A_1),insert_a(B_1,B))))) # label(fact_185_insertCI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.31 391 (all B_1_1 all B_2 is_fun_a_bool(image_fun_nat_bool_a(B_1_1,B_2))) # label(gsy_c_Set_Oimage_000tc__fun_Itc__Nat__Onat_Mtc__HOL__Obool_J_000t__a) # label(axiom) # label(non_clause). [assumption]. 1.00/1.31 392 (all A_1 all A (hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,insert_fun_nat_bool(A_1,A))) <-> hBOOL(hAPP_f1637334154l_bool(finite2012431853t_bool,A)))) # label(fact_138_finite__insert) # label(axiom) # label(non_clause). [assumption]. 1.00/1.31 393 (all B_1_1 all B_2 (is_bool(B_2) -> is_bool(hAPP_bool_bool(B_1_1,B_2)))) # label(gsy_c_hAPP_000tc__HOL__Obool_000tc__HOL__Obool) # label(axiom) # label(non_clause). [assumption]. 1.00/1.31 394 (all A all B (is_fun_a_bool(A) & is_fun_a_bool(B) -> (hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B)) -> (hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),A)) -> B = A)))) # label(fact_179_equalityI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.31 395 (all B_1_1 all B_2 (is_fun_pname_bool(B_2) -> is_fun949378684l_bool(image_112932426a_bool(B_1_1,B_2)))) # label(gsy_c_Set_Oimage_000tc__Com__Opname_000tc__fun_It__a_Mtc__HOL__Obool_J) # label(axiom) # label(non_clause). [assumption]. 1.00/1.31 396 (all F all A (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f2009550088ol_nat(finite1306199131a_bool,image_nat_fun_a_bool(F,A))),hAPP_f22106695ol_nat(finite_card_nat,A))))) # label(fact_76_card__image__le) # label(axiom) # label(non_clause). [assumption]. 1.00/1.31 397 (all X_2 all A all B (hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B)) -> (hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A)) -> hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),B))))) # label(fact_239_in__mono) # label(axiom) # label(non_clause). [assumption]. 1.00/1.31 398 (all H all F_1 (hBOOL(hAPP_f389811538l_bool(finite786885583l_bool,F_1)) -> hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,image_1705983821_pname(H,F_1))))) # label(fact_19_finite__imageI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.31 399 (all Q_1 all Pa (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(Q_1))) | hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(Pa))) -> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,collect_nat(cOMBS_nat_bool_bool(cOMBB_1015721476ol_nat(fconj,Pa),Q_1)))))) # label(fact_115_finite__Collect__conjI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.31 400 (all A all B (hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,B)) -> (hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_fun_a_bool_nat(finite_card_a,A)),hAPP_fun_a_bool_nat(finite_card_a,B)))))) # label(fact_84_card__mono) # label(axiom) # label(non_clause). [assumption]. 1.00/1.31 401 (all X_2 hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,X_2),X_2))) # label(fact_298_order__refl) # label(axiom) # label(non_clause). [assumption]. 1.00/1.31 402 (all P all Q all R hAPP_fun_a_bool_bool(cOMBB_2140588453a_bool(P,Q),R) = hAPP_bool_bool(P,hAPP_fun_a_bool_bool(Q,R))) # label(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000tc__fun_It__a_Mtc__H) # label(axiom) # label(non_clause). [assumption]. 1.00/1.31 403 (all X all Y (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(fequal_nat,X),Y)) | Y != X)) # label(help_fequal_2_1_fequal_000tc__Nat__Onat_T) # label(axiom) # label(non_clause). [assumption]. 1.00/1.31 404 (all A all B (A = B -> -(hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)) -> -hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),A))))) # label(fact_249_equalityE) # label(axiom) # label(non_clause). [assumption]. 1.00/1.31 405 (all X_1 all Xa insert_a(X_1,Xa) = collect_a(cOMBS_a_bool_bool(cOMBB_1972296269bool_a(fdisj,hAPP_a_fun_a_bool(cOMBC_a_a_bool(fequal_a),X_1)),hAPP_f2050579477a_bool(cOMBC_1355376034l_bool(member_a),Xa)))) # label(fact_265_insert__compr__raw) # label(axiom) # label(non_clause). [assumption]. 1.00/1.31 406 (all Q all P (hBOOL(hAPP_bool_bool(hAPP_b589554111l_bool(fconj,P),Q)) | -hBOOL(Q) | -hBOOL(P))) # label(help_fconj_1_1_U) # label(axiom) # label(non_clause). [assumption]. 1.00/1.31 407 (all N_1 hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,N_1),N_1))) # label(fact_127_le__refl) # label(axiom) # label(non_clause). [assumption]. 1.00/1.31 408 (all B all X_2 all A (-hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A)) -> (-hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),B)) -> (A = B <-> insert_nat(X_2,B) = insert_nat(X_2,A))))) # label(fact_216_insert__ident) # label(axiom) # label(non_clause). [assumption]. 1.00/1.31 409 (all A_1 all B_1 all A (is_pname(B_1) & is_pname(A_1) -> (hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),insert_pname(B_1,A))) -> (A_1 != B_1 -> hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A_1),A)))))) # label(fact_187_insertE) # label(axiom) # label(non_clause). [assumption]. 1.00/1.31 410 (all B_1_1 all B_2 (is_fun_pname_bool(B_2) -> is_fun1661590463l_bool(hAPP_f434788991l_bool(B_1_1,B_2)))) # label(gsy_c_hAPP_000tc__fun_Itc__Com__Opname_Mtc__HOL__Obool_J_000tc__fun_Itc__fun_Itc) # label(axiom) # label(non_clause). [assumption]. 1.00/1.31 411 (all A (hBOOL(hAPP_f389811538l_bool(finite786885583l_bool,A)) -> hBOOL(hAPP_f595608956l_bool(finite1491191519l_bool,collec792590109l_bool(hAPP_f1759205631l_bool(cOMBC_336095980l_bool(ord_le1375671464l_bool),A)))))) # label(fact_5_finite__Collect__subsets) # label(axiom) # label(non_clause). [assumption]. 1.00/1.31 412 (all X_1 all Xa collect_pname(cOMBS_568398431l_bool(cOMBB_675860798_pname(fdisj,hAPP_p61793385e_bool(cOMBC_1149511130e_bool(fequal_pname),X_1)),hAPP_f759274231e_bool(cOMBC_1058051404l_bool(member_pname),Xa))) = insert_pname(X_1,Xa)) # label(fact_264_insert__compr__raw) # label(axiom) # label(non_clause). [assumption]. 1.00/1.31 413 (all H all F_1 (hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,F_1)) -> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,image_fun_a_bool_nat(H,F_1))))) # label(fact_14_finite__imageI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.31 414 (all X_2 all A (hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A)) -> (-hBOOL(hAPP_f1935102916l_bool(hAPP_f556039215l_bool(member799430823e_bool,X_2),A)) -> hAPP_f55526627ol_nat(finite1340463720e_bool,insert1325755072e_bool(X_2,A)) = hAPP_nat_nat(suc,hAPP_f55526627ol_nat(finite1340463720e_bool,A))))) # label(fact_105_card__insert__disjoint) # label(axiom) # label(non_clause). [assumption]. 1.00/1.31 415 (all B_1_1 all B_2 (is_fun_pname_bool(B_2) -> is_fun_a_bool(image_pname_a(B_1_1,B_2)))) # label(gsy_c_Set_Oimage_000tc__Com__Opname_000t__a) # label(hypothesis) # label(non_clause). [assumption]. 1.00/1.31 416 (all C all A all B (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)) -> (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,B),C)) -> hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),C))))) # label(fact_246_subset__trans) # label(axiom) # label(non_clause). [assumption]. 1.00/1.31 417 (all H all F_1 (hBOOL(hAPP_f1664156314l_bool(finite_finite_pname,F_1)) -> hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,image_112932426a_bool(H,F_1))))) # label(fact_35_finite__imageI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.31 418 (all B_1_1 all B_2 is_fun949378684l_bool(image_nat_fun_a_bool(B_1_1,B_2))) # label(gsy_c_Set_Oimage_000tc__Nat__Onat_000tc__fun_It__a_Mtc__HOL__Obool_J) # label(axiom) # label(non_clause). [assumption]. 1.00/1.31 419 (all X_2 all A all B (hBOOL(hAPP_f54304608l_bool(hAPP_f103356543l_bool(ord_le1568362934t_bool,A),B)) -> (hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),A)) -> hBOOL(hAPP_f54304608l_bool(hAPP_n215258509l_bool(member_nat,X_2),B))))) # label(fact_237_in__mono) # label(axiom) # label(non_clause). [assumption]. 1.00/1.31 420 (all A_1 all A (hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,A)) <-> hBOOL(hAPP_fun_a_bool_bool(finite_finite_a,insert_a(A_1,A))))) # label(fact_137_finite__insert) # label(axiom) # label(non_clause). [assumption]. 1.00/1.31 421 (all M_2 all N_1 (hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),hAPP_nat_nat(suc,N_1))) -> (-hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,M_2),N_1)) -> hAPP_nat_nat(suc,N_1) = M_2))) # label(fact_154_le__SucE) # label(axiom) # label(non_clause). [assumption]. 1.00/1.47 422 (all B_1_1 all B_2 (is_fun_a_bool(B_2) -> is_fun_a_bool(cOMBB_bool_bool_a(B_1_1,B_2)))) # label(gsy_c_COMBB_000tc__HOL__Obool_000tc__HOL__Obool_000t__a) # label(axiom) # label(non_clause). [assumption]. 1.00/1.47 423 (all F all A (hBOOL(hAPP_f54304608l_bool(finite_finite_nat,A)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f696928925ol_nat(finite346522414t_bool,image_26036933t_bool(F,A))),hAPP_f22106695ol_nat(finite_card_nat,A))))) # label(fact_74_card__image__le) # label(axiom) # label(non_clause). [assumption]. 1.00/1.47 424 (all X_2 all A (hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A)) -> hBOOL(hAPP_nat_bool(hAPP_n1699378549t_bool(ord_less_eq_nat,hAPP_f2009550088ol_nat(finite1306199131a_bool,A)),hAPP_f2009550088ol_nat(finite1306199131a_bool,insert_fun_a_bool(X_2,A)))))) # label(fact_94_card__insert__le) # label(axiom) # label(non_clause). [assumption]. 1.00/1.47 425 (all A all B (B = A -> hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B)))) # label(fact_232_equalityD1) # label(axiom) # label(non_clause). [assumption]. 1.00/1.47 426 (all A_1 all Pa collect_fun_a_bool(cOMBS_1035972772l_bool(cOMBB_338059395a_bool(fimplies,cOMBB_2140588453a_bool(fNot,hAPP_f1631501043l_bool(cOMBC_1732670874l_bool(fequal_fun_a_bool),A_1))),Pa)) = insert_fun_a_bool(A_1,collect_fun_a_bool(Pa))) # label(fact_203_insert__Collect) # label(axiom) # label(non_clause). [assumption]. 1.00/1.47 427 (all F all A all B (hBOOL(hAPP_f1664156314l_bool(hAPP_f434788991l_bool(ord_le313189616e_bool,A),B)) -> hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,image_pname_a(F,A)),image_pname_a(F,B))))) # label(fact_287_image__mono) # label(axiom) # label(non_clause). [assumption]. 1.00/1.47 428 (all A (hBOOL(hAPP_f1935102916l_bool(finite595471783e_bool,A)) -> hBOOL(hAPP_f389811538l_bool(finite786885583l_bool,collec1613912337l_bool(hAPP_f510955609l_bool(cOMBC_7971162l_bool(ord_le675606854l_bool),A)))))) # label(fact_8_finite__Collect__subsets) # label(axiom) # label(non_clause). [assumption]. 1.00/1.47 429 (all A_1 all A (hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,insert_fun_a_bool(A_1,A))) <-> hBOOL(hAPP_f621171935l_bool(finite347923420a_bool,A)))) # label(fact_140_finite__insert) # label(axiom) # label(non_clause). [assumption]. 1.00/1.47 430 (all P all Q all R hAPP_b589554111l_bool(P,hAPP_fun_a_bool_bool(Q,R)) = hAPP_f198738859l_bool(cOMBB_338059395a_bool(P,Q),R)) # label(help_COMBB_1_1_COMBB_000tc__HOL__Obool_000tc__fun_Itc__HOL__Obool_Mtc__HOL__Oboo_008) # label(axiom) # label(non_clause). [assumption]. 1.00/1.47 431 (all H all F_1 (hBOOL(hAPP_f292226953l_bool(finite1381704300l_bool,F_1)) -> hBOOL(hAPP_f54304608l_bool(finite_finite_nat,image_1802975832ol_nat(H,F_1))))) # label(fact_13_finite__imageI) # label(axiom) # label(non_clause). [assumption]. 1.00/1.47 1.00/1.47 ============================== end of process non-clausal formulas === 1.00/1.47 1.00/1.47 ============================== PROCESS INITIAL CLAUSES =============== 1.00/1.47 1.00/1.47 ============================== PREDICATE ELIMINATION ================= 1.00/1.47 1.00/1.47 ============================== end predicate elimination ============= 1.00/1.47 1.00/1.47 Auto_denials: (non-Horn, no changes). 1.00/1.47 1.00/1.47 Term ordering decisions: 1.00/1.47 Function symbol KB weights: ord_less_eq_nat=1. finite_finite_pname=1. ord_le1311769555a_bool=1. finite_finite_nat=1. member_a=1. member_pname=1. finite_finite_a=1. suc=1. ord_le313189616e_bool=1. ord_le1568362934t_bool=1. member_nat=1. finite2012431853t_bool=1. finite347923420a_bool=1. finite595471783e_bool=1. fdisj=1. finite_card_a=1. finite_card_pname=1. finite_card_nat=1. finite1306199131a_bool=1. finite1340463720e_bool=1. finite346522414t_bool=1. fconj=1. fimplies=1. finite1381704300l_bool=1. finite1701474069l_bool=1. finite786885583l_bool=1. fNot=1. bool=1. fequal_a=1. u=1. fequal533582459e_bool=1. fequal_fun_a_bool=1. fequal_fun_nat_bool=1. fequal_nat=1. fequal_pname=1. member799430823e_bool=1. member_fun_a_bool=1. member_fun_nat_bool=1. ord_le1375614389l_bool=1. ord_le1454342156l_bool=1. ord_le675606854l_bool=1. g=1. mgt_call=1. pname=1. x_a=1. na=1. pn=1. finite1343359508l_bool=1. finite1352710292l_bool=1. finite1491191519l_bool=1. finite1659325229l_bool=1. finite269641166l_bool=1. finite719726885l_bool=1. ord_le1375671464l_bool=1. ord_le65145710l_bool=1. ord_le967226251l_bool=1. hAPP_fun_a_bool_bool=1. hAPP_f1664156314l_bool=1. hAPP_f54304608l_bool=1. hAPP_nat_bool=1. hAPP_n1699378549t_bool=1. hAPP_nat_nat=1. hAPP_f1631501043l_bool=1. hAPP_f434788991l_bool=1. hAPP_a85458249l_bool=1. hAPP_f103356543l_bool=1. hAPP_p338031245l_bool=1. hAPP_f1637334154l_bool=1. hAPP_f1935102916l_bool=1. hAPP_f621171935l_bool=1. insert_a=1. insert_pname=1. insert_nat=1. hAPP_n215258509l_bool=1. image_pname_a=1. hAPP_fun_a_bool_nat=1. hAPP_f921600141ol_nat=1. hAPP_bool_bool=1. hAPP_f22106695ol_nat=1. hAPP_a_bool=1. hAPP_pname_bool=1. hAPP_f2009550088ol_nat=1. hAPP_f55526627ol_nat=1. hAPP_f696928925ol_nat=1. hAPP_b589554111l_bool=1. hAPP_f292226953l_bool=1. hAPP_f389811538l_bool=1. hAPP_f937997336l_bool=1. hAPP_pname_a=1. cOMBS_568398431l_bool=1. insert1325755072e_bool=1. insert_fun_a_bool=1. cOMBB_675860798_pname=1. cOMBS_1035972772l_bool=1. cOMBS_350070575l_bool=1. cOMBS_a_bool_bool=1. insert_fun_nat_bool=1. cOMBB_1015721476ol_nat=1. cOMBB_1972296269bool_a=1. cOMBB_2095475776e_bool=1. cOMBB_338059395a_bool=1. cOMBB_444170502t_bool=1. cOMBS_1187019125l_bool=1. cOMBS_nat_bool_bool=1. hAPP_a_fun_a_bool=1. hAPP_p61793385e_bool=1. hAPP_f1434722111l_bool=1. hAPP_f1772781669l_bool=1. hAPP_f510955609l_bool=1. fun=1. hAPP_f759274231e_bool=1. hAPP_n1025906991e_bool=1. hAPP_nat_fun_a_bool=1. hAPP_f1951378235l_bool=1. hAPP_f2050579477a_bool=1. hAPP_f2117159681l_bool=1. hAPP_f285962445l_bool=1. hAPP_f556039215l_bool=1. hAPP_f559147733l_bool=1. cOMBB_2140588453a_bool=1. cOMBB_307249310e_bool=1. cOMBB_647938656_pname=1. cOMBB_bool_bool_a=1. hAPP_a93125764e_bool=1. hAPP_f1050622307l_bool=1. hAPP_f1246832597l_bool=1. hAPP_f1759205631l_bool=1. hAPP_f760187903l_bool=1. hAPP_f800510211t_bool=1. hAPP_p1534023578a_bool=1. image_112932426a_bool=1. image_1283814551_pname=1. image_1655916159e_bool=1. image_1854862208_pname=1. image_1921560913_pname=1. image_47868345e_bool=1. image_876012084bool_a=1. image_a_a=1. image_a_pname=1. image_fun_a_bool_a=1. image_fun_nat_bool_a=1. image_nat_a=1. image_nat_fun_a_bool=1. image_nat_pname=1. image_pname_pname=1. cOMBB_1897541054_pname=1. cOMBB_238756964t_bool=1. cOMBB_bool_bool_nat=1. hAPP_a_fun_bool_bool=1. hAPP_f1295398978l_bool=1. hAPP_f1363661463l_bool=1. hAPP_f1476298914l_bool=1. hAPP_f1748468828l_bool=1. hAPP_f198738859l_bool=1. hAPP_f595608956l_bool=1. hAPP_n1006566506l_bool=1. hAPP_p393069232l_bool=1. image_1551609309ol_nat=1. image_1604018183_pname=1. image_1705983821_pname=1. image_2129980159t_bool=1. image_26036933t_bool=1. image_349102846bool_a=1. image_496248727ol_nat=1. image_526090948bool_a=1. image_573985017bool_a=1. image_990671762_pname=1. image_a_nat=1. image_fun_a_bool_nat=1. image_pname_nat=1. hAPP_f1253658590ol_nat=1. hAPP_f1690079119ol_nat=1. hAPP_f98387925ol_nat=1. image_1079571347ol_nat=1. image_1154884483l_bool=1. image_1208015684l_bool=1. image_1420695166l_bool=1. image_1607900221l_bool=1. image_1642285373l_bool=1. image_1802975832ol_nat=1. image_1874789623l_bool=1. image_2089570637ol_nat=1. insert1117693814l_bool=1. insert1457093509l_bool=1. insert2003652156l_bool=1. f3=1. f4=1. f9=1. f11=1. f12=1. f16=1. minus_minus_nat=1. collec1974731493e_bool=1. collect_fun_a_bool=1. collect_nat=1. collect_pname=1. collect_fun_nat_bool=1. collect_a=1. cOMBC_nat_nat_bool=1. cOMBC_1284144636l_bool=1. cOMBC_1693257480l_bool=1. cOMBC_1732670874l_bool=1. cOMBC_1058051404l_bool=1. cOMBC_1149511130e_bool=1. cOMBC_a_a_bool=1. cOMBC_1245412066l_bool=1. cOMBC_1355376034l_bool=1. cOMBC_1880041174l_bool=1. cOMBC_1988546018l_bool=1. cOMBC_226598744l_bool=1. cOMBC_1269652216l_bool=1. cOMBC_331553030l_bool=1. cOMBC_336095980l_bool=1. cOMBC_595898202l_bool=1. cOMBC_636888218l_bool=1. cOMBC_7971162l_bool=1. cOMBC_pname_a_bool=1. p=1. collec1015864663l_bool=1. collec1613912337l_bool=1. collec1635217238l_bool=1. collec1874991203l_bool=1. collec707592106l_bool=1. collec792590109l_bool=1. undefi1699038445l_bool=1. undefi17486888e_bool=1. undefi64961550l_bool=1. undefined_a=1. undefined_fun_a_bool=1. undefined_pname=1. f10=1. f1=1. f2=1. f5=1. f6=1. f7=1. f8=1. f13=1. f14=1. f15=1. f17=1. 13.71/13.95 13.71/13.95 ============================== end of process initial clauses ======== 13.71/13.95 13.71/13.95 ============================== CLAUSES FOR SEARCH ==================== 13.71/13.95 13.71/13.95 ============================== end of clauses for search ============= 13.71/13.95 13.71/13.95 ============================== SEARCH ================================ 13.71/13.95 13.71/13.95 % Starting search at 0.24 seconds. 13.71/13.95 13.71/13.95 Low Water (keep): wt=54.000, iters=3333 13.71/13.95 13.71/13.95 Low Water (keep): wt=45.000, iters=3371 13.71/13.95 13.71/13.95 Low Water (keep): wt=43.000, iters=3405 13.71/13.95 13.71/13.95 Low Water (keep): wt=41.000, iters=3339 13.71/13.95 13.71/13.95 Low Water (keep): wt=37.000, iters=3341 13.71/13.95 13.71/13.95 Low Water (keep): wt=36.000, iters=3348 13.71/13.95 13.71/13.95 Low Water (keep): wt=33.000, iters=3336 13.71/13.95 13.71/13.95 Low Water (keep): wt=32.000, iters=3343 13.71/13.95 13.71/13.95 Low Water (keep): wt=31.000, iters=3414 13.71/13.95 13.71/13.95 Low Water (keep): wt=29.000, iters=3346 13.71/13.95 13.71/13.95 Low Water (keep): wt=28.000, iters=3339 13.71/13.95 13.71/13.95 Low Water (keep): wt=27.000, iters=3485 13.71/13.95 13.71/13.95 Low Water (keep): wt=25.000, iters=3353 13.71/13.95 13.71/13.95 Low Water (keep): wt=24.000, iters=3340 13.71/13.95 13.71/13.95 Low Water (keep): wt=23.000, iters=3348 13.71/13.95 13.71/13.95 Low Water (keep): wt=22.000, iters=3366 13.71/13.95 13.71/13.95 Low Water (keep): wt=21.000, iters=3339 13.71/13.95 13.71/13.95 Low Water (keep): wt=19.000, iters=3337 13.71/13.95 13.71/13.95 Low Water (keep): wt=18.000, iters=3366 13.71/13.95 13.71/13.95 Low Water (keep): wt=17.000, iters=3337 13.71/13.95 13.71/13.95 Low Water (displace): id=12261, wt=14.000 13.71/13.95 13.71/13.95 Low Water (displace): id=12262, wt=9.000 13.71/13.95 13.71/13.95 Low Water (displace): id=12271, wt=8.000 13.71/13.95 13.71/13.95 Low Water (keep): wt=16.000, iters=5909 13.71/13.95 13.71/13.95 Low Water (keep): wt=14.000, iters=5581 13.71/13.95 13.71/13.95 Low Water (keep): wt=11.000, iters=4871 13.71/13.95 13.71/13.95 Low Water (keep): wt=9.000, iters=4406 13.71/13.95 13.71/13.95 Low Water (keep): wt=8.000, iters=3373 13.71/13.95 13.71/13.95 ============================== PROOF ================================= 13.71/13.95 % SZS status Theorem 13.71/13.95 % SZS output start Refutation 13.71/13.95 13.71/13.95 % Proof 1 at 12.58 (+ 0.15) seconds. 13.71/13.95 % Length of proof is 15. 13.71/13.95 % Level of proof is 5. 13.71/13.95 % Maximum clause weight is 29.000. 13.71/13.95 % Given clauses 5672. 13.71/13.95 13.71/13.95 31 (all F all X_2 all A (hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,X_2),A)) -> hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,hAPP_pname_a(F,X_2)),image_pname_a(F,A))))) # label(fact_261_imageI) # label(axiom) # label(non_clause). [assumption]. 13.71/13.95 238 (all X_2 all A all B (hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,insert_a(X_2,A)),B)) <-> hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,A),B)) & hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,X_2),B)))) # label(fact_274_insert__subset) # label(axiom) # label(non_clause). [assumption]. 13.71/13.95 370 (all A_1 all B insert_a(A_1,B) = collect_a(cOMBS_a_bool_bool(cOMBB_1972296269bool_a(fdisj,hAPP_a_fun_a_bool(cOMBC_a_a_bool(fequal_a),A_1)),hAPP_f2050579477a_bool(cOMBC_1355376034l_bool(member_a),B)))) # label(fact_194_insert__compr) # label(axiom) # label(non_clause). [assumption]. 13.71/13.95 470 -hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,A),B)) | hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,hAPP_pname_a(C,A)),image_pname_a(C,B))) # label(fact_261_imageI) # label(axiom). [clausify(31)]. 13.71/13.95 500 hBOOL(hAPP_f1664156314l_bool(hAPP_p338031245l_bool(member_pname,pn),u)) # label(conj_4) # label(hypothesis). [assumption]. 13.71/13.95 529 hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,g),image_pname_a(mgt_call,u))) # label(conj_1) # label(hypothesis). [assumption]. 13.71/13.95 774 hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,insert_a(A,B)),C)) | -hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),C)) | -hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A),C)) # label(fact_274_insert__subset) # label(axiom). [clausify(238)]. 13.71/13.95 951 collect_a(cOMBS_a_bool_bool(cOMBB_1972296269bool_a(fdisj,hAPP_a_fun_a_bool(cOMBC_a_a_bool(fequal_a),A)),hAPP_f2050579477a_bool(cOMBC_1355376034l_bool(member_a),B))) = insert_a(A,B) # label(fact_194_insert__compr) # label(axiom). [clausify(370)]. 13.71/13.95 952 insert_a(A,B) = collect_a(cOMBS_a_bool_bool(cOMBB_1972296269bool_a(fdisj,hAPP_a_fun_a_bool(cOMBC_a_a_bool(fequal_a),A)),hAPP_f2050579477a_bool(cOMBC_1355376034l_bool(member_a),B))). [copy(951),flip(a)]. 13.71/13.95 1031 -hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,insert_a(hAPP_pname_a(mgt_call,pn),g)),image_pname_a(mgt_call,u))) # label(conj_6) # label(negated_conjecture). [assumption]. 13.71/13.95 1032 -hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,collect_a(cOMBS_a_bool_bool(cOMBB_1972296269bool_a(fdisj,hAPP_a_fun_a_bool(cOMBC_a_a_bool(fequal_a),hAPP_pname_a(mgt_call,pn))),hAPP_f2050579477a_bool(cOMBC_1355376034l_bool(member_a),g)))),image_pname_a(mgt_call,u))). [copy(1031),rewrite([952(6)])]. 13.71/13.95 1086 hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,collect_a(cOMBS_a_bool_bool(cOMBB_1972296269bool_a(fdisj,hAPP_a_fun_a_bool(cOMBC_a_a_bool(fequal_a),A)),hAPP_f2050579477a_bool(cOMBC_1355376034l_bool(member_a),B)))),C)) | -hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,B),C)) | -hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A),C)). [back_rewrite(774),rewrite([952(2)])]. 13.71/13.95 1122 hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,hAPP_pname_a(A,pn)),image_pname_a(A,u))). [resolve(500,a,470,a)]. 13.71/13.95 2240 hBOOL(hAPP_fun_a_bool_bool(hAPP_f1631501043l_bool(ord_le1311769555a_bool,collect_a(cOMBS_a_bool_bool(cOMBB_1972296269bool_a(fdisj,hAPP_a_fun_a_bool(cOMBC_a_a_bool(fequal_a),A)),hAPP_f2050579477a_bool(cOMBC_1355376034l_bool(member_a),g)))),image_pname_a(mgt_call,u))) | -hBOOL(hAPP_fun_a_bool_bool(hAPP_a85458249l_bool(member_a,A),image_pname_a(mgt_call,u))). [resolve(1086,b,529,a)]. 13.71/13.95 45855 $F. [resolve(2240,b,1122,a),unit_del(a,1032)]. 13.71/13.95 13.71/13.95 % SZS output end Refutation 13.71/13.95 ============================== end of proof ========================== 13.71/13.95 13.71/13.95 ============================== STATISTICS ============================ 13.71/13.95 13.71/13.95 Given=5672. Generated=234159. Kept=45307. proofs=1. 13.71/13.95 Usable=5526. Sos=9998. Demods=601. Limbo=0, Disabled=30331. Hints=0. 13.71/13.95 Megabytes=80.42. 13.71/13.95 User_CPU=12.58, System_CPU=0.15, Wall_clock=13. 13.71/13.95 13.71/13.95 ============================== end of statistics ===================== 13.71/13.95 13.71/13.95 ============================== end of search ========================= 13.71/13.95 13.71/13.95 THEOREM PROVED 13.71/13.95 % SZS status Theorem 13.71/13.95 13.71/13.95 Exiting with 1 proof. 13.71/13.95 13.71/13.95 Process 20486 exit (max_proofs) Wed Jul 14 16:44:53 2021 13.71/13.95 Prover9 interrupted 13.71/13.95 EOF