% TIMEFORMAT='%3R'; { time (exec 2>&1; /home/martin/bin/satallax -E /home/martin/.isabelle/contrib/e-2.5-1/x86_64-linux/eprover -p tstp -t 5 /home/martin/judgement-day/tptp-thf/tptp/StrongNorm/prob_187__5211256_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:38:12.263

% Could-be-implicit typings (6)
thf(ty_n_t__List__Olist_It__List__Olist_It__Lambda__OdB_J_J, type,
    list_list_dB : $tType).
thf(ty_n_t__Set__Oset_It__List__Olist_It__Lambda__OdB_J_J, type,
    set_list_dB : $tType).
thf(ty_n_t__List__Olist_It__Lambda__OdB_J, type,
    list_dB : $tType).
thf(ty_n_t__Set__Oset_It__Lambda__OdB_J, type,
    set_dB : $tType).
thf(ty_n_t__Lambda__OdB, type,
    dB : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).

% Explicit typings (40)
thf(sy_c_BNF__Greatest__Fixpoint_OShift_001t__Lambda__OdB, type,
    bNF_Gr1110975684ift_dB : set_list_dB > dB > set_list_dB).
thf(sy_c_BNF__Greatest__Fixpoint_OSucc_001t__Lambda__OdB, type,
    bNF_Greatest_Succ_dB : set_list_dB > list_dB > set_dB).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_InductTermi_OIT, type,
    it : dB > $o).
thf(sy_c_Lambda_OdB_OAbs, type,
    abs : dB > dB).
thf(sy_c_Lambda_OdB_OApp, type,
    app : dB > dB > dB).
thf(sy_c_Lambda_OdB_OVar, type,
    var : nat > dB).
thf(sy_c_Lambda_OdB_Osize__dB, type,
    size_dB : dB > nat).
thf(sy_c_Lambda_Olift, type,
    lift : dB > nat > dB).
thf(sy_c_Lambda_Osubst, type,
    subst : dB > dB > nat > dB).
thf(sy_c_List_Oappend_001t__Lambda__OdB, type,
    append_dB : list_dB > list_dB > list_dB).
thf(sy_c_List_Oappend_001t__List__Olist_It__Lambda__OdB_J, type,
    append_list_dB : list_list_dB > list_list_dB > list_list_dB).
thf(sy_c_List_Obind_001t__Lambda__OdB_001t__Lambda__OdB, type,
    bind_dB_dB : list_dB > (dB > list_dB) > list_dB).
thf(sy_c_List_Oconcat_001t__Lambda__OdB, type,
    concat_dB : list_list_dB > list_dB).
thf(sy_c_List_Ofoldl_001t__Lambda__OdB_001t__Lambda__OdB, type,
    foldl_dB_dB : (dB > dB > dB) > dB > list_dB > dB).
thf(sy_c_List_Oinsert_001t__Lambda__OdB, type,
    insert_dB : dB > list_dB > list_dB).
thf(sy_c_List_Olist_OCons_001t__Lambda__OdB, type,
    cons_dB : dB > list_dB > list_dB).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Lambda__OdB_J, type,
    cons_list_dB : list_dB > list_list_dB > list_list_dB).
thf(sy_c_List_Olist_ONil_001t__Lambda__OdB, type,
    nil_dB : list_dB).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Lambda__OdB_J, type,
    nil_list_dB : list_list_dB).
thf(sy_c_List_Olist_Omap_001t__Lambda__OdB_001t__Lambda__OdB, type,
    map_dB_dB : (dB > dB) > list_dB > list_dB).
thf(sy_c_List_Olist_Omap_001t__List__Olist_It__Lambda__OdB_J_001t__List__Olist_It__Lambda__OdB_J, type,
    map_list_dB_list_dB : (list_dB > list_dB) > list_list_dB > list_list_dB).
thf(sy_c_List_Olist_Orec__list_001t__List__Olist_It__Lambda__OdB_J_001t__Lambda__OdB, type,
    rec_list_list_dB_dB : list_dB > (dB > list_dB > list_dB > list_dB) > list_dB > list_dB).
thf(sy_c_List_Olistsp_001t__Lambda__OdB, type,
    listsp_dB : (dB > $o) > list_dB > $o).
thf(sy_c_List_Omap__tailrec_001t__Lambda__OdB_001t__Lambda__OdB, type,
    map_tailrec_dB_dB : (dB > dB) > list_dB > list_dB).
thf(sy_c_List_On__lists_001t__Lambda__OdB, type,
    n_lists_dB : nat > list_dB > list_list_dB).
thf(sy_c_List_Oproduct__lists_001t__Lambda__OdB, type,
    product_lists_dB : list_list_dB > list_list_dB).
thf(sy_c_List_Osubseqs_001t__Lambda__OdB, type,
    subseqs_dB : list_dB > list_list_dB).
thf(sy_c_Set_OCollect_001t__Lambda__OdB, type,
    collect_dB : (dB > $o) > set_dB).
thf(sy_c_Set_OCollect_001t__List__Olist_It__Lambda__OdB_J, type,
    collect_list_dB : (list_dB > $o) > set_list_dB).
thf(sy_c_member_001t__Lambda__OdB, type,
    member_dB : dB > set_dB > $o).
thf(sy_c_member_001t__List__Olist_It__Lambda__OdB_J, type,
    member_list_dB : list_dB > set_list_dB > $o).
thf(sy_v_a____, type,
    a : dB).
thf(sy_v_as____, type,
    as : list_dB).
thf(sy_v_i____, type,
    i : nat).
thf(sy_v_n____, type,
    n : nat).
thf(sy_v_rs____, type,
    rs : list_dB).
thf(sy_v_t____, type,
    t : dB).
thf(sy_v_u1____, type,
    u1 : dB).
thf(sy_v_u____, type,
    u : dB).

% Relevant facts (134)
thf(fact_0__092_060open_062IT_At_092_060close_062, axiom,
    ((it @ t))). % \<open>IT t\<close>
thf(fact_1_Var_Oprems_I2_J, axiom,
    ((it @ u1))). % Var.prems(2)
thf(fact_2_local_OCons, axiom,
    ((rs = (cons_dB @ a @ as)))). % local.Cons
thf(fact_3_True, axiom,
    ((n = i))). % True
thf(fact_4_uIT, axiom,
    ((it @ u))). % uIT
thf(fact_5_Var__IT, axiom,
    ((![N : nat]: (it @ (var @ N))))). % Var_IT
thf(fact_6_app__Var__IT, axiom,
    ((![T : dB, I : nat]: ((it @ T) => (it @ (app @ T @ (var @ I))))))). % app_Var_IT
thf(fact_7_subst__Var__IT, axiom,
    ((![R : dB, I : nat, J : nat]: ((it @ R) => (it @ (subst @ R @ (var @ I) @ J)))))). % subst_Var_IT
thf(fact_8_lift__IT, axiom,
    ((![T : dB, I : nat]: ((it @ T) => (it @ (lift @ T @ I)))))). % lift_IT
thf(fact_9_Var__apps__eq__Var__apps__conv, axiom,
    ((![M : nat, Rs : list_dB, N : nat, Ss : list_dB]: (((foldl_dB_dB @ app @ (var @ M) @ Rs) = (foldl_dB_dB @ app @ (var @ N) @ Ss)) = (((M = N)) & ((Rs = Ss))))))). % Var_apps_eq_Var_apps_conv
thf(fact_10_subst__eq, axiom,
    ((![K : nat, U : dB]: ((subst @ (var @ K) @ U @ K) = U)))). % subst_eq
thf(fact_11_apps__eq__tail__conv, axiom,
    ((![R : dB, Ts : list_dB, S : dB]: (((foldl_dB_dB @ app @ R @ Ts) = (foldl_dB_dB @ app @ S @ Ts)) = (R = S))))). % apps_eq_tail_conv
thf(fact_12_dB_Oinject_I1_J, axiom,
    ((![X1 : nat, Y1 : nat]: (((var @ X1) = (var @ Y1)) = (X1 = Y1))))). % dB.inject(1)
thf(fact_13_dB_Oinject_I2_J, axiom,
    ((![X21 : dB, X22 : dB, Y21 : dB, Y22 : dB]: (((app @ X21 @ X22) = (app @ Y21 @ Y22)) = (((X21 = Y21)) & ((X22 = Y22))))))). % dB.inject(2)
thf(fact_14_subst__App, axiom,
    ((![T : dB, U : dB, S : dB, K : nat]: ((subst @ (app @ T @ U) @ S @ K) = (app @ (subst @ T @ S @ K) @ (subst @ U @ S @ K)))))). % subst_App
thf(fact_15_dB_Odistinct_I1_J, axiom,
    ((![X1 : nat, X21 : dB, X22 : dB]: (~ (((var @ X1) = (app @ X21 @ X22))))))). % dB.distinct(1)
thf(fact_16__092_060open_062IT_A_I_IVar_A0_A_092_060degree_062_092_060degree_062_Amap_A_I_092_060lambda_062t_O_Alift_At_A0_J_A_Imap_A_I_092_060lambda_062t_O_At_091u_Pi_093_J_Aas_J_J_091u_A_092_060degree_062_Aa_091u_Pi_093_P0_093_J_092_060close_062, axiom,
    ((it @ (subst @ (foldl_dB_dB @ app @ (var @ zero_zero_nat) @ (map_dB_dB @ (^[T2 : dB]: (lift @ T2 @ zero_zero_nat)) @ (map_dB_dB @ (^[T2 : dB]: (subst @ T2 @ u @ i)) @ as))) @ (app @ u @ (subst @ a @ u @ i)) @ zero_zero_nat)))). % \<open>IT ((Var 0 \<degree>\<degree> map (\<lambda>t. lift t 0) (map (\<lambda>t. t[u/i]) as))[u \<degree> a[u/i]/0])\<close>
thf(fact_17_IT_OVar, axiom,
    ((![Rs : list_dB, N : nat]: ((listsp_dB @ it @ Rs) => (it @ (foldl_dB_dB @ app @ (var @ N) @ Rs)))))). % IT.Var
thf(fact_18_subst__lift, axiom,
    ((![T : dB, K : nat, S : dB]: ((subst @ (lift @ T @ K) @ S @ K) = T)))). % subst_lift
thf(fact_19_Var__eq__apps__conv, axiom,
    ((![M : nat, S : dB, Ss : list_dB]: (((var @ M) = (foldl_dB_dB @ app @ S @ Ss)) = ((((var @ M) = S)) & ((Ss = nil_dB))))))). % Var_eq_apps_conv
thf(fact_20_subst__map, axiom,
    ((![T : dB, Ts : list_dB, U : dB, I : nat]: ((subst @ (foldl_dB_dB @ app @ T @ Ts) @ U @ I) = (foldl_dB_dB @ app @ (subst @ T @ U @ I) @ (map_dB_dB @ (^[T2 : dB]: (subst @ T2 @ U @ I)) @ Ts)))))). % subst_map
thf(fact_21_lift__map, axiom,
    ((![T : dB, Ts : list_dB, I : nat]: ((lift @ (foldl_dB_dB @ app @ T @ Ts) @ I) = (foldl_dB_dB @ app @ (lift @ T @ I) @ (map_dB_dB @ (^[T2 : dB]: (lift @ T2 @ I)) @ Ts)))))). % lift_map
thf(fact_22_lifts__IT, axiom,
    ((![Ts : list_dB]: ((listsp_dB @ it @ Ts) => (listsp_dB @ it @ (map_dB_dB @ (^[T2 : dB]: (lift @ T2 @ zero_zero_nat)) @ Ts)))))). % lifts_IT
thf(fact_23_lift_Osimps_I2_J, axiom,
    ((![S : dB, T : dB, K : nat]: ((lift @ (app @ S @ T) @ K) = (app @ (lift @ S @ K) @ (lift @ T @ K)))))). % lift.simps(2)
thf(fact_24_listsp__simps_I1_J, axiom,
    ((![A : dB > $o]: (listsp_dB @ A @ nil_dB)))). % listsp_simps(1)
thf(fact_25_Nil__is__map__conv, axiom,
    ((![F : dB > dB, Xs : list_dB]: ((nil_dB = (map_dB_dB @ F @ Xs)) = (Xs = nil_dB))))). % Nil_is_map_conv
thf(fact_26_map__is__Nil__conv, axiom,
    ((![F : dB > dB, Xs : list_dB]: (((map_dB_dB @ F @ Xs) = nil_dB) = (Xs = nil_dB))))). % map_is_Nil_conv
thf(fact_27_list_Omap__disc__iff, axiom,
    ((![F : dB > dB, A2 : list_dB]: (((map_dB_dB @ F @ A2) = nil_dB) = (A2 = nil_dB))))). % list.map_disc_iff
thf(fact_28_listsp__conj__eq, axiom,
    ((![A : dB > $o, B : dB > $o]: ((listsp_dB @ (^[X : dB]: (((A @ X)) & ((B @ X))))) = (^[X : list_dB]: (((listsp_dB @ A @ X)) & ((listsp_dB @ B @ X)))))))). % listsp_conj_eq
thf(fact_29_map__ident, axiom,
    (((map_dB_dB @ (^[X : dB]: X)) = (^[Xs2 : list_dB]: Xs2)))). % map_ident
thf(fact_30_listsp_Ocases, axiom,
    ((![A : dB > $o, A2 : list_dB]: ((listsp_dB @ A @ A2) => ((~ ((A2 = nil_dB))) => (~ ((![A3 : dB, L : list_dB]: ((A2 = (cons_dB @ A3 @ L)) => ((A @ A3) => (~ ((listsp_dB @ A @ L))))))))))))). % listsp.cases
thf(fact_31_listsp_Osimps, axiom,
    ((listsp_dB = (^[A4 : dB > $o]: (^[A5 : list_dB]: (((A5 = nil_dB)) | ((?[B2 : dB]: (?[L2 : list_dB]: (((A5 = (cons_dB @ B2 @ L2))) & ((((A4 @ B2)) & ((listsp_dB @ A4 @ L2)))))))))))))). % listsp.simps
thf(fact_32_listsp_Oinducts, axiom,
    ((![A : dB > $o, X2 : list_dB, P : list_dB > $o]: ((listsp_dB @ A @ X2) => ((P @ nil_dB) => ((![A3 : dB, L : list_dB]: ((A @ A3) => ((listsp_dB @ A @ L) => ((P @ L) => (P @ (cons_dB @ A3 @ L)))))) => (P @ X2))))))). % listsp.inducts
thf(fact_33_list_Oinject, axiom,
    ((![X21 : dB, X22 : list_dB, Y21 : dB, Y22 : list_dB]: (((cons_dB @ X21 @ X22) = (cons_dB @ Y21 @ Y22)) = (((X21 = Y21)) & ((X22 = Y22))))))). % list.inject
thf(fact_34_not__Cons__self2, axiom,
    ((![X2 : dB, Xs : list_dB]: (~ (((cons_dB @ X2 @ Xs) = Xs)))))). % not_Cons_self2
thf(fact_35_list_Omap__ident, axiom,
    ((![T : list_dB]: ((map_dB_dB @ (^[X : dB]: X) @ T) = T)))). % list.map_ident
thf(fact_36_map__tailrec__rev_Oinduct, axiom,
    ((![P : (dB > dB) > list_dB > list_dB > $o, A0 : dB > dB, A1 : list_dB, A22 : list_dB]: ((![F2 : dB > dB, X_1 : list_dB]: (P @ F2 @ nil_dB @ X_1)) => ((![F2 : dB > dB, A3 : dB, As : list_dB, Bs : list_dB]: ((P @ F2 @ As @ (cons_dB @ (F2 @ A3) @ Bs)) => (P @ F2 @ (cons_dB @ A3 @ As) @ Bs))) => (P @ A0 @ A1 @ A22)))))). % map_tailrec_rev.induct
thf(fact_37_list__nonempty__induct, axiom,
    ((![Xs : list_dB, P : list_dB > $o]: ((~ ((Xs = nil_dB))) => ((![X3 : dB]: (P @ (cons_dB @ X3 @ nil_dB))) => ((![X3 : dB, Xs3 : list_dB]: ((~ ((Xs3 = nil_dB))) => ((P @ Xs3) => (P @ (cons_dB @ X3 @ Xs3))))) => (P @ Xs))))))). % list_nonempty_induct
thf(fact_38_successively_Oinduct, axiom,
    ((![P : (dB > dB > $o) > list_dB > $o, A0 : dB > dB > $o, A1 : list_dB]: ((![P2 : dB > dB > $o]: (P @ P2 @ nil_dB)) => ((![P2 : dB > dB > $o, X3 : dB]: (P @ P2 @ (cons_dB @ X3 @ nil_dB))) => ((![P2 : dB > dB > $o, X3 : dB, Y : dB, Xs3 : list_dB]: ((P @ P2 @ (cons_dB @ Y @ Xs3)) => (P @ P2 @ (cons_dB @ X3 @ (cons_dB @ Y @ Xs3))))) => (P @ A0 @ A1))))))). % successively.induct
thf(fact_39_remdups__adj_Oinduct, axiom,
    ((![P : list_dB > $o, A0 : list_dB]: ((P @ nil_dB) => ((![X3 : dB]: (P @ (cons_dB @ X3 @ nil_dB))) => ((![X3 : dB, Y : dB, Xs3 : list_dB]: (((X3 = Y) => (P @ (cons_dB @ X3 @ Xs3))) => (((~ ((X3 = Y))) => (P @ (cons_dB @ Y @ Xs3))) => (P @ (cons_dB @ X3 @ (cons_dB @ Y @ Xs3)))))) => (P @ A0))))))). % remdups_adj.induct
thf(fact_40_sorted__wrt_Oinduct, axiom,
    ((![P : (dB > dB > $o) > list_dB > $o, A0 : dB > dB > $o, A1 : list_dB]: ((![P2 : dB > dB > $o]: (P @ P2 @ nil_dB)) => ((![P2 : dB > dB > $o, X3 : dB, Ys : list_dB]: ((P @ P2 @ Ys) => (P @ P2 @ (cons_dB @ X3 @ Ys)))) => (P @ A0 @ A1)))))). % sorted_wrt.induct
thf(fact_41_remdups__adj_Ocases, axiom,
    ((![X2 : list_dB]: ((~ ((X2 = nil_dB))) => ((![X3 : dB]: (~ ((X2 = (cons_dB @ X3 @ nil_dB))))) => (~ ((![X3 : dB, Y : dB, Xs3 : list_dB]: (~ ((X2 = (cons_dB @ X3 @ (cons_dB @ Y @ Xs3))))))))))))). % remdups_adj.cases
thf(fact_42_transpose_Ocases, axiom,
    ((![X2 : list_list_dB]: ((~ ((X2 = nil_list_dB))) => ((![Xss : list_list_dB]: (~ ((X2 = (cons_list_dB @ nil_dB @ Xss))))) => (~ ((![X3 : dB, Xs3 : list_dB, Xss : list_list_dB]: (~ ((X2 = (cons_list_dB @ (cons_dB @ X3 @ Xs3) @ Xss)))))))))))). % transpose.cases
thf(fact_43_shuffles_Oinduct, axiom,
    ((![P : list_dB > list_dB > $o, A0 : list_dB, A1 : list_dB]: ((![X_1 : list_dB]: (P @ nil_dB @ X_1)) => ((![Xs3 : list_dB]: (P @ Xs3 @ nil_dB)) => ((![X3 : dB, Xs3 : list_dB, Y : dB, Ys : list_dB]: ((P @ Xs3 @ (cons_dB @ Y @ Ys)) => ((P @ (cons_dB @ X3 @ Xs3) @ Ys) => (P @ (cons_dB @ X3 @ Xs3) @ (cons_dB @ Y @ Ys))))) => (P @ A0 @ A1))))))). % shuffles.induct
thf(fact_44_induct__list012, axiom,
    ((![P : list_dB > $o, Xs : list_dB]: ((P @ nil_dB) => ((![X3 : dB]: (P @ (cons_dB @ X3 @ nil_dB))) => ((![X3 : dB, Y : dB, Zs : list_dB]: ((P @ Zs) => ((P @ (cons_dB @ Y @ Zs)) => (P @ (cons_dB @ X3 @ (cons_dB @ Y @ Zs)))))) => (P @ Xs))))))). % induct_list012
thf(fact_45_splice_Oinduct, axiom,
    ((![P : list_dB > list_dB > $o, A0 : list_dB, A1 : list_dB]: ((![X_1 : list_dB]: (P @ nil_dB @ X_1)) => ((![X3 : dB, Xs3 : list_dB, Ys : list_dB]: ((P @ Ys @ Xs3) => (P @ (cons_dB @ X3 @ Xs3) @ Ys))) => (P @ A0 @ A1)))))). % splice.induct
thf(fact_46_list__induct2_H, axiom,
    ((![P : list_dB > list_dB > $o, Xs : list_dB, Ys2 : list_dB]: ((P @ nil_dB @ nil_dB) => ((![X3 : dB, Xs3 : list_dB]: (P @ (cons_dB @ X3 @ Xs3) @ nil_dB)) => ((![Y : dB, Ys : list_dB]: (P @ nil_dB @ (cons_dB @ Y @ Ys))) => ((![X3 : dB, Xs3 : list_dB, Y : dB, Ys : list_dB]: ((P @ Xs3 @ Ys) => (P @ (cons_dB @ X3 @ Xs3) @ (cons_dB @ Y @ Ys)))) => (P @ Xs @ Ys2)))))))). % list_induct2'
thf(fact_47_neq__Nil__conv, axiom,
    ((![Xs : list_dB]: ((~ ((Xs = nil_dB))) = (?[Y2 : dB]: (?[Ys3 : list_dB]: (Xs = (cons_dB @ Y2 @ Ys3)))))))). % neq_Nil_conv
thf(fact_48_list_Oinducts, axiom,
    ((![P : list_dB > $o, List : list_dB]: ((P @ nil_dB) => ((![X12 : dB, X23 : list_dB]: ((P @ X23) => (P @ (cons_dB @ X12 @ X23)))) => (P @ List)))))). % list.inducts
thf(fact_49_list_Oexhaust, axiom,
    ((![Y3 : list_dB]: ((~ ((Y3 = nil_dB))) => (~ ((![X212 : dB, X222 : list_dB]: (~ ((Y3 = (cons_dB @ X212 @ X222))))))))))). % list.exhaust
thf(fact_50_list_OdiscI, axiom,
    ((![List : list_dB, X21 : dB, X22 : list_dB]: ((List = (cons_dB @ X21 @ X22)) => (~ ((List = nil_dB))))))). % list.discI
thf(fact_51_list_Odistinct_I1_J, axiom,
    ((![X21 : dB, X22 : list_dB]: (~ ((nil_dB = (cons_dB @ X21 @ X22))))))). % list.distinct(1)
thf(fact_52_map__eq__Cons__conv, axiom,
    ((![F : dB > dB, Xs : list_dB, Y3 : dB, Ys2 : list_dB]: (((map_dB_dB @ F @ Xs) = (cons_dB @ Y3 @ Ys2)) = (?[Z : dB]: (?[Zs2 : list_dB]: (((Xs = (cons_dB @ Z @ Zs2))) & (((((F @ Z) = Y3)) & (((map_dB_dB @ F @ Zs2) = Ys2))))))))))). % map_eq_Cons_conv
thf(fact_53_Cons__eq__map__conv, axiom,
    ((![X2 : dB, Xs : list_dB, F : dB > dB, Ys2 : list_dB]: (((cons_dB @ X2 @ Xs) = (map_dB_dB @ F @ Ys2)) = (?[Z : dB]: (?[Zs2 : list_dB]: (((Ys2 = (cons_dB @ Z @ Zs2))) & ((((X2 = (F @ Z))) & ((Xs = (map_dB_dB @ F @ Zs2)))))))))))). % Cons_eq_map_conv
thf(fact_54_map__eq__Cons__D, axiom,
    ((![F : dB > dB, Xs : list_dB, Y3 : dB, Ys2 : list_dB]: (((map_dB_dB @ F @ Xs) = (cons_dB @ Y3 @ Ys2)) => (?[Z2 : dB, Zs : list_dB]: ((Xs = (cons_dB @ Z2 @ Zs)) & (((F @ Z2) = Y3) & ((map_dB_dB @ F @ Zs) = Ys2)))))))). % map_eq_Cons_D
thf(fact_55_Cons__eq__map__D, axiom,
    ((![X2 : dB, Xs : list_dB, F : dB > dB, Ys2 : list_dB]: (((cons_dB @ X2 @ Xs) = (map_dB_dB @ F @ Ys2)) => (?[Z2 : dB, Zs : list_dB]: ((Ys2 = (cons_dB @ Z2 @ Zs)) & ((X2 = (F @ Z2)) & (Xs = (map_dB_dB @ F @ Zs))))))))). % Cons_eq_map_D
thf(fact_56_list_Osimps_I9_J, axiom,
    ((![F : dB > dB, X21 : dB, X22 : list_dB]: ((map_dB_dB @ F @ (cons_dB @ X21 @ X22)) = (cons_dB @ (F @ X21) @ (map_dB_dB @ F @ X22)))))). % list.simps(9)
thf(fact_57_list_Osimps_I8_J, axiom,
    ((![F : dB > dB]: ((map_dB_dB @ F @ nil_dB) = nil_dB)))). % list.simps(8)
thf(fact_58_foldl__Cons, axiom,
    ((![F : dB > dB > dB, A2 : dB, X2 : dB, Xs : list_dB]: ((foldl_dB_dB @ F @ A2 @ (cons_dB @ X2 @ Xs)) = (foldl_dB_dB @ F @ (F @ A2 @ X2) @ Xs))))). % foldl_Cons
thf(fact_59_foldl__Nil, axiom,
    ((![F : dB > dB > dB, A2 : dB]: ((foldl_dB_dB @ F @ A2 @ nil_dB) = A2)))). % foldl_Nil
thf(fact_60_listsp_OCons, axiom,
    ((![A : dB > $o, A2 : dB, L3 : list_dB]: ((A @ A2) => ((listsp_dB @ A @ L3) => (listsp_dB @ A @ (cons_dB @ A2 @ L3))))))). % listsp.Cons
thf(fact_61_listspE, axiom,
    ((![A : dB > $o, X2 : dB, L3 : list_dB]: ((listsp_dB @ A @ (cons_dB @ X2 @ L3)) => (~ (((A @ X2) => (~ ((listsp_dB @ A @ L3)))))))))). % listspE
thf(fact_62_listsp__simps_I2_J, axiom,
    ((![A : dB > $o, X2 : dB, Xs : list_dB]: ((listsp_dB @ A @ (cons_dB @ X2 @ Xs)) = (((A @ X2)) & ((listsp_dB @ A @ Xs))))))). % listsp_simps(2)
thf(fact_63_listsp_ONil, axiom,
    ((![A : dB > $o]: (listsp_dB @ A @ nil_dB)))). % listsp.Nil
thf(fact_64_foldl__map, axiom,
    ((![G : dB > dB > dB, A2 : dB, F : dB > dB, Xs : list_dB]: ((foldl_dB_dB @ G @ A2 @ (map_dB_dB @ F @ Xs)) = (foldl_dB_dB @ (^[A5 : dB]: (^[X : dB]: (G @ A5 @ (F @ X)))) @ A2 @ Xs))))). % foldl_map
thf(fact_65_IT_Oinducts, axiom,
    ((![X2 : dB, P : dB > $o]: ((it @ X2) => ((![Rs2 : list_dB, N2 : nat]: ((listsp_dB @ (^[X : dB]: (((it @ X)) & ((P @ X)))) @ Rs2) => (P @ (foldl_dB_dB @ app @ (var @ N2) @ Rs2)))) => ((![R2 : dB]: ((it @ R2) => ((P @ R2) => (P @ (abs @ R2))))) => ((![R2 : dB, S2 : dB, Ss2 : list_dB]: ((it @ (foldl_dB_dB @ app @ (subst @ R2 @ S2 @ zero_zero_nat) @ Ss2)) => ((P @ (foldl_dB_dB @ app @ (subst @ R2 @ S2 @ zero_zero_nat) @ Ss2)) => ((it @ S2) => ((P @ S2) => (P @ (foldl_dB_dB @ app @ (app @ (abs @ R2) @ S2) @ Ss2))))))) => (P @ X2)))))))). % IT.inducts
thf(fact_66_IT_Ocases, axiom,
    ((![A2 : dB]: ((it @ A2) => ((![Rs2 : list_dB]: ((?[N2 : nat]: (A2 = (foldl_dB_dB @ app @ (var @ N2) @ Rs2))) => (~ ((listsp_dB @ it @ Rs2))))) => ((![R2 : dB]: ((A2 = (abs @ R2)) => (~ ((it @ R2))))) => (~ ((![R2 : dB, S2 : dB, Ss2 : list_dB]: ((A2 = (foldl_dB_dB @ app @ (app @ (abs @ R2) @ S2) @ Ss2)) => ((it @ (foldl_dB_dB @ app @ (subst @ R2 @ S2 @ zero_zero_nat) @ Ss2)) => (~ ((it @ S2)))))))))))))). % IT.cases
thf(fact_67_IT_Osimps, axiom,
    ((it = (^[A5 : dB]: (((?[Rs3 : list_dB]: (?[N3 : nat]: (((A5 = (foldl_dB_dB @ app @ (var @ N3) @ Rs3))) & ((listsp_dB @ it @ Rs3)))))) | ((((?[R3 : dB]: (((A5 = (abs @ R3))) & ((it @ R3))))) | ((?[R3 : dB]: (?[S3 : dB]: (?[Ss3 : list_dB]: (((A5 = (foldl_dB_dB @ app @ (app @ (abs @ R3) @ S3) @ Ss3))) & ((((it @ (foldl_dB_dB @ app @ (subst @ R3 @ S3 @ zero_zero_nat) @ Ss3))) & ((it @ S3)))))))))))))))). % IT.simps
thf(fact_68_n__lists__Nil, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((n_lists_dB @ N @ nil_dB) = (cons_list_dB @ nil_dB @ nil_list_dB))) & ((~ ((N = zero_zero_nat))) => ((n_lists_dB @ N @ nil_dB) = nil_list_dB)))))). % n_lists_Nil
thf(fact_69_map__rec, axiom,
    ((map_dB_dB = (^[F3 : dB > dB]: (rec_list_list_dB_dB @ nil_dB @ (^[X : dB]: (^[Uu : list_dB]: (cons_dB @ (F3 @ X))))))))). % map_rec
thf(fact_70_Beta, axiom,
    ((![R : dB, S : dB, Ss : list_dB]: ((it @ (foldl_dB_dB @ app @ (subst @ R @ S @ zero_zero_nat) @ Ss)) => ((it @ S) => (it @ (foldl_dB_dB @ app @ (app @ (abs @ R) @ S) @ Ss))))))). % Beta
thf(fact_71_dB_Oinject_I3_J, axiom,
    ((![X32 : dB, Y32 : dB]: (((abs @ X32) = (abs @ Y32)) = (X32 = Y32))))). % dB.inject(3)
thf(fact_72_Abs__apps__eq__Abs__apps__conv, axiom,
    ((![R : dB, Rs : list_dB, S : dB, Ss : list_dB]: (((foldl_dB_dB @ app @ (abs @ R) @ Rs) = (foldl_dB_dB @ app @ (abs @ S) @ Ss)) = (((R = S)) & ((Rs = Ss))))))). % Abs_apps_eq_Abs_apps_conv
thf(fact_73_apps__eq__Abs__conv, axiom,
    ((![S : dB, Ss : list_dB, R : dB]: (((foldl_dB_dB @ app @ S @ Ss) = (abs @ R)) = (((S = (abs @ R))) & ((Ss = nil_dB))))))). % apps_eq_Abs_conv
thf(fact_74_Abs__eq__apps__conv, axiom,
    ((![R : dB, S : dB, Ss : list_dB]: (((abs @ R) = (foldl_dB_dB @ app @ S @ Ss)) = ((((abs @ R) = S)) & ((Ss = nil_dB))))))). % Abs_eq_apps_conv
thf(fact_75_dB_Odistinct_I5_J, axiom,
    ((![X21 : dB, X22 : dB, X32 : dB]: (~ (((app @ X21 @ X22) = (abs @ X32))))))). % dB.distinct(5)
thf(fact_76_dB_Odistinct_I3_J, axiom,
    ((![X1 : nat, X32 : dB]: (~ (((var @ X1) = (abs @ X32))))))). % dB.distinct(3)
thf(fact_77_Lambda, axiom,
    ((![R : dB]: ((it @ R) => (it @ (abs @ R)))))). % Lambda
thf(fact_78_dB_Oexhaust, axiom,
    ((![Y3 : dB]: ((![X12 : nat]: (~ ((Y3 = (var @ X12))))) => ((![X212 : dB, X222 : dB]: (~ ((Y3 = (app @ X212 @ X222))))) => (~ ((![X33 : dB]: (~ ((Y3 = (abs @ X33)))))))))))). % dB.exhaust
thf(fact_79_dB_Oinduct, axiom,
    ((![P : dB > $o, DB : dB]: ((![X3 : nat]: (P @ (var @ X3))) => ((![X1a : dB, X23 : dB]: ((P @ X1a) => ((P @ X23) => (P @ (app @ X1a @ X23))))) => ((![X3 : dB]: ((P @ X3) => (P @ (abs @ X3)))) => (P @ DB))))))). % dB.induct
thf(fact_80_zero__reorient, axiom,
    ((![X2 : nat]: ((zero_zero_nat = X2) = (X2 = zero_zero_nat))))). % zero_reorient
thf(fact_81_Var__apps__neq__Abs__apps, axiom,
    ((![N : nat, Ts : list_dB, R : dB, Ss : list_dB]: (~ (((foldl_dB_dB @ app @ (var @ N) @ Ts) = (foldl_dB_dB @ app @ (abs @ R) @ Ss))))))). % Var_apps_neq_Abs_apps
thf(fact_82_Abs__App__neq__Var__apps, axiom,
    ((![S : dB, T : dB, N : nat, Ss : list_dB]: (~ (((app @ (abs @ S) @ T) = (foldl_dB_dB @ app @ (var @ N) @ Ss))))))). % Abs_App_neq_Var_apps
thf(fact_83_ex__head__tail, axiom,
    ((![T : dB]: (?[Ts2 : list_dB, H : dB]: ((T = (foldl_dB_dB @ app @ H @ Ts2)) & ((?[N2 : nat]: (H = (var @ N2))) | (?[U2 : dB]: (H = (abs @ U2))))))))). % ex_head_tail
thf(fact_84_n__lists_Osimps_I1_J, axiom,
    ((![Xs : list_dB]: ((n_lists_dB @ zero_zero_nat @ Xs) = (cons_list_dB @ nil_dB @ nil_list_dB))))). % n_lists.simps(1)
thf(fact_85_product__lists_Osimps_I1_J, axiom,
    (((product_lists_dB @ nil_list_dB) = (cons_list_dB @ nil_dB @ nil_list_dB)))). % product_lists.simps(1)
thf(fact_86_subseqs_Osimps_I1_J, axiom,
    (((subseqs_dB @ nil_dB) = (cons_list_dB @ nil_dB @ nil_list_dB)))). % subseqs.simps(1)
thf(fact_87_dB_Osize__gen_I1_J, axiom,
    ((![X1 : nat]: ((size_dB @ (var @ X1)) = zero_zero_nat)))). % dB.size_gen(1)
thf(fact_88_insert__Nil, axiom,
    ((![X2 : dB]: ((insert_dB @ X2 @ nil_dB) = (cons_dB @ X2 @ nil_dB))))). % insert_Nil
thf(fact_89_App__eq__foldl__conv, axiom,
    ((![R : dB, S : dB, T : dB, Ts : list_dB]: (((app @ R @ S) = (foldl_dB_dB @ app @ T @ Ts)) = (((((Ts = nil_dB)) => (((app @ R @ S) = T)))) & ((((~ ((Ts = nil_dB)))) => ((?[Ss3 : list_dB]: (((Ts = (append_dB @ Ss3 @ (cons_dB @ S @ nil_dB)))) & ((R = (foldl_dB_dB @ app @ T @ Ss3))))))))))))). % App_eq_foldl_conv
thf(fact_90_app__last, axiom,
    ((![T : dB, Ts : list_dB, U : dB]: ((app @ (foldl_dB_dB @ app @ T @ Ts) @ U) = (foldl_dB_dB @ app @ T @ (append_dB @ Ts @ (cons_dB @ U @ nil_dB))))))). % app_last
thf(fact_91_map__eq__map__tailrec, axiom,
    ((map_dB_dB = map_tailrec_dB_dB))). % map_eq_map_tailrec
thf(fact_92_append_Oassoc, axiom,
    ((![A2 : list_dB, B3 : list_dB, C : list_dB]: ((append_dB @ (append_dB @ A2 @ B3) @ C) = (append_dB @ A2 @ (append_dB @ B3 @ C)))))). % append.assoc
thf(fact_93_append__assoc, axiom,
    ((![Xs : list_dB, Ys2 : list_dB, Zs3 : list_dB]: ((append_dB @ (append_dB @ Xs @ Ys2) @ Zs3) = (append_dB @ Xs @ (append_dB @ Ys2 @ Zs3)))))). % append_assoc
thf(fact_94_append__same__eq, axiom,
    ((![Ys2 : list_dB, Xs : list_dB, Zs3 : list_dB]: (((append_dB @ Ys2 @ Xs) = (append_dB @ Zs3 @ Xs)) = (Ys2 = Zs3))))). % append_same_eq
thf(fact_95_same__append__eq, axiom,
    ((![Xs : list_dB, Ys2 : list_dB, Zs3 : list_dB]: (((append_dB @ Xs @ Ys2) = (append_dB @ Xs @ Zs3)) = (Ys2 = Zs3))))). % same_append_eq
thf(fact_96_append__Nil2, axiom,
    ((![Xs : list_dB]: ((append_dB @ Xs @ nil_dB) = Xs)))). % append_Nil2
thf(fact_97_append__self__conv, axiom,
    ((![Xs : list_dB, Ys2 : list_dB]: (((append_dB @ Xs @ Ys2) = Xs) = (Ys2 = nil_dB))))). % append_self_conv
thf(fact_98_self__append__conv, axiom,
    ((![Xs : list_dB, Ys2 : list_dB]: ((Xs = (append_dB @ Xs @ Ys2)) = (Ys2 = nil_dB))))). % self_append_conv
thf(fact_99_append__self__conv2, axiom,
    ((![Xs : list_dB, Ys2 : list_dB]: (((append_dB @ Xs @ Ys2) = Ys2) = (Xs = nil_dB))))). % append_self_conv2
thf(fact_100_self__append__conv2, axiom,
    ((![Ys2 : list_dB, Xs : list_dB]: ((Ys2 = (append_dB @ Xs @ Ys2)) = (Xs = nil_dB))))). % self_append_conv2
thf(fact_101_Nil__is__append__conv, axiom,
    ((![Xs : list_dB, Ys2 : list_dB]: ((nil_dB = (append_dB @ Xs @ Ys2)) = (((Xs = nil_dB)) & ((Ys2 = nil_dB))))))). % Nil_is_append_conv
thf(fact_102_append__is__Nil__conv, axiom,
    ((![Xs : list_dB, Ys2 : list_dB]: (((append_dB @ Xs @ Ys2) = nil_dB) = (((Xs = nil_dB)) & ((Ys2 = nil_dB))))))). % append_is_Nil_conv
thf(fact_103_append_Oright__neutral, axiom,
    ((![A2 : list_dB]: ((append_dB @ A2 @ nil_dB) = A2)))). % append.right_neutral
thf(fact_104_map__append, axiom,
    ((![F : dB > dB, Xs : list_dB, Ys2 : list_dB]: ((map_dB_dB @ F @ (append_dB @ Xs @ Ys2)) = (append_dB @ (map_dB_dB @ F @ Xs) @ (map_dB_dB @ F @ Ys2)))))). % map_append
thf(fact_105_foldl__append, axiom,
    ((![F : dB > dB > dB, A2 : dB, Xs : list_dB, Ys2 : list_dB]: ((foldl_dB_dB @ F @ A2 @ (append_dB @ Xs @ Ys2)) = (foldl_dB_dB @ F @ (foldl_dB_dB @ F @ A2 @ Xs) @ Ys2))))). % foldl_append
thf(fact_106_append__in__listsp__conv, axiom,
    ((![A : dB > $o, Xs : list_dB, Ys2 : list_dB]: ((listsp_dB @ A @ (append_dB @ Xs @ Ys2)) = (((listsp_dB @ A @ Xs)) & ((listsp_dB @ A @ Ys2))))))). % append_in_listsp_conv
thf(fact_107_append1__eq__conv, axiom,
    ((![Xs : list_dB, X2 : dB, Ys2 : list_dB, Y3 : dB]: (((append_dB @ Xs @ (cons_dB @ X2 @ nil_dB)) = (append_dB @ Ys2 @ (cons_dB @ Y3 @ nil_dB))) = (((Xs = Ys2)) & ((X2 = Y3))))))). % append1_eq_conv
thf(fact_108_rev__induct, axiom,
    ((![P : list_dB > $o, Xs : list_dB]: ((P @ nil_dB) => ((![X3 : dB, Xs3 : list_dB]: ((P @ Xs3) => (P @ (append_dB @ Xs3 @ (cons_dB @ X3 @ nil_dB))))) => (P @ Xs)))))). % rev_induct
thf(fact_109_rev__exhaust, axiom,
    ((![Xs : list_dB]: ((~ ((Xs = nil_dB))) => (~ ((![Ys : list_dB, Y : dB]: (~ ((Xs = (append_dB @ Ys @ (cons_dB @ Y @ nil_dB)))))))))))). % rev_exhaust
thf(fact_110_Cons__eq__append__conv, axiom,
    ((![X2 : dB, Xs : list_dB, Ys2 : list_dB, Zs3 : list_dB]: (((cons_dB @ X2 @ Xs) = (append_dB @ Ys2 @ Zs3)) = (((((Ys2 = nil_dB)) & (((cons_dB @ X2 @ Xs) = Zs3)))) | ((?[Ys4 : list_dB]: ((((cons_dB @ X2 @ Ys4) = Ys2)) & ((Xs = (append_dB @ Ys4 @ Zs3))))))))))). % Cons_eq_append_conv
thf(fact_111_append__eq__Cons__conv, axiom,
    ((![Ys2 : list_dB, Zs3 : list_dB, X2 : dB, Xs : list_dB]: (((append_dB @ Ys2 @ Zs3) = (cons_dB @ X2 @ Xs)) = (((((Ys2 = nil_dB)) & ((Zs3 = (cons_dB @ X2 @ Xs))))) | ((?[Ys4 : list_dB]: (((Ys2 = (cons_dB @ X2 @ Ys4))) & (((append_dB @ Ys4 @ Zs3) = Xs)))))))))). % append_eq_Cons_conv
thf(fact_112_rev__nonempty__induct, axiom,
    ((![Xs : list_dB, P : list_dB > $o]: ((~ ((Xs = nil_dB))) => ((![X3 : dB]: (P @ (cons_dB @ X3 @ nil_dB))) => ((![X3 : dB, Xs3 : list_dB]: ((~ ((Xs3 = nil_dB))) => ((P @ Xs3) => (P @ (append_dB @ Xs3 @ (cons_dB @ X3 @ nil_dB)))))) => (P @ Xs))))))). % rev_nonempty_induct
thf(fact_113_append__Cons, axiom,
    ((![X2 : dB, Xs : list_dB, Ys2 : list_dB]: ((append_dB @ (cons_dB @ X2 @ Xs) @ Ys2) = (cons_dB @ X2 @ (append_dB @ Xs @ Ys2)))))). % append_Cons
thf(fact_114_Cons__eq__appendI, axiom,
    ((![X2 : dB, Xs1 : list_dB, Ys2 : list_dB, Xs : list_dB, Zs3 : list_dB]: (((cons_dB @ X2 @ Xs1) = Ys2) => ((Xs = (append_dB @ Xs1 @ Zs3)) => ((cons_dB @ X2 @ Xs) = (append_dB @ Ys2 @ Zs3))))))). % Cons_eq_appendI
thf(fact_115_append_Oleft__neutral, axiom,
    ((![A2 : list_dB]: ((append_dB @ nil_dB @ A2) = A2)))). % append.left_neutral
thf(fact_116_append__Nil, axiom,
    ((![Ys2 : list_dB]: ((append_dB @ nil_dB @ Ys2) = Ys2)))). % append_Nil
thf(fact_117_eq__Nil__appendI, axiom,
    ((![Xs : list_dB, Ys2 : list_dB]: ((Xs = Ys2) => (Xs = (append_dB @ nil_dB @ Ys2)))))). % eq_Nil_appendI
thf(fact_118_map__eq__append__conv, axiom,
    ((![F : dB > dB, Xs : list_dB, Ys2 : list_dB, Zs3 : list_dB]: (((map_dB_dB @ F @ Xs) = (append_dB @ Ys2 @ Zs3)) = (?[Us : list_dB]: (?[Vs : list_dB]: (((Xs = (append_dB @ Us @ Vs))) & ((((Ys2 = (map_dB_dB @ F @ Us))) & ((Zs3 = (map_dB_dB @ F @ Vs)))))))))))). % map_eq_append_conv
thf(fact_119_append__eq__map__conv, axiom,
    ((![Ys2 : list_dB, Zs3 : list_dB, F : dB > dB, Xs : list_dB]: (((append_dB @ Ys2 @ Zs3) = (map_dB_dB @ F @ Xs)) = (?[Us : list_dB]: (?[Vs : list_dB]: (((Xs = (append_dB @ Us @ Vs))) & ((((Ys2 = (map_dB_dB @ F @ Us))) & ((Zs3 = (map_dB_dB @ F @ Vs)))))))))))). % append_eq_map_conv
thf(fact_120_append__eq__appendI, axiom,
    ((![Xs : list_dB, Xs1 : list_dB, Zs3 : list_dB, Ys2 : list_dB, Us2 : list_dB]: (((append_dB @ Xs @ Xs1) = Zs3) => ((Ys2 = (append_dB @ Xs1 @ Us2)) => ((append_dB @ Xs @ Ys2) = (append_dB @ Zs3 @ Us2))))))). % append_eq_appendI
thf(fact_121_append__eq__append__conv2, axiom,
    ((![Xs : list_dB, Ys2 : list_dB, Zs3 : list_dB, Ts : list_dB]: (((append_dB @ Xs @ Ys2) = (append_dB @ Zs3 @ Ts)) = (?[Us : list_dB]: (((((Xs = (append_dB @ Zs3 @ Us))) & (((append_dB @ Us @ Ys2) = Ts)))) | (((((append_dB @ Xs @ Us) = Zs3)) & ((Ys2 = (append_dB @ Us @ Ts))))))))))). % append_eq_append_conv2
thf(fact_122_rev__exhaust2, axiom,
    ((![Xs : list_dB]: ((~ ((Xs = nil_dB))) => (~ ((![Ys : list_dB, Y : dB]: (~ ((Xs = (append_dB @ Ys @ (cons_dB @ Y @ nil_dB)))))))))))). % rev_exhaust2
thf(fact_123_Succ__def, axiom,
    ((bNF_Greatest_Succ_dB = (^[Kl : set_list_dB]: (^[Kl2 : list_dB]: (collect_dB @ (^[K2 : dB]: (member_list_dB @ (append_dB @ Kl2 @ (cons_dB @ K2 @ nil_dB)) @ Kl)))))))). % Succ_def
thf(fact_124_SuccI, axiom,
    ((![Kl3 : list_dB, K : dB, Kl4 : set_list_dB]: ((member_list_dB @ (append_dB @ Kl3 @ (cons_dB @ K @ nil_dB)) @ Kl4) => (member_dB @ K @ (bNF_Greatest_Succ_dB @ Kl4 @ Kl3)))))). % SuccI
thf(fact_125_subseqs_Osimps_I2_J, axiom,
    ((![X2 : dB, Xs : list_dB]: ((subseqs_dB @ (cons_dB @ X2 @ Xs)) = (append_list_dB @ (map_list_dB_list_dB @ (cons_dB @ X2) @ (subseqs_dB @ Xs)) @ (subseqs_dB @ Xs)))))). % subseqs.simps(2)
thf(fact_126_SuccD, axiom,
    ((![K : dB, Kl4 : set_list_dB, Kl3 : list_dB]: ((member_dB @ K @ (bNF_Greatest_Succ_dB @ Kl4 @ Kl3)) => (member_list_dB @ (append_dB @ Kl3 @ (cons_dB @ K @ nil_dB)) @ Kl4))))). % SuccD
thf(fact_127_empty__Shift, axiom,
    ((![Kl4 : set_list_dB, K : dB]: ((member_list_dB @ nil_dB @ Kl4) => ((member_dB @ K @ (bNF_Greatest_Succ_dB @ Kl4 @ nil_dB)) => (member_list_dB @ nil_dB @ (bNF_Gr1110975684ift_dB @ Kl4 @ K))))))). % empty_Shift
thf(fact_128_Succ__Shift, axiom,
    ((![Kl4 : set_list_dB, K : dB, Kl3 : list_dB]: ((bNF_Greatest_Succ_dB @ (bNF_Gr1110975684ift_dB @ Kl4 @ K) @ Kl3) = (bNF_Greatest_Succ_dB @ Kl4 @ (cons_dB @ K @ Kl3)))))). % Succ_Shift
thf(fact_129_ShiftD, axiom,
    ((![Kl3 : list_dB, Kl4 : set_list_dB, K : dB]: ((member_list_dB @ Kl3 @ (bNF_Gr1110975684ift_dB @ Kl4 @ K)) => (member_list_dB @ (cons_dB @ K @ Kl3) @ Kl4))))). % ShiftD
thf(fact_130_Shift__def, axiom,
    ((bNF_Gr1110975684ift_dB = (^[Kl : set_list_dB]: (^[K2 : dB]: (collect_list_dB @ (^[Kl2 : list_dB]: (member_list_dB @ (cons_dB @ K2 @ Kl2) @ Kl)))))))). % Shift_def
thf(fact_131_concat__eq__append__conv, axiom,
    ((![Xss2 : list_list_dB, Ys2 : list_dB, Zs3 : list_dB]: (((concat_dB @ Xss2) = (append_dB @ Ys2 @ Zs3)) = (((((Xss2 = nil_list_dB)) => ((((Ys2 = nil_dB)) & ((Zs3 = nil_dB)))))) & ((((~ ((Xss2 = nil_list_dB)))) => ((?[Xss1 : list_list_dB]: (?[Xs2 : list_dB]: (?[Xs4 : list_dB]: (?[Xss22 : list_list_dB]: (((Xss2 = (append_list_dB @ Xss1 @ (cons_list_dB @ (append_dB @ Xs2 @ Xs4) @ Xss22)))) & ((((Ys2 = (append_dB @ (concat_dB @ Xss1) @ Xs2))) & ((Zs3 = (append_dB @ Xs4 @ (concat_dB @ Xss22))))))))))))))))))). % concat_eq_append_conv
thf(fact_132_bind__simps_I2_J, axiom,
    ((![X2 : dB, Xs : list_dB, F : dB > list_dB]: ((bind_dB_dB @ (cons_dB @ X2 @ Xs) @ F) = (append_dB @ (F @ X2) @ (bind_dB_dB @ Xs @ F)))))). % bind_simps(2)
thf(fact_133_bind__simps_I1_J, axiom,
    ((![F : dB > list_dB]: ((bind_dB_dB @ nil_dB @ F) = nil_dB)))). % bind_simps(1)

% Conjectures (1)
thf(conj_0, conjecture,
    ((it @ (subst @ (foldl_dB_dB @ app @ (var @ n) @ rs) @ u @ i)))).
