% 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_82__5210024_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:37:26.628

% 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__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__Set__Oset_It__Nat__Onat_J, type,
    set_nat : $tType).
thf(ty_n_t__Lambda__OdB, type,
    dB : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).

% Explicit typings (36)
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_If_001t__List__Olist_It__Lambda__OdB_J, type,
    if_list_dB : $o > list_dB > list_dB > list_dB).
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_Osubst, type,
    subst : dB > dB > nat > dB).
thf(sy_c_Lambda_Osubstn, type,
    substn : 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_Obutlast_001t__Lambda__OdB, type,
    butlast_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_Olast_001t__Lambda__OdB, type,
    last_dB : list_dB > 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_Oset_001t__Lambda__OdB, type,
    set_dB2 : list_dB > set_dB).
thf(sy_c_List_Olist__ex1_001t__Lambda__OdB, type,
    list_ex1_dB : (dB > $o) > list_dB > $o).
thf(sy_c_List_Olistsp_001t__Lambda__OdB, type,
    listsp_dB : (dB > $o) > list_dB > $o).
thf(sy_c_List_Omaps_001t__Lambda__OdB_001t__Lambda__OdB, type,
    maps_dB_dB : (dB > list_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_Onths_001t__Lambda__OdB, type,
    nths_dB : list_dB > set_nat > 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_Orotate1_001t__Lambda__OdB, type,
    rotate1_dB : list_dB > list_dB).
thf(sy_c_List_Osubseqs_001t__Lambda__OdB, type,
    subseqs_dB : list_dB > list_list_dB).
thf(sy_c_member_001t__Lambda__OdB, type,
    member_dB : dB > set_dB > $o).
thf(sy_c_member_001t__Nat__Onat, type,
    member_nat : nat > set_nat > $o).
thf(sy_v_i, type,
    i : nat).
thf(sy_v_r, type,
    r : dB).
thf(sy_v_s, type,
    s : dB).
thf(sy_v_ss, type,
    ss : list_dB).

% Relevant facts (138)
thf(fact_0_Var__IT, axiom,
    ((![N : nat]: (it @ (var @ N))))). % Var_IT
thf(fact_1_subst__Var__IT, axiom,
    ((![R : dB, I : nat, J : nat]: ((it @ R) => (it @ (subst @ R @ (var @ I) @ J)))))). % subst_Var_IT
thf(fact_2_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_3_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_4_append1__eq__conv, axiom,
    ((![Xs : list_dB, X : dB, Ys : list_dB, Y : dB]: (((append_dB @ Xs @ (cons_dB @ X @ nil_dB)) = (append_dB @ Ys @ (cons_dB @ Y @ nil_dB))) = (((Xs = Ys)) & ((X = Y))))))). % append1_eq_conv
thf(fact_5_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_6_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)))) => ((?[Ss2 : list_dB]: (((Ts = (append_dB @ Ss2 @ (cons_dB @ S @ nil_dB)))) & ((R = (foldl_dB_dB @ app @ T @ Ss2))))))))))))). % App_eq_foldl_conv
thf(fact_7_subst__eq, axiom,
    ((![K : nat, U : dB]: ((subst @ (var @ K) @ U @ K) = U)))). % subst_eq
thf(fact_8_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_9_foldl__append, axiom,
    ((![F : dB > dB > dB, A : dB, Xs : list_dB, Ys : list_dB]: ((foldl_dB_dB @ F @ A @ (append_dB @ Xs @ Ys)) = (foldl_dB_dB @ F @ (foldl_dB_dB @ F @ A @ Xs) @ Ys))))). % foldl_append
thf(fact_10_append__Nil2, axiom,
    ((![Xs : list_dB]: ((append_dB @ Xs @ nil_dB) = Xs)))). % append_Nil2
thf(fact_11_append__self__conv, axiom,
    ((![Xs : list_dB, Ys : list_dB]: (((append_dB @ Xs @ Ys) = Xs) = (Ys = nil_dB))))). % append_self_conv
thf(fact_12_self__append__conv, axiom,
    ((![Xs : list_dB, Ys : list_dB]: ((Xs = (append_dB @ Xs @ Ys)) = (Ys = nil_dB))))). % self_append_conv
thf(fact_13_append__self__conv2, axiom,
    ((![Xs : list_dB, Ys : list_dB]: (((append_dB @ Xs @ Ys) = Ys) = (Xs = nil_dB))))). % append_self_conv2
thf(fact_14_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_15_same__append__eq, axiom,
    ((![Xs : list_dB, Ys : list_dB, Zs : list_dB]: (((append_dB @ Xs @ Ys) = (append_dB @ Xs @ Zs)) = (Ys = Zs))))). % same_append_eq
thf(fact_16_append__same__eq, axiom,
    ((![Ys : list_dB, Xs : list_dB, Zs : list_dB]: (((append_dB @ Ys @ Xs) = (append_dB @ Zs @ Xs)) = (Ys = Zs))))). % append_same_eq
thf(fact_17_append__assoc, axiom,
    ((![Xs : list_dB, Ys : list_dB, Zs : list_dB]: ((append_dB @ (append_dB @ Xs @ Ys) @ Zs) = (append_dB @ Xs @ (append_dB @ Ys @ Zs)))))). % append_assoc
thf(fact_18_append_Oassoc, axiom,
    ((![A : list_dB, B : list_dB, C : list_dB]: ((append_dB @ (append_dB @ A @ B) @ C) = (append_dB @ A @ (append_dB @ B @ C)))))). % append.assoc
thf(fact_19_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_20_dB_Oinject_I1_J, axiom,
    ((![X1 : nat, Y1 : nat]: (((var @ X1) = (var @ Y1)) = (X1 = Y1))))). % dB.inject(1)
thf(fact_21_append_Oright__neutral, axiom,
    ((![A : list_dB]: ((append_dB @ A @ nil_dB) = A)))). % append.right_neutral
thf(fact_22_append__is__Nil__conv, axiom,
    ((![Xs : list_dB, Ys : list_dB]: (((append_dB @ Xs @ Ys) = nil_dB) = (((Xs = nil_dB)) & ((Ys = nil_dB))))))). % append_is_Nil_conv
thf(fact_23_Nil__is__append__conv, axiom,
    ((![Xs : list_dB, Ys : list_dB]: ((nil_dB = (append_dB @ Xs @ Ys)) = (((Xs = nil_dB)) & ((Ys = nil_dB))))))). % Nil_is_append_conv
thf(fact_24_self__append__conv2, axiom,
    ((![Ys : list_dB, Xs : list_dB]: ((Ys = (append_dB @ Xs @ Ys)) = (Xs = nil_dB))))). % self_append_conv2
thf(fact_25_not__Cons__self2, axiom,
    ((![X : dB, Xs : list_dB]: (~ (((cons_dB @ X @ Xs) = Xs)))))). % not_Cons_self2
thf(fact_26_append__eq__append__conv2, axiom,
    ((![Xs : list_dB, Ys : list_dB, Zs : list_dB, Ts : list_dB]: (((append_dB @ Xs @ Ys) = (append_dB @ Zs @ Ts)) = (?[Us : list_dB]: (((((Xs = (append_dB @ Zs @ Us))) & (((append_dB @ Us @ Ys) = Ts)))) | (((((append_dB @ Xs @ Us) = Zs)) & ((Ys = (append_dB @ Us @ Ts))))))))))). % append_eq_append_conv2
thf(fact_27_append__eq__appendI, axiom,
    ((![Xs : list_dB, Xs1 : list_dB, Zs : list_dB, Ys : list_dB, Us2 : list_dB]: (((append_dB @ Xs @ Xs1) = Zs) => ((Ys = (append_dB @ Xs1 @ Us2)) => ((append_dB @ Xs @ Ys) = (append_dB @ Zs @ Us2))))))). % append_eq_appendI
thf(fact_28_map__tailrec__rev_Oinduct, axiom,
    ((![P : (dB > dB) > list_dB > list_dB > $o, A0 : dB > dB, A1 : list_dB, A2 : 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 @ A2)))))). % map_tailrec_rev.induct
thf(fact_29_list__nonempty__induct, axiom,
    ((![Xs : list_dB, P : list_dB > $o]: ((~ ((Xs = nil_dB))) => ((![X2 : dB]: (P @ (cons_dB @ X2 @ nil_dB))) => ((![X2 : dB, Xs2 : list_dB]: ((~ ((Xs2 = nil_dB))) => ((P @ Xs2) => (P @ (cons_dB @ X2 @ Xs2))))) => (P @ Xs))))))). % list_nonempty_induct
thf(fact_30_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, X2 : dB]: (P @ P2 @ (cons_dB @ X2 @ nil_dB))) => ((![P2 : dB > dB > $o, X2 : dB, Y2 : dB, Xs2 : list_dB]: ((P @ P2 @ (cons_dB @ Y2 @ Xs2)) => (P @ P2 @ (cons_dB @ X2 @ (cons_dB @ Y2 @ Xs2))))) => (P @ A0 @ A1))))))). % successively.induct
thf(fact_31_remdups__adj_Oinduct, axiom,
    ((![P : list_dB > $o, A0 : list_dB]: ((P @ nil_dB) => ((![X2 : dB]: (P @ (cons_dB @ X2 @ nil_dB))) => ((![X2 : dB, Y2 : dB, Xs2 : list_dB]: (((X2 = Y2) => (P @ (cons_dB @ X2 @ Xs2))) => (((~ ((X2 = Y2))) => (P @ (cons_dB @ Y2 @ Xs2))) => (P @ (cons_dB @ X2 @ (cons_dB @ Y2 @ Xs2)))))) => (P @ A0))))))). % remdups_adj.induct
thf(fact_32_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, X2 : dB, Ys2 : list_dB]: ((P @ P2 @ Ys2) => (P @ P2 @ (cons_dB @ X2 @ Ys2)))) => (P @ A0 @ A1)))))). % sorted_wrt.induct
thf(fact_33_remdups__adj_Ocases, axiom,
    ((![X : list_dB]: ((~ ((X = nil_dB))) => ((![X2 : dB]: (~ ((X = (cons_dB @ X2 @ nil_dB))))) => (~ ((![X2 : dB, Y2 : dB, Xs2 : list_dB]: (~ ((X = (cons_dB @ X2 @ (cons_dB @ Y2 @ Xs2))))))))))))). % remdups_adj.cases
thf(fact_34_transpose_Ocases, axiom,
    ((![X : list_list_dB]: ((~ ((X = nil_list_dB))) => ((![Xss : list_list_dB]: (~ ((X = (cons_list_dB @ nil_dB @ Xss))))) => (~ ((![X2 : dB, Xs2 : list_dB, Xss : list_list_dB]: (~ ((X = (cons_list_dB @ (cons_dB @ X2 @ Xs2) @ Xss)))))))))))). % transpose.cases
thf(fact_35_shuffles_Oinduct, axiom,
    ((![P : list_dB > list_dB > $o, A0 : list_dB, A1 : list_dB]: ((![X_1 : list_dB]: (P @ nil_dB @ X_1)) => ((![Xs2 : list_dB]: (P @ Xs2 @ nil_dB)) => ((![X2 : dB, Xs2 : list_dB, Y2 : dB, Ys2 : list_dB]: ((P @ Xs2 @ (cons_dB @ Y2 @ Ys2)) => ((P @ (cons_dB @ X2 @ Xs2) @ Ys2) => (P @ (cons_dB @ X2 @ Xs2) @ (cons_dB @ Y2 @ Ys2))))) => (P @ A0 @ A1))))))). % shuffles.induct
thf(fact_36_induct__list012, axiom,
    ((![P : list_dB > $o, Xs : list_dB]: ((P @ nil_dB) => ((![X2 : dB]: (P @ (cons_dB @ X2 @ nil_dB))) => ((![X2 : dB, Y2 : dB, Zs2 : list_dB]: ((P @ Zs2) => ((P @ (cons_dB @ Y2 @ Zs2)) => (P @ (cons_dB @ X2 @ (cons_dB @ Y2 @ Zs2)))))) => (P @ Xs))))))). % induct_list012
thf(fact_37_splice_Oinduct, axiom,
    ((![P : list_dB > list_dB > $o, A0 : list_dB, A1 : list_dB]: ((![X_1 : list_dB]: (P @ nil_dB @ X_1)) => ((![X2 : dB, Xs2 : list_dB, Ys2 : list_dB]: ((P @ Ys2 @ Xs2) => (P @ (cons_dB @ X2 @ Xs2) @ Ys2))) => (P @ A0 @ A1)))))). % splice.induct
thf(fact_38_list__induct2_H, axiom,
    ((![P : list_dB > list_dB > $o, Xs : list_dB, Ys : list_dB]: ((P @ nil_dB @ nil_dB) => ((![X2 : dB, Xs2 : list_dB]: (P @ (cons_dB @ X2 @ Xs2) @ nil_dB)) => ((![Y2 : dB, Ys2 : list_dB]: (P @ nil_dB @ (cons_dB @ Y2 @ Ys2))) => ((![X2 : dB, Xs2 : list_dB, Y2 : dB, Ys2 : list_dB]: ((P @ Xs2 @ Ys2) => (P @ (cons_dB @ X2 @ Xs2) @ (cons_dB @ Y2 @ Ys2)))) => (P @ Xs @ Ys)))))))). % list_induct2'
thf(fact_39_neq__Nil__conv, axiom,
    ((![Xs : list_dB]: ((~ ((Xs = nil_dB))) = (?[Y3 : dB]: (?[Ys3 : list_dB]: (Xs = (cons_dB @ Y3 @ Ys3)))))))). % neq_Nil_conv
thf(fact_40_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_41_list_Oexhaust, axiom,
    ((![Y : list_dB]: ((~ ((Y = nil_dB))) => (~ ((![X212 : dB, X222 : list_dB]: (~ ((Y = (cons_dB @ X212 @ X222))))))))))). % list.exhaust
thf(fact_42_list_OdiscI, axiom,
    ((![List : list_dB, X21 : dB, X22 : list_dB]: ((List = (cons_dB @ X21 @ X22)) => (~ ((List = nil_dB))))))). % list.discI
thf(fact_43_list_Odistinct_I1_J, axiom,
    ((![X21 : dB, X22 : list_dB]: (~ ((nil_dB = (cons_dB @ X21 @ X22))))))). % list.distinct(1)
thf(fact_44_append__Cons, axiom,
    ((![X : dB, Xs : list_dB, Ys : list_dB]: ((append_dB @ (cons_dB @ X @ Xs) @ Ys) = (cons_dB @ X @ (append_dB @ Xs @ Ys)))))). % append_Cons
thf(fact_45_Cons__eq__appendI, axiom,
    ((![X : dB, Xs1 : list_dB, Ys : list_dB, Xs : list_dB, Zs : list_dB]: (((cons_dB @ X @ Xs1) = Ys) => ((Xs = (append_dB @ Xs1 @ Zs)) => ((cons_dB @ X @ Xs) = (append_dB @ Ys @ Zs))))))). % Cons_eq_appendI
thf(fact_46_append_Oleft__neutral, axiom,
    ((![A : list_dB]: ((append_dB @ nil_dB @ A) = A)))). % append.left_neutral
thf(fact_47_append__Nil, axiom,
    ((![Ys : list_dB]: ((append_dB @ nil_dB @ Ys) = Ys)))). % append_Nil
thf(fact_48_eq__Nil__appendI, axiom,
    ((![Xs : list_dB, Ys : list_dB]: ((Xs = Ys) => (Xs = (append_dB @ nil_dB @ Ys)))))). % eq_Nil_appendI
thf(fact_49_foldl__Cons, axiom,
    ((![F : dB > dB > dB, A : dB, X : dB, Xs : list_dB]: ((foldl_dB_dB @ F @ A @ (cons_dB @ X @ Xs)) = (foldl_dB_dB @ F @ (F @ A @ X) @ Xs))))). % foldl_Cons
thf(fact_50_foldl__Nil, axiom,
    ((![F : dB > dB > dB, A : dB]: ((foldl_dB_dB @ F @ A @ nil_dB) = A)))). % foldl_Nil
thf(fact_51_dB_Odistinct_I1_J, axiom,
    ((![X1 : nat, X21 : dB, X22 : dB]: (~ (((var @ X1) = (app @ X21 @ X22))))))). % dB.distinct(1)
thf(fact_52_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_53_rev__nonempty__induct, axiom,
    ((![Xs : list_dB, P : list_dB > $o]: ((~ ((Xs = nil_dB))) => ((![X2 : dB]: (P @ (cons_dB @ X2 @ nil_dB))) => ((![X2 : dB, Xs2 : list_dB]: ((~ ((Xs2 = nil_dB))) => ((P @ Xs2) => (P @ (append_dB @ Xs2 @ (cons_dB @ X2 @ nil_dB)))))) => (P @ Xs))))))). % rev_nonempty_induct
thf(fact_54_append__eq__Cons__conv, axiom,
    ((![Ys : list_dB, Zs : list_dB, X : dB, Xs : list_dB]: (((append_dB @ Ys @ Zs) = (cons_dB @ X @ Xs)) = (((((Ys = nil_dB)) & ((Zs = (cons_dB @ X @ Xs))))) | ((?[Ys4 : list_dB]: (((Ys = (cons_dB @ X @ Ys4))) & (((append_dB @ Ys4 @ Zs) = Xs)))))))))). % append_eq_Cons_conv
thf(fact_55_Cons__eq__append__conv, axiom,
    ((![X : dB, Xs : list_dB, Ys : list_dB, Zs : list_dB]: (((cons_dB @ X @ Xs) = (append_dB @ Ys @ Zs)) = (((((Ys = nil_dB)) & (((cons_dB @ X @ Xs) = Zs)))) | ((?[Ys4 : list_dB]: ((((cons_dB @ X @ Ys4) = Ys)) & ((Xs = (append_dB @ Ys4 @ Zs))))))))))). % Cons_eq_append_conv
thf(fact_56_rev__exhaust, axiom,
    ((![Xs : list_dB]: ((~ ((Xs = nil_dB))) => (~ ((![Ys2 : list_dB, Y2 : dB]: (~ ((Xs = (append_dB @ Ys2 @ (cons_dB @ Y2 @ nil_dB)))))))))))). % rev_exhaust
thf(fact_57_rev__induct, axiom,
    ((![P : list_dB > $o, Xs : list_dB]: ((P @ nil_dB) => ((![X2 : dB, Xs2 : list_dB]: ((P @ Xs2) => (P @ (append_dB @ Xs2 @ (cons_dB @ X2 @ nil_dB))))) => (P @ Xs)))))). % rev_induct
thf(fact_58_rev__exhaust2, axiom,
    ((![Xs : list_dB]: ((~ ((Xs = nil_dB))) => (~ ((![Ys2 : list_dB, Y2 : dB]: (~ ((Xs = (append_dB @ Ys2 @ (cons_dB @ Y2 @ nil_dB)))))))))))). % rev_exhaust2
thf(fact_59_bind__simps_I2_J, axiom,
    ((![X : dB, Xs : list_dB, F : dB > list_dB]: ((bind_dB_dB @ (cons_dB @ X @ Xs) @ F) = (append_dB @ (F @ X) @ (bind_dB_dB @ Xs @ F)))))). % bind_simps(2)
thf(fact_60_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_61_dB_Osize__gen_I1_J, axiom,
    ((![X1 : nat]: ((size_dB @ (var @ X1)) = zero_zero_nat)))). % dB.size_gen(1)
thf(fact_62_maps__simps_I1_J, axiom,
    ((![F : dB > list_dB, X : dB, Xs : list_dB]: ((maps_dB_dB @ F @ (cons_dB @ X @ Xs)) = (append_dB @ (F @ X) @ (maps_dB_dB @ F @ Xs)))))). % maps_simps(1)
thf(fact_63_insert__Nil, axiom,
    ((![X : dB]: ((insert_dB @ X @ nil_dB) = (cons_dB @ X @ nil_dB))))). % insert_Nil
thf(fact_64_nths__singleton, axiom,
    ((![A4 : set_nat, X : dB]: (((member_nat @ zero_zero_nat @ A4) => ((nths_dB @ (cons_dB @ X @ nil_dB) @ A4) = (cons_dB @ X @ nil_dB))) & ((~ ((member_nat @ zero_zero_nat @ A4))) => ((nths_dB @ (cons_dB @ X @ nil_dB) @ A4) = nil_dB)))))). % nths_singleton
thf(fact_65_rotate1_Osimps_I2_J, axiom,
    ((![X : dB, Xs : list_dB]: ((rotate1_dB @ (cons_dB @ X @ Xs)) = (append_dB @ Xs @ (cons_dB @ X @ nil_dB)))))). % rotate1.simps(2)
thf(fact_66_butlast__snoc, axiom,
    ((![Xs : list_dB, X : dB]: ((butlast_dB @ (append_dB @ Xs @ (cons_dB @ X @ nil_dB))) = Xs)))). % butlast_snoc
thf(fact_67_nths__nil, axiom,
    ((![A4 : set_nat]: ((nths_dB @ nil_dB @ A4) = nil_dB)))). % nths_nil
thf(fact_68_rotate1__is__Nil__conv, axiom,
    ((![Xs : list_dB]: (((rotate1_dB @ Xs) = nil_dB) = (Xs = nil_dB))))). % rotate1_is_Nil_conv
thf(fact_69_bind__simps_I1_J, axiom,
    ((![F : dB > list_dB]: ((bind_dB_dB @ nil_dB @ F) = nil_dB)))). % bind_simps(1)
thf(fact_70_butlast_Osimps_I1_J, axiom,
    (((butlast_dB @ nil_dB) = nil_dB))). % butlast.simps(1)
thf(fact_71_rotate1_Osimps_I1_J, axiom,
    (((rotate1_dB @ nil_dB) = nil_dB))). % rotate1.simps(1)
thf(fact_72_butlast_Osimps_I2_J, axiom,
    ((![Xs : list_dB, X : dB]: (((Xs = nil_dB) => ((butlast_dB @ (cons_dB @ X @ Xs)) = nil_dB)) & ((~ ((Xs = nil_dB))) => ((butlast_dB @ (cons_dB @ X @ Xs)) = (cons_dB @ X @ (butlast_dB @ Xs)))))))). % butlast.simps(2)
thf(fact_73_butlast__append, axiom,
    ((![Ys : list_dB, Xs : list_dB]: (((Ys = nil_dB) => ((butlast_dB @ (append_dB @ Xs @ Ys)) = (butlast_dB @ Xs))) & ((~ ((Ys = nil_dB))) => ((butlast_dB @ (append_dB @ Xs @ Ys)) = (append_dB @ Xs @ (butlast_dB @ Ys)))))))). % butlast_append
thf(fact_74_maps__simps_I2_J, axiom,
    ((![F : dB > list_dB]: ((maps_dB_dB @ F @ nil_dB) = nil_dB)))). % maps_simps(2)
thf(fact_75_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_76_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_77_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_78_append__butlast__last__id, axiom,
    ((![Xs : list_dB]: ((~ ((Xs = nil_dB))) => ((append_dB @ (butlast_dB @ Xs) @ (cons_dB @ (last_dB @ Xs) @ nil_dB)) = Xs))))). % append_butlast_last_id
thf(fact_79_subseqs_Osimps_I1_J, axiom,
    (((subseqs_dB @ nil_dB) = (cons_list_dB @ nil_dB @ nil_list_dB)))). % subseqs.simps(1)
thf(fact_80_concat__eq__append__conv, axiom,
    ((![Xss2 : list_list_dB, Ys : list_dB, Zs : list_dB]: (((concat_dB @ Xss2) = (append_dB @ Ys @ Zs)) = (((((Xss2 = nil_list_dB)) => ((((Ys = nil_dB)) & ((Zs = nil_dB)))))) & ((((~ ((Xss2 = nil_list_dB)))) => ((?[Xss1 : list_list_dB]: (?[Xs3 : list_dB]: (?[Xs4 : list_dB]: (?[Xss22 : list_list_dB]: (((Xss2 = (append_list_dB @ Xss1 @ (cons_list_dB @ (append_dB @ Xs3 @ Xs4) @ Xss22)))) & ((((Ys = (append_dB @ (concat_dB @ Xss1) @ Xs3))) & ((Zs = (append_dB @ Xs4 @ (concat_dB @ Xss22))))))))))))))))))). % concat_eq_append_conv
thf(fact_81_last__appendR, axiom,
    ((![Ys : list_dB, Xs : list_dB]: ((~ ((Ys = nil_dB))) => ((last_dB @ (append_dB @ Xs @ Ys)) = (last_dB @ Ys)))))). % last_appendR
thf(fact_82_last__appendL, axiom,
    ((![Ys : list_dB, Xs : list_dB]: ((Ys = nil_dB) => ((last_dB @ (append_dB @ Xs @ Ys)) = (last_dB @ Xs)))))). % last_appendL
thf(fact_83_concat__append, axiom,
    ((![Xs : list_list_dB, Ys : list_list_dB]: ((concat_dB @ (append_list_dB @ Xs @ Ys)) = (append_dB @ (concat_dB @ Xs) @ (concat_dB @ Ys)))))). % concat_append
thf(fact_84_last__snoc, axiom,
    ((![Xs : list_dB, X : dB]: ((last_dB @ (append_dB @ Xs @ (cons_dB @ X @ nil_dB))) = X)))). % last_snoc
thf(fact_85_last__ConsR, axiom,
    ((![Xs : list_dB, X : dB]: ((~ ((Xs = nil_dB))) => ((last_dB @ (cons_dB @ X @ Xs)) = (last_dB @ Xs)))))). % last_ConsR
thf(fact_86_last__ConsL, axiom,
    ((![Xs : list_dB, X : dB]: ((Xs = nil_dB) => ((last_dB @ (cons_dB @ X @ Xs)) = X))))). % last_ConsL
thf(fact_87_last_Osimps, axiom,
    ((![Xs : list_dB, X : dB]: (((Xs = nil_dB) => ((last_dB @ (cons_dB @ X @ Xs)) = X)) & ((~ ((Xs = nil_dB))) => ((last_dB @ (cons_dB @ X @ Xs)) = (last_dB @ Xs))))))). % last.simps
thf(fact_88_longest__common__suffix, axiom,
    ((![Xs : list_dB, Ys : list_dB]: (?[Ss3 : list_dB, Xs5 : list_dB, Ys5 : list_dB]: ((Xs = (append_dB @ Xs5 @ Ss3)) & ((Ys = (append_dB @ Ys5 @ Ss3)) & ((Xs5 = nil_dB) | ((Ys5 = nil_dB) | (~ (((last_dB @ Xs5) = (last_dB @ Ys5)))))))))))). % longest_common_suffix
thf(fact_89_last__append, axiom,
    ((![Ys : list_dB, Xs : list_dB]: (((Ys = nil_dB) => ((last_dB @ (append_dB @ Xs @ Ys)) = (last_dB @ Xs))) & ((~ ((Ys = nil_dB))) => ((last_dB @ (append_dB @ Xs @ Ys)) = (last_dB @ Ys))))))). % last_append
thf(fact_90_concat_Osimps_I1_J, axiom,
    (((concat_dB @ nil_list_dB) = nil_dB))). % concat.simps(1)
thf(fact_91_concat_Osimps_I2_J, axiom,
    ((![X : list_dB, Xs : list_list_dB]: ((concat_dB @ (cons_list_dB @ X @ Xs)) = (append_dB @ X @ (concat_dB @ Xs)))))). % concat.simps(2)
thf(fact_92_concat__eq__appendD, axiom,
    ((![Xss2 : list_list_dB, Ys : list_dB, Zs : list_dB]: (((concat_dB @ Xss2) = (append_dB @ Ys @ Zs)) => ((~ ((Xss2 = nil_list_dB))) => (?[Xss12 : list_list_dB, Xs2 : list_dB, Xs5 : list_dB, Xss23 : list_list_dB]: ((Xss2 = (append_list_dB @ Xss12 @ (cons_list_dB @ (append_dB @ Xs2 @ Xs5) @ Xss23))) & ((Ys = (append_dB @ (concat_dB @ Xss12) @ Xs2)) & (Zs = (append_dB @ Xs5 @ (concat_dB @ Xss23))))))))))). % concat_eq_appendD
thf(fact_93_snoc__eq__iff__butlast, axiom,
    ((![Xs : list_dB, X : dB, Ys : list_dB]: (((append_dB @ Xs @ (cons_dB @ X @ nil_dB)) = Ys) = (((~ ((Ys = nil_dB)))) & (((((butlast_dB @ Ys) = Xs)) & (((last_dB @ Ys) = X))))))))). % snoc_eq_iff_butlast
thf(fact_94_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_95_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_96_list__ex1__simps_I1_J, axiom,
    ((![P : dB > $o]: (~ ((list_ex1_dB @ P @ nil_dB)))))). % list_ex1_simps(1)
thf(fact_97_dB_Oinject_I3_J, axiom,
    ((![X3 : dB, Y32 : dB]: (((abs @ X3) = (abs @ Y32)) = (X3 = Y32))))). % dB.inject(3)
thf(fact_98_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_99_dB_Odistinct_I3_J, axiom,
    ((![X1 : nat, X3 : dB]: (~ (((var @ X1) = (abs @ X3))))))). % dB.distinct(3)
thf(fact_100_dB_Odistinct_I5_J, axiom,
    ((![X21 : dB, X22 : dB, X3 : dB]: (~ (((app @ X21 @ X22) = (abs @ X3))))))). % dB.distinct(5)
thf(fact_101_dB_Oinduct, axiom,
    ((![P : dB > $o, DB : dB]: ((![X2 : nat]: (P @ (var @ X2))) => ((![X1a : dB, X23 : dB]: ((P @ X1a) => ((P @ X23) => (P @ (app @ X1a @ X23))))) => ((![X2 : dB]: ((P @ X2) => (P @ (abs @ X2)))) => (P @ DB))))))). % dB.induct
thf(fact_102_dB_Oexhaust, axiom,
    ((![Y : dB]: ((![X12 : nat]: (~ ((Y = (var @ X12))))) => ((![X212 : dB, X222 : dB]: (~ ((Y = (app @ X212 @ X222))))) => (~ ((![X32 : dB]: (~ ((Y = (abs @ X32)))))))))))). % dB.exhaust
thf(fact_103_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_104_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_105_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_106_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_107_Lambda, axiom,
    ((![R : dB]: ((it @ R) => (it @ (abs @ R)))))). % Lambda
thf(fact_108_IT_Ocases, axiom,
    ((![A : dB]: ((it @ A) => ((![Rs2 : list_dB]: ((?[N2 : nat]: (A = (foldl_dB_dB @ app @ (var @ N2) @ Rs2))) => (~ ((listsp_dB @ it @ Rs2))))) => ((![R2 : dB]: ((A = (abs @ R2)) => (~ ((it @ R2))))) => (~ ((![R2 : dB, S2 : dB, Ss3 : list_dB]: ((A = (foldl_dB_dB @ app @ (app @ (abs @ R2) @ S2) @ Ss3)) => ((it @ (foldl_dB_dB @ app @ (subst @ R2 @ S2 @ zero_zero_nat) @ Ss3)) => (~ ((it @ S2)))))))))))))). % IT.cases
thf(fact_109_listsp__simps_I1_J, axiom,
    ((![A4 : dB > $o]: (listsp_dB @ A4 @ nil_dB)))). % listsp_simps(1)
thf(fact_110_append__in__listsp__conv, axiom,
    ((![A4 : dB > $o, Xs : list_dB, Ys : list_dB]: ((listsp_dB @ A4 @ (append_dB @ Xs @ Ys)) = (((listsp_dB @ A4 @ Xs)) & ((listsp_dB @ A4 @ Ys))))))). % append_in_listsp_conv
thf(fact_111_listsp__simps_I2_J, axiom,
    ((![A4 : dB > $o, X : dB, Xs : list_dB]: ((listsp_dB @ A4 @ (cons_dB @ X @ Xs)) = (((A4 @ X)) & ((listsp_dB @ A4 @ Xs))))))). % listsp_simps(2)
thf(fact_112_listspE, axiom,
    ((![A4 : dB > $o, X : dB, L : list_dB]: ((listsp_dB @ A4 @ (cons_dB @ X @ L)) => (~ (((A4 @ X) => (~ ((listsp_dB @ A4 @ L)))))))))). % listspE
thf(fact_113_listsp_OCons, axiom,
    ((![A4 : dB > $o, A : dB, L : list_dB]: ((A4 @ A) => ((listsp_dB @ A4 @ L) => (listsp_dB @ A4 @ (cons_dB @ A @ L))))))). % listsp.Cons
thf(fact_114_listsp_ONil, axiom,
    ((![A4 : dB > $o]: (listsp_dB @ A4 @ nil_dB)))). % listsp.Nil
thf(fact_115_listsp_Oinducts, axiom,
    ((![A4 : dB > $o, X : list_dB, P : list_dB > $o]: ((listsp_dB @ A4 @ X) => ((P @ nil_dB) => ((![A3 : dB, L2 : list_dB]: ((A4 @ A3) => ((listsp_dB @ A4 @ L2) => ((P @ L2) => (P @ (cons_dB @ A3 @ L2)))))) => (P @ X))))))). % listsp.inducts
thf(fact_116_listsp_Osimps, axiom,
    ((listsp_dB = (^[A5 : dB > $o]: (^[A6 : list_dB]: (((A6 = nil_dB)) | ((?[B2 : dB]: (?[L3 : list_dB]: (((A6 = (cons_dB @ B2 @ L3))) & ((((A5 @ B2)) & ((listsp_dB @ A5 @ L3)))))))))))))). % listsp.simps
thf(fact_117_listsp_Ocases, axiom,
    ((![A4 : dB > $o, A : list_dB]: ((listsp_dB @ A4 @ A) => ((~ ((A = nil_dB))) => (~ ((![A3 : dB, L2 : list_dB]: ((A = (cons_dB @ A3 @ L2)) => ((A4 @ A3) => (~ ((listsp_dB @ A4 @ L2))))))))))))). % listsp.cases
thf(fact_118_IT_OVar, axiom,
    ((![Rs : list_dB, N : nat]: ((listsp_dB @ it @ Rs) => (it @ (foldl_dB_dB @ app @ (var @ N) @ Rs)))))). % IT.Var
thf(fact_119_IT_Osimps, axiom,
    ((it = (^[A6 : dB]: (((?[Rs3 : list_dB]: (?[N3 : nat]: (((A6 = (foldl_dB_dB @ app @ (var @ N3) @ Rs3))) & ((listsp_dB @ it @ Rs3)))))) | ((((?[R3 : dB]: (((A6 = (abs @ R3))) & ((it @ R3))))) | ((?[R3 : dB]: (?[S3 : dB]: (?[Ss2 : list_dB]: (((A6 = (foldl_dB_dB @ app @ (app @ (abs @ R3) @ S3) @ Ss2))) & ((((it @ (foldl_dB_dB @ app @ (subst @ R3 @ S3 @ zero_zero_nat) @ Ss2))) & ((it @ S3)))))))))))))))). % IT.simps
thf(fact_120_substn__subst__0, axiom,
    ((![T : dB, S : dB]: ((substn @ T @ S @ zero_zero_nat) = (subst @ T @ S @ zero_zero_nat))))). % substn_subst_0
thf(fact_121_Apps__dB__induct, axiom,
    ((![P : dB > $o, T : dB]: ((![N2 : nat, Ts2 : list_dB]: ((![X4 : dB]: ((member_dB @ X4 @ (set_dB2 @ Ts2)) => (P @ X4))) => (P @ (foldl_dB_dB @ app @ (var @ N2) @ Ts2)))) => ((![U2 : dB]: ((P @ U2) => (![Ts2 : list_dB]: ((![X4 : dB]: ((member_dB @ X4 @ (set_dB2 @ Ts2)) => (P @ X4))) => (P @ (foldl_dB_dB @ app @ (abs @ U2) @ Ts2)))))) => (P @ T)))))). % Apps_dB_induct
thf(fact_122_in__listspI, axiom,
    ((![Xs : list_dB, A4 : dB > $o]: ((![X2 : dB]: ((member_dB @ X2 @ (set_dB2 @ Xs)) => (A4 @ X2))) => (listsp_dB @ A4 @ Xs))))). % in_listspI
thf(fact_123_set__rotate1, axiom,
    ((![Xs : list_dB]: ((set_dB2 @ (rotate1_dB @ Xs)) = (set_dB2 @ Xs))))). % set_rotate1
thf(fact_124_in__set__insert, axiom,
    ((![X : dB, Xs : list_dB]: ((member_dB @ X @ (set_dB2 @ Xs)) => ((insert_dB @ X @ Xs) = Xs))))). % in_set_insert
thf(fact_125_not__in__set__insert, axiom,
    ((![X : dB, Xs : list_dB]: ((~ ((member_dB @ X @ (set_dB2 @ Xs)))) => ((insert_dB @ X @ Xs) = (cons_dB @ X @ Xs)))))). % not_in_set_insert
thf(fact_126_in__listsp__conv__set, axiom,
    ((listsp_dB = (^[A5 : dB > $o]: (^[Xs3 : list_dB]: (![X5 : dB]: (((member_dB @ X5 @ (set_dB2 @ Xs3))) => ((A5 @ X5))))))))). % in_listsp_conv_set
thf(fact_127_in__listspD, axiom,
    ((![A4 : dB > $o, Xs : list_dB]: ((listsp_dB @ A4 @ Xs) => (![X4 : dB]: ((member_dB @ X4 @ (set_dB2 @ Xs)) => (A4 @ X4))))))). % in_listspD
thf(fact_128_last__in__set, axiom,
    ((![As2 : list_dB]: ((~ ((As2 = nil_dB))) => (member_dB @ (last_dB @ As2) @ (set_dB2 @ As2)))))). % last_in_set
thf(fact_129_in__set__butlast__appendI, axiom,
    ((![X : dB, Xs : list_dB, Ys : list_dB]: (((member_dB @ X @ (set_dB2 @ (butlast_dB @ Xs))) | (member_dB @ X @ (set_dB2 @ (butlast_dB @ Ys)))) => (member_dB @ X @ (set_dB2 @ (butlast_dB @ (append_dB @ Xs @ Ys)))))))). % in_set_butlast_appendI
thf(fact_130_List_Oinsert__def, axiom,
    ((insert_dB = (^[X5 : dB]: (^[Xs3 : list_dB]: (if_list_dB @ (member_dB @ X5 @ (set_dB2 @ Xs3)) @ Xs3 @ (cons_dB @ X5 @ Xs3))))))). % List.insert_def
thf(fact_131_foldl__cong, axiom,
    ((![A : dB, B : dB, L : list_dB, K : list_dB, F : dB > dB > dB, G : dB > dB > dB]: ((A = B) => ((L = K) => ((![A3 : dB, X2 : dB]: ((member_dB @ X2 @ (set_dB2 @ L)) => ((F @ A3 @ X2) = (G @ A3 @ X2)))) => ((foldl_dB_dB @ F @ A @ L) = (foldl_dB_dB @ G @ B @ K)))))))). % foldl_cong
thf(fact_132_list_Oset__cases, axiom,
    ((![E : dB, A : list_dB]: ((member_dB @ E @ (set_dB2 @ A)) => ((![Z2 : list_dB]: (~ ((A = (cons_dB @ E @ Z2))))) => (~ ((![Z1 : dB, Z2 : list_dB]: ((A = (cons_dB @ Z1 @ Z2)) => (~ ((member_dB @ E @ (set_dB2 @ Z2))))))))))))). % list.set_cases
thf(fact_133_set__ConsD, axiom,
    ((![Y : dB, X : dB, Xs : list_dB]: ((member_dB @ Y @ (set_dB2 @ (cons_dB @ X @ Xs))) => ((Y = X) | (member_dB @ Y @ (set_dB2 @ Xs))))))). % set_ConsD
thf(fact_134_list_Oset__intros_I1_J, axiom,
    ((![X21 : dB, X22 : list_dB]: (member_dB @ X21 @ (set_dB2 @ (cons_dB @ X21 @ X22)))))). % list.set_intros(1)
thf(fact_135_list_Oset__intros_I2_J, axiom,
    ((![Y : dB, X22 : list_dB, X21 : dB]: ((member_dB @ Y @ (set_dB2 @ X22)) => (member_dB @ Y @ (set_dB2 @ (cons_dB @ X21 @ X22))))))). % list.set_intros(2)
thf(fact_136_notin__set__nthsI, axiom,
    ((![X : dB, Xs : list_dB, I2 : set_nat]: ((~ ((member_dB @ X @ (set_dB2 @ Xs)))) => (~ ((member_dB @ X @ (set_dB2 @ (nths_dB @ Xs @ I2))))))))). % notin_set_nthsI
thf(fact_137_in__set__butlastD, axiom,
    ((![X : dB, Xs : list_dB]: ((member_dB @ X @ (set_dB2 @ (butlast_dB @ Xs))) => (member_dB @ X @ (set_dB2 @ Xs)))))). % in_set_butlastD

% Helper facts (3)
thf(help_If_3_1_If_001t__List__Olist_It__Lambda__OdB_J_T, axiom,
    ((![P : $o]: ((P = $true) | (P = $false))))).
thf(help_If_2_1_If_001t__List__Olist_It__Lambda__OdB_J_T, axiom,
    ((![X : list_dB, Y : list_dB]: ((if_list_dB @ $false @ X @ Y) = Y)))).
thf(help_If_1_1_If_001t__List__Olist_It__Lambda__OdB_J_T, axiom,
    ((![X : list_dB, Y : list_dB]: ((if_list_dB @ $true @ X @ Y) = X)))).

% Conjectures (5)
thf(conj_0, hypothesis,
    ((it @ (foldl_dB_dB @ app @ (subst @ r @ s @ zero_zero_nat) @ ss)))).
thf(conj_1, hypothesis,
    ((it @ (app @ (foldl_dB_dB @ app @ (subst @ r @ s @ zero_zero_nat) @ ss) @ (var @ i))))).
thf(conj_2, hypothesis,
    ((it @ s))).
thf(conj_3, hypothesis,
    ((it @ (app @ s @ (var @ i))))).
thf(conj_4, conjecture,
    ((it @ (foldl_dB_dB @ app @ (subst @ r @ s @ zero_zero_nat) @ (append_dB @ ss @ (cons_dB @ (var @ i) @ nil_dB)))))).
