% 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_119__5210340_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:37:35.600

% Could-be-implicit typings (7)
thf(ty_n_t__List__Olist_It__List__Olist_It__LambdaType__Otype_J_J, type,
    list_list_type : $tType).
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__LambdaType__Otype_J, type,
    list_type : $tType).
thf(ty_n_t__List__Olist_It__Lambda__OdB_J, type,
    list_dB : $tType).
thf(ty_n_t__LambdaType__Otype, type,
    type : $tType).
thf(ty_n_t__Lambda__OdB, type,
    dB : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).

% Explicit typings (63)
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_LambdaType_Oshift_001t__LambdaType__Otype, type,
    shift_type : (nat > type) > nat > type > nat > type).
thf(sy_c_LambdaType_Otype_OFun, type,
    fun : type > type > type).
thf(sy_c_LambdaType_Otyping, type,
    typing : (nat > type) > dB > type > $o).
thf(sy_c_LambdaType_Otypings, type,
    typings : (nat > type) > list_dB > list_type > $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_List_Oappend_001t__LambdaType__Otype, type,
    append_type : list_type > list_type > list_type).
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__LambdaType__Otype_J, type,
    append_list_type : list_list_type > list_list_type > list_list_type).
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__LambdaType__Otype_001t__LambdaType__Otype, type,
    bind_type_type : list_type > (type > list_type) > list_type).
thf(sy_c_List_Obind_001t__LambdaType__Otype_001t__Lambda__OdB, type,
    bind_type_dB : list_type > (type > list_dB) > list_dB).
thf(sy_c_List_Obind_001t__Lambda__OdB_001t__LambdaType__Otype, type,
    bind_dB_type : list_dB > (dB > list_type) > list_type).
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__LambdaType__Otype, type,
    concat_type : list_list_type > list_type).
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_Ofoldr_001t__LambdaType__Otype_001t__LambdaType__Otype, type,
    foldr_type_type : (type > type > type) > list_type > type > type).
thf(sy_c_List_Oinsert_001t__LambdaType__Otype, type,
    insert_type : type > list_type > list_type).
thf(sy_c_List_Oinsert_001t__Lambda__OdB, type,
    insert_dB : dB > list_dB > list_dB).
thf(sy_c_List_Olist_OCons_001t__LambdaType__Otype, type,
    cons_type : type > list_type > list_type).
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__LambdaType__Otype_J, type,
    cons_list_type : list_type > list_list_type > list_list_type).
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__LambdaType__Otype, type,
    nil_type : list_type).
thf(sy_c_List_Olist_ONil_001t__Lambda__OdB, type,
    nil_dB : list_dB).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__LambdaType__Otype_J, type,
    nil_list_type : list_list_type).
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_Ocase__list_001_Eo_001t__LambdaType__Otype, type,
    case_list_o_type : $o > (type > list_type > $o) > list_type > $o).
thf(sy_c_List_Olist_Ocase__list_001_Eo_001t__Lambda__OdB, type,
    case_list_o_dB : $o > (dB > list_dB > $o) > list_dB > $o).
thf(sy_c_List_Olist__ex1_001t__LambdaType__Otype, type,
    list_ex1_type : (type > $o) > list_type > $o).
thf(sy_c_List_Olist__ex1_001t__Lambda__OdB, type,
    list_ex1_dB : (dB > $o) > list_dB > $o).
thf(sy_c_List_Olistsp_001t__LambdaType__Otype, type,
    listsp_type : (type > $o) > list_type > $o).
thf(sy_c_List_Olistsp_001t__Lambda__OdB, type,
    listsp_dB : (dB > $o) > list_dB > $o).
thf(sy_c_List_Omaps_001t__LambdaType__Otype_001t__LambdaType__Otype, type,
    maps_type_type : (type > list_type) > list_type > list_type).
thf(sy_c_List_Omaps_001t__LambdaType__Otype_001t__Lambda__OdB, type,
    maps_type_dB : (type > list_dB) > list_type > list_dB).
thf(sy_c_List_Omaps_001t__Lambda__OdB_001t__LambdaType__Otype, type,
    maps_dB_type : (dB > list_type) > list_dB > list_type).
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__LambdaType__Otype, type,
    n_lists_type : nat > list_type > list_list_type).
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__LambdaType__Otype, type,
    product_lists_type : list_list_type > list_list_type).
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__LambdaType__Otype, type,
    subseqs_type : list_type > list_list_type).
thf(sy_c_List_Osubseqs_001t__Lambda__OdB, type,
    subseqs_dB : list_dB > list_list_dB).
thf(sy_v_T_H1____, type,
    t_1 : type).
thf(sy_v_T_H____, type,
    t : type).
thf(sy_v_T____, type,
    t2 : type).
thf(sy_v_a____, type,
    a : dB).
thf(sy_v_as____, type,
    as : list_dB).
thf(sy_v_e1____, type,
    e1 : nat > type).
thf(sy_v_e____, type,
    e : nat > type).
thf(sy_v_i1____, type,
    i1 : nat).
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,
    t3 : dB).
thf(sy_v_u1____, type,
    u1 : dB).
thf(sy_v_u____, type,
    u : dB).

% Relevant facts (218)
thf(fact_0_True, axiom,
    ((n = i))). % True
thf(fact_1_local_OCons, axiom,
    ((rs = (cons_dB @ a @ as)))). % local.Cons
thf(fact_2_uT, axiom,
    ((typing @ e @ u @ t2))). % uT
thf(fact_3_nT, axiom,
    ((typing @ (shift_type @ e @ i @ t2) @ (foldl_dB_dB @ app @ (var @ n) @ rs) @ t))). % nT
thf(fact_4__092_060open_062IT_At_092_060close_062, axiom,
    ((it @ t3))). % \<open>IT t\<close>
thf(fact_5_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_6_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_7_var__app__type__eq, axiom,
    ((![E : nat > type, I : nat, Ts : list_dB, T : type, U : type]: ((typing @ E @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ T) => ((typing @ E @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ U) => (T = U)))))). % var_app_type_eq
thf(fact_8_shift__eq, axiom,
    ((![I : nat, J : nat, E : nat > type, T : type]: ((I = J) => ((shift_type @ E @ I @ T @ J) = T))))). % shift_eq
thf(fact_9_dB_Oinject_I1_J, axiom,
    ((![X1 : nat, Y1 : nat]: (((var @ X1) = (var @ Y1)) = (X1 = Y1))))). % dB.inject(1)
thf(fact_10_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_11_typing__elims_I1_J, axiom,
    ((![E : nat > type, I : nat, T : type]: ((typing @ E @ (var @ I) @ T) => ((E @ I) = T))))). % typing_elims(1)
thf(fact_12_typing_OVar, axiom,
    ((![Env : nat > type, X : nat, T : type]: (((Env @ X) = T) => (typing @ Env @ (var @ X) @ T))))). % typing.Var
thf(fact_13_dB_Odistinct_I1_J, axiom,
    ((![X1 : nat, X21 : dB, X22 : dB]: (~ (((var @ X1) = (app @ X21 @ X22))))))). % dB.distinct(1)
thf(fact_14_Var_Oprems_I1_J, axiom,
    ((typing @ (shift_type @ e1 @ i1 @ t2) @ (foldl_dB_dB @ app @ (var @ n) @ rs) @ t_1))). % Var.prems(1)
thf(fact_15_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_16_Var_Oprems_I2_J, axiom,
    ((it @ u1))). % Var.prems(2)
thf(fact_17_uIT, axiom,
    ((it @ u))). % uIT
thf(fact_18_Var_Oprems_I3_J, axiom,
    ((typing @ e1 @ u1 @ t2))). % Var.prems(3)
thf(fact_19_Var__IT, axiom,
    ((![N : nat]: (it @ (var @ N))))). % Var_IT
thf(fact_20_app__Var__IT, axiom,
    ((![T2 : dB, I : nat]: ((it @ T2) => (it @ (app @ T2 @ (var @ I))))))). % app_Var_IT
thf(fact_21_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_22_list_Oinject, axiom,
    ((![X21 : type, X22 : list_type, Y21 : type, Y22 : list_type]: (((cons_type @ X21 @ X22) = (cons_type @ Y21 @ Y22)) = (((X21 = Y21)) & ((X22 = Y22))))))). % list.inject
thf(fact_23_foldl__Nil, axiom,
    ((![F : dB > dB > dB, A : dB]: ((foldl_dB_dB @ F @ A @ nil_dB) = A)))). % foldl_Nil
thf(fact_24_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_25_list_Odistinct_I1_J, axiom,
    ((![X21 : dB, X22 : list_dB]: (~ ((nil_dB = (cons_dB @ X21 @ X22))))))). % list.distinct(1)
thf(fact_26_list_Odistinct_I1_J, axiom,
    ((![X21 : type, X22 : list_type]: (~ ((nil_type = (cons_type @ X21 @ X22))))))). % list.distinct(1)
thf(fact_27_list_OdiscI, axiom,
    ((![List : list_dB, X21 : dB, X22 : list_dB]: ((List = (cons_dB @ X21 @ X22)) => (~ ((List = nil_dB))))))). % list.discI
thf(fact_28_list_OdiscI, axiom,
    ((![List : list_type, X21 : type, X22 : list_type]: ((List = (cons_type @ X21 @ X22)) => (~ ((List = nil_type))))))). % list.discI
thf(fact_29_list_Oexhaust, axiom,
    ((![Y : list_dB]: ((~ ((Y = nil_dB))) => (~ ((![X212 : dB, X222 : list_dB]: (~ ((Y = (cons_dB @ X212 @ X222))))))))))). % list.exhaust
thf(fact_30_list_Oexhaust, axiom,
    ((![Y : list_type]: ((~ ((Y = nil_type))) => (~ ((![X212 : type, X222 : list_type]: (~ ((Y = (cons_type @ X212 @ X222))))))))))). % list.exhaust
thf(fact_31_list_Oinducts, axiom,
    ((![P : list_dB > $o, List : list_dB]: ((P @ nil_dB) => ((![X12 : dB, X2 : list_dB]: ((P @ X2) => (P @ (cons_dB @ X12 @ X2)))) => (P @ List)))))). % list.inducts
thf(fact_32_list_Oinducts, axiom,
    ((![P : list_type > $o, List : list_type]: ((P @ nil_type) => ((![X12 : type, X2 : list_type]: ((P @ X2) => (P @ (cons_type @ X12 @ X2)))) => (P @ List)))))). % list.inducts
thf(fact_33_neq__Nil__conv, axiom,
    ((![Xs : list_dB]: ((~ ((Xs = nil_dB))) = (?[Y2 : dB]: (?[Ys : list_dB]: (Xs = (cons_dB @ Y2 @ Ys)))))))). % neq_Nil_conv
thf(fact_34_neq__Nil__conv, axiom,
    ((![Xs : list_type]: ((~ ((Xs = nil_type))) = (?[Y2 : type]: (?[Ys : list_type]: (Xs = (cons_type @ Y2 @ Ys)))))))). % neq_Nil_conv
thf(fact_35_list__induct2_H, axiom,
    ((![P : list_dB > list_dB > $o, Xs : list_dB, Ys2 : list_dB]: ((P @ nil_dB @ nil_dB) => ((![X3 : dB, Xs2 : list_dB]: (P @ (cons_dB @ X3 @ Xs2) @ nil_dB)) => ((![Y3 : dB, Ys3 : list_dB]: (P @ nil_dB @ (cons_dB @ Y3 @ Ys3))) => ((![X3 : dB, Xs2 : list_dB, Y3 : dB, Ys3 : list_dB]: ((P @ Xs2 @ Ys3) => (P @ (cons_dB @ X3 @ Xs2) @ (cons_dB @ Y3 @ Ys3)))) => (P @ Xs @ Ys2)))))))). % list_induct2'
thf(fact_36_list__induct2_H, axiom,
    ((![P : list_dB > list_type > $o, Xs : list_dB, Ys2 : list_type]: ((P @ nil_dB @ nil_type) => ((![X3 : dB, Xs2 : list_dB]: (P @ (cons_dB @ X3 @ Xs2) @ nil_type)) => ((![Y3 : type, Ys3 : list_type]: (P @ nil_dB @ (cons_type @ Y3 @ Ys3))) => ((![X3 : dB, Xs2 : list_dB, Y3 : type, Ys3 : list_type]: ((P @ Xs2 @ Ys3) => (P @ (cons_dB @ X3 @ Xs2) @ (cons_type @ Y3 @ Ys3)))) => (P @ Xs @ Ys2)))))))). % list_induct2'
thf(fact_37_list__induct2_H, axiom,
    ((![P : list_type > list_dB > $o, Xs : list_type, Ys2 : list_dB]: ((P @ nil_type @ nil_dB) => ((![X3 : type, Xs2 : list_type]: (P @ (cons_type @ X3 @ Xs2) @ nil_dB)) => ((![Y3 : dB, Ys3 : list_dB]: (P @ nil_type @ (cons_dB @ Y3 @ Ys3))) => ((![X3 : type, Xs2 : list_type, Y3 : dB, Ys3 : list_dB]: ((P @ Xs2 @ Ys3) => (P @ (cons_type @ X3 @ Xs2) @ (cons_dB @ Y3 @ Ys3)))) => (P @ Xs @ Ys2)))))))). % list_induct2'
thf(fact_38_list__induct2_H, axiom,
    ((![P : list_type > list_type > $o, Xs : list_type, Ys2 : list_type]: ((P @ nil_type @ nil_type) => ((![X3 : type, Xs2 : list_type]: (P @ (cons_type @ X3 @ Xs2) @ nil_type)) => ((![Y3 : type, Ys3 : list_type]: (P @ nil_type @ (cons_type @ Y3 @ Ys3))) => ((![X3 : type, Xs2 : list_type, Y3 : type, Ys3 : list_type]: ((P @ Xs2 @ Ys3) => (P @ (cons_type @ X3 @ Xs2) @ (cons_type @ Y3 @ Ys3)))) => (P @ Xs @ Ys2)))))))). % list_induct2'
thf(fact_39_not__Cons__self2, axiom,
    ((![X : dB, Xs : list_dB]: (~ (((cons_dB @ X @ Xs) = Xs)))))). % not_Cons_self2
thf(fact_40_not__Cons__self2, axiom,
    ((![X : type, Xs : list_type]: (~ (((cons_type @ X @ Xs) = Xs)))))). % not_Cons_self2
thf(fact_41_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_42_map__tailrec__rev_Oinduct, axiom,
    ((![P : (type > dB) > list_type > list_dB > $o, A0 : type > dB, A1 : list_type, A2 : list_dB]: ((![F2 : type > dB, X_1 : list_dB]: (P @ F2 @ nil_type @ X_1)) => ((![F2 : type > dB, A3 : type, As : list_type, Bs : list_dB]: ((P @ F2 @ As @ (cons_dB @ (F2 @ A3) @ Bs)) => (P @ F2 @ (cons_type @ A3 @ As) @ Bs))) => (P @ A0 @ A1 @ A2)))))). % map_tailrec_rev.induct
thf(fact_43_map__tailrec__rev_Oinduct, axiom,
    ((![P : (dB > type) > list_dB > list_type > $o, A0 : dB > type, A1 : list_dB, A2 : list_type]: ((![F2 : dB > type, X_1 : list_type]: (P @ F2 @ nil_dB @ X_1)) => ((![F2 : dB > type, A3 : dB, As : list_dB, Bs : list_type]: ((P @ F2 @ As @ (cons_type @ (F2 @ A3) @ Bs)) => (P @ F2 @ (cons_dB @ A3 @ As) @ Bs))) => (P @ A0 @ A1 @ A2)))))). % map_tailrec_rev.induct
thf(fact_44_map__tailrec__rev_Oinduct, axiom,
    ((![P : (type > type) > list_type > list_type > $o, A0 : type > type, A1 : list_type, A2 : list_type]: ((![F2 : type > type, X_1 : list_type]: (P @ F2 @ nil_type @ X_1)) => ((![F2 : type > type, A3 : type, As : list_type, Bs : list_type]: ((P @ F2 @ As @ (cons_type @ (F2 @ A3) @ Bs)) => (P @ F2 @ (cons_type @ A3 @ As) @ Bs))) => (P @ A0 @ A1 @ A2)))))). % map_tailrec_rev.induct
thf(fact_45_list__nonempty__induct, axiom,
    ((![Xs : list_dB, P : list_dB > $o]: ((~ ((Xs = nil_dB))) => ((![X3 : dB]: (P @ (cons_dB @ X3 @ nil_dB))) => ((![X3 : dB, Xs2 : list_dB]: ((~ ((Xs2 = nil_dB))) => ((P @ Xs2) => (P @ (cons_dB @ X3 @ Xs2))))) => (P @ Xs))))))). % list_nonempty_induct
thf(fact_46_list__nonempty__induct, axiom,
    ((![Xs : list_type, P : list_type > $o]: ((~ ((Xs = nil_type))) => ((![X3 : type]: (P @ (cons_type @ X3 @ nil_type))) => ((![X3 : type, Xs2 : list_type]: ((~ ((Xs2 = nil_type))) => ((P @ Xs2) => (P @ (cons_type @ X3 @ Xs2))))) => (P @ Xs))))))). % list_nonempty_induct
thf(fact_47_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, Y3 : dB, Xs2 : list_dB]: ((P @ P2 @ (cons_dB @ Y3 @ Xs2)) => (P @ P2 @ (cons_dB @ X3 @ (cons_dB @ Y3 @ Xs2))))) => (P @ A0 @ A1))))))). % successively.induct
thf(fact_48_successively_Oinduct, axiom,
    ((![P : (type > type > $o) > list_type > $o, A0 : type > type > $o, A1 : list_type]: ((![P2 : type > type > $o]: (P @ P2 @ nil_type)) => ((![P2 : type > type > $o, X3 : type]: (P @ P2 @ (cons_type @ X3 @ nil_type))) => ((![P2 : type > type > $o, X3 : type, Y3 : type, Xs2 : list_type]: ((P @ P2 @ (cons_type @ Y3 @ Xs2)) => (P @ P2 @ (cons_type @ X3 @ (cons_type @ Y3 @ Xs2))))) => (P @ A0 @ A1))))))). % successively.induct
thf(fact_49_remdups__adj_Oinduct, axiom,
    ((![P : list_dB > $o, A0 : list_dB]: ((P @ nil_dB) => ((![X3 : dB]: (P @ (cons_dB @ X3 @ nil_dB))) => ((![X3 : dB, Y3 : dB, Xs2 : list_dB]: (((X3 = Y3) => (P @ (cons_dB @ X3 @ Xs2))) => (((~ ((X3 = Y3))) => (P @ (cons_dB @ Y3 @ Xs2))) => (P @ (cons_dB @ X3 @ (cons_dB @ Y3 @ Xs2)))))) => (P @ A0))))))). % remdups_adj.induct
thf(fact_50_remdups__adj_Oinduct, axiom,
    ((![P : list_type > $o, A0 : list_type]: ((P @ nil_type) => ((![X3 : type]: (P @ (cons_type @ X3 @ nil_type))) => ((![X3 : type, Y3 : type, Xs2 : list_type]: (((X3 = Y3) => (P @ (cons_type @ X3 @ Xs2))) => (((~ ((X3 = Y3))) => (P @ (cons_type @ Y3 @ Xs2))) => (P @ (cons_type @ X3 @ (cons_type @ Y3 @ Xs2)))))) => (P @ A0))))))). % remdups_adj.induct
thf(fact_51_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, Ys3 : list_dB]: ((P @ P2 @ Ys3) => (P @ P2 @ (cons_dB @ X3 @ Ys3)))) => (P @ A0 @ A1)))))). % sorted_wrt.induct
thf(fact_52_sorted__wrt_Oinduct, axiom,
    ((![P : (type > type > $o) > list_type > $o, A0 : type > type > $o, A1 : list_type]: ((![P2 : type > type > $o]: (P @ P2 @ nil_type)) => ((![P2 : type > type > $o, X3 : type, Ys3 : list_type]: ((P @ P2 @ Ys3) => (P @ P2 @ (cons_type @ X3 @ Ys3)))) => (P @ A0 @ A1)))))). % sorted_wrt.induct
thf(fact_53_remdups__adj_Ocases, axiom,
    ((![X : list_dB]: ((~ ((X = nil_dB))) => ((![X3 : dB]: (~ ((X = (cons_dB @ X3 @ nil_dB))))) => (~ ((![X3 : dB, Y3 : dB, Xs2 : list_dB]: (~ ((X = (cons_dB @ X3 @ (cons_dB @ Y3 @ Xs2))))))))))))). % remdups_adj.cases
thf(fact_54_remdups__adj_Ocases, axiom,
    ((![X : list_type]: ((~ ((X = nil_type))) => ((![X3 : type]: (~ ((X = (cons_type @ X3 @ nil_type))))) => (~ ((![X3 : type, Y3 : type, Xs2 : list_type]: (~ ((X = (cons_type @ X3 @ (cons_type @ Y3 @ Xs2))))))))))))). % remdups_adj.cases
thf(fact_55_transpose_Ocases, axiom,
    ((![X : list_list_dB]: ((~ ((X = nil_list_dB))) => ((![Xss : list_list_dB]: (~ ((X = (cons_list_dB @ nil_dB @ Xss))))) => (~ ((![X3 : dB, Xs2 : list_dB, Xss : list_list_dB]: (~ ((X = (cons_list_dB @ (cons_dB @ X3 @ Xs2) @ Xss)))))))))))). % transpose.cases
thf(fact_56_transpose_Ocases, axiom,
    ((![X : list_list_type]: ((~ ((X = nil_list_type))) => ((![Xss : list_list_type]: (~ ((X = (cons_list_type @ nil_type @ Xss))))) => (~ ((![X3 : type, Xs2 : list_type, Xss : list_list_type]: (~ ((X = (cons_list_type @ (cons_type @ X3 @ Xs2) @ Xss)))))))))))). % transpose.cases
thf(fact_57_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)) => ((![X3 : dB, Xs2 : list_dB, Y3 : dB, Ys3 : list_dB]: ((P @ Xs2 @ (cons_dB @ Y3 @ Ys3)) => ((P @ (cons_dB @ X3 @ Xs2) @ Ys3) => (P @ (cons_dB @ X3 @ Xs2) @ (cons_dB @ Y3 @ Ys3))))) => (P @ A0 @ A1))))))). % shuffles.induct
thf(fact_58_shuffles_Oinduct, axiom,
    ((![P : list_type > list_type > $o, A0 : list_type, A1 : list_type]: ((![X_1 : list_type]: (P @ nil_type @ X_1)) => ((![Xs2 : list_type]: (P @ Xs2 @ nil_type)) => ((![X3 : type, Xs2 : list_type, Y3 : type, Ys3 : list_type]: ((P @ Xs2 @ (cons_type @ Y3 @ Ys3)) => ((P @ (cons_type @ X3 @ Xs2) @ Ys3) => (P @ (cons_type @ X3 @ Xs2) @ (cons_type @ Y3 @ Ys3))))) => (P @ A0 @ A1))))))). % shuffles.induct
thf(fact_59_induct__list012, axiom,
    ((![P : list_dB > $o, Xs : list_dB]: ((P @ nil_dB) => ((![X3 : dB]: (P @ (cons_dB @ X3 @ nil_dB))) => ((![X3 : dB, Y3 : dB, Zs : list_dB]: ((P @ Zs) => ((P @ (cons_dB @ Y3 @ Zs)) => (P @ (cons_dB @ X3 @ (cons_dB @ Y3 @ Zs)))))) => (P @ Xs))))))). % induct_list012
thf(fact_60_induct__list012, axiom,
    ((![P : list_type > $o, Xs : list_type]: ((P @ nil_type) => ((![X3 : type]: (P @ (cons_type @ X3 @ nil_type))) => ((![X3 : type, Y3 : type, Zs : list_type]: ((P @ Zs) => ((P @ (cons_type @ Y3 @ Zs)) => (P @ (cons_type @ X3 @ (cons_type @ Y3 @ Zs)))))) => (P @ Xs))))))). % induct_list012
thf(fact_61_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, Xs2 : list_dB, Ys3 : list_dB]: ((P @ Ys3 @ Xs2) => (P @ (cons_dB @ X3 @ Xs2) @ Ys3))) => (P @ A0 @ A1)))))). % splice.induct
thf(fact_62_splice_Oinduct, axiom,
    ((![P : list_type > list_type > $o, A0 : list_type, A1 : list_type]: ((![X_1 : list_type]: (P @ nil_type @ X_1)) => ((![X3 : type, Xs2 : list_type, Ys3 : list_type]: ((P @ Ys3 @ Xs2) => (P @ (cons_type @ X3 @ Xs2) @ Ys3))) => (P @ A0 @ A1)))))). % splice.induct
thf(fact_63_MI1, axiom,
    ((![T1 : type, T22 : type, T2 : dB, E : nat > type, I : nat, T : type, U2 : dB]: ((t2 = (fun @ T1 @ T22)) => ((it @ T2) => ((typing @ (shift_type @ E @ I @ T1) @ T2 @ T) => ((it @ U2) => ((typing @ E @ U2 @ T1) => (it @ (subst @ T2 @ U2 @ I)))))))))). % MI1
thf(fact_64_MI2, axiom,
    ((![T1 : type, T22 : type, T2 : dB, E : nat > type, I : nat, T : type, U2 : dB]: ((t2 = (fun @ T1 @ T22)) => ((it @ T2) => ((typing @ (shift_type @ E @ I @ T22) @ T2 @ T) => ((it @ U2) => ((typing @ E @ U2 @ T22) => (it @ (subst @ T2 @ U2 @ I)))))))))). % MI2
thf(fact_65_insert__Nil, axiom,
    ((![X : dB]: ((insert_dB @ X @ nil_dB) = (cons_dB @ X @ nil_dB))))). % insert_Nil
thf(fact_66_insert__Nil, axiom,
    ((![X : type]: ((insert_type @ X @ nil_type) = (cons_type @ X @ nil_type))))). % insert_Nil
thf(fact_67_app__last, axiom,
    ((![T2 : dB, Ts : list_dB, U2 : dB]: ((app @ (foldl_dB_dB @ app @ T2 @ Ts) @ U2) = (foldl_dB_dB @ app @ T2 @ (append_dB @ Ts @ (cons_dB @ U2 @ nil_dB))))))). % app_last
thf(fact_68_App__eq__foldl__conv, axiom,
    ((![R : dB, S : dB, T2 : dB, Ts : list_dB]: (((app @ R @ S) = (foldl_dB_dB @ app @ T2 @ Ts)) = (((((Ts = nil_dB)) => (((app @ R @ S) = T2)))) & ((((~ ((Ts = nil_dB)))) => ((?[Ss2 : list_dB]: (((Ts = (append_dB @ Ss2 @ (cons_dB @ S @ nil_dB)))) & ((R = (foldl_dB_dB @ app @ T2 @ Ss2))))))))))))). % App_eq_foldl_conv
thf(fact_69_list__ex1__simps_I1_J, axiom,
    ((![P : dB > $o]: (~ ((list_ex1_dB @ P @ nil_dB)))))). % list_ex1_simps(1)
thf(fact_70_list__ex1__simps_I1_J, axiom,
    ((![P : type > $o]: (~ ((list_ex1_type @ P @ nil_type)))))). % list_ex1_simps(1)
thf(fact_71_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_72_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_73_same__append__eq, axiom,
    ((![Xs : list_dB, Ys2 : list_dB, Zs2 : list_dB]: (((append_dB @ Xs @ Ys2) = (append_dB @ Xs @ Zs2)) = (Ys2 = Zs2))))). % same_append_eq
thf(fact_74_same__append__eq, axiom,
    ((![Xs : list_type, Ys2 : list_type, Zs2 : list_type]: (((append_type @ Xs @ Ys2) = (append_type @ Xs @ Zs2)) = (Ys2 = Zs2))))). % same_append_eq
thf(fact_75_append__same__eq, axiom,
    ((![Ys2 : list_dB, Xs : list_dB, Zs2 : list_dB]: (((append_dB @ Ys2 @ Xs) = (append_dB @ Zs2 @ Xs)) = (Ys2 = Zs2))))). % append_same_eq
thf(fact_76_append__same__eq, axiom,
    ((![Ys2 : list_type, Xs : list_type, Zs2 : list_type]: (((append_type @ Ys2 @ Xs) = (append_type @ Zs2 @ Xs)) = (Ys2 = Zs2))))). % append_same_eq
thf(fact_77_append__assoc, axiom,
    ((![Xs : list_dB, Ys2 : list_dB, Zs2 : list_dB]: ((append_dB @ (append_dB @ Xs @ Ys2) @ Zs2) = (append_dB @ Xs @ (append_dB @ Ys2 @ Zs2)))))). % append_assoc
thf(fact_78_append__assoc, axiom,
    ((![Xs : list_type, Ys2 : list_type, Zs2 : list_type]: ((append_type @ (append_type @ Xs @ Ys2) @ Zs2) = (append_type @ Xs @ (append_type @ Ys2 @ Zs2)))))). % append_assoc
thf(fact_79_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_80_append_Oassoc, axiom,
    ((![A : list_type, B : list_type, C : list_type]: ((append_type @ (append_type @ A @ B) @ C) = (append_type @ A @ (append_type @ B @ C)))))). % append.assoc
thf(fact_81_dB_Oinject_I3_J, axiom,
    ((![X32 : dB, Y32 : dB]: (((abs @ X32) = (abs @ Y32)) = (X32 = Y32))))). % dB.inject(3)
thf(fact_82_type_Oinject_I2_J, axiom,
    ((![X21 : type, X22 : type, Y21 : type, Y22 : type]: (((fun @ X21 @ X22) = (fun @ Y21 @ Y22)) = (((X21 = Y21)) & ((X22 = Y22))))))). % type.inject(2)
thf(fact_83_append__Nil2, axiom,
    ((![Xs : list_dB]: ((append_dB @ Xs @ nil_dB) = Xs)))). % append_Nil2
thf(fact_84_append__Nil2, axiom,
    ((![Xs : list_type]: ((append_type @ Xs @ nil_type) = Xs)))). % append_Nil2
thf(fact_85_append__self__conv, axiom,
    ((![Xs : list_dB, Ys2 : list_dB]: (((append_dB @ Xs @ Ys2) = Xs) = (Ys2 = nil_dB))))). % append_self_conv
thf(fact_86_append__self__conv, axiom,
    ((![Xs : list_type, Ys2 : list_type]: (((append_type @ Xs @ Ys2) = Xs) = (Ys2 = nil_type))))). % append_self_conv
thf(fact_87_self__append__conv, axiom,
    ((![Xs : list_dB, Ys2 : list_dB]: ((Xs = (append_dB @ Xs @ Ys2)) = (Ys2 = nil_dB))))). % self_append_conv
thf(fact_88_self__append__conv, axiom,
    ((![Xs : list_type, Ys2 : list_type]: ((Xs = (append_type @ Xs @ Ys2)) = (Ys2 = nil_type))))). % self_append_conv
thf(fact_89_append__self__conv2, axiom,
    ((![Xs : list_dB, Ys2 : list_dB]: (((append_dB @ Xs @ Ys2) = Ys2) = (Xs = nil_dB))))). % append_self_conv2
thf(fact_90_append__self__conv2, axiom,
    ((![Xs : list_type, Ys2 : list_type]: (((append_type @ Xs @ Ys2) = Ys2) = (Xs = nil_type))))). % append_self_conv2
thf(fact_91_self__append__conv2, axiom,
    ((![Ys2 : list_dB, Xs : list_dB]: ((Ys2 = (append_dB @ Xs @ Ys2)) = (Xs = nil_dB))))). % self_append_conv2
thf(fact_92_self__append__conv2, axiom,
    ((![Ys2 : list_type, Xs : list_type]: ((Ys2 = (append_type @ Xs @ Ys2)) = (Xs = nil_type))))). % self_append_conv2
thf(fact_93_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_94_Nil__is__append__conv, axiom,
    ((![Xs : list_type, Ys2 : list_type]: ((nil_type = (append_type @ Xs @ Ys2)) = (((Xs = nil_type)) & ((Ys2 = nil_type))))))). % Nil_is_append_conv
thf(fact_95_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_96_append__is__Nil__conv, axiom,
    ((![Xs : list_type, Ys2 : list_type]: (((append_type @ Xs @ Ys2) = nil_type) = (((Xs = nil_type)) & ((Ys2 = nil_type))))))). % append_is_Nil_conv
thf(fact_97_append_Oright__neutral, axiom,
    ((![A : list_dB]: ((append_dB @ A @ nil_dB) = A)))). % append.right_neutral
thf(fact_98_append_Oright__neutral, axiom,
    ((![A : list_type]: ((append_type @ A @ nil_type) = A)))). % append.right_neutral
thf(fact_99_foldl__append, axiom,
    ((![F : dB > dB > dB, A : dB, Xs : list_dB, Ys2 : list_dB]: ((foldl_dB_dB @ F @ A @ (append_dB @ Xs @ Ys2)) = (foldl_dB_dB @ F @ (foldl_dB_dB @ F @ A @ Xs) @ Ys2))))). % foldl_append
thf(fact_100_subst__eq, axiom,
    ((![K : nat, U2 : dB]: ((subst @ (var @ K) @ U2 @ K) = U2)))). % subst_eq
thf(fact_101_append1__eq__conv, axiom,
    ((![Xs : list_dB, X : dB, Ys2 : list_dB, Y : dB]: (((append_dB @ Xs @ (cons_dB @ X @ nil_dB)) = (append_dB @ Ys2 @ (cons_dB @ Y @ nil_dB))) = (((Xs = Ys2)) & ((X = Y))))))). % append1_eq_conv
thf(fact_102_append1__eq__conv, axiom,
    ((![Xs : list_type, X : type, Ys2 : list_type, Y : type]: (((append_type @ Xs @ (cons_type @ X @ nil_type)) = (append_type @ Ys2 @ (cons_type @ Y @ nil_type))) = (((Xs = Ys2)) & ((X = Y))))))). % append1_eq_conv
thf(fact_103_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_104_append__eq__append__conv2, axiom,
    ((![Xs : list_dB, Ys2 : list_dB, Zs2 : list_dB, Ts : list_dB]: (((append_dB @ Xs @ Ys2) = (append_dB @ Zs2 @ Ts)) = (?[Us : list_dB]: (((((Xs = (append_dB @ Zs2 @ Us))) & (((append_dB @ Us @ Ys2) = Ts)))) | (((((append_dB @ Xs @ Us) = Zs2)) & ((Ys2 = (append_dB @ Us @ Ts))))))))))). % append_eq_append_conv2
thf(fact_105_append__eq__append__conv2, axiom,
    ((![Xs : list_type, Ys2 : list_type, Zs2 : list_type, Ts : list_type]: (((append_type @ Xs @ Ys2) = (append_type @ Zs2 @ Ts)) = (?[Us : list_type]: (((((Xs = (append_type @ Zs2 @ Us))) & (((append_type @ Us @ Ys2) = Ts)))) | (((((append_type @ Xs @ Us) = Zs2)) & ((Ys2 = (append_type @ Us @ Ts))))))))))). % append_eq_append_conv2
thf(fact_106_append__eq__appendI, axiom,
    ((![Xs : list_dB, Xs1 : list_dB, Zs2 : list_dB, Ys2 : list_dB, Us2 : list_dB]: (((append_dB @ Xs @ Xs1) = Zs2) => ((Ys2 = (append_dB @ Xs1 @ Us2)) => ((append_dB @ Xs @ Ys2) = (append_dB @ Zs2 @ Us2))))))). % append_eq_appendI
thf(fact_107_append__eq__appendI, axiom,
    ((![Xs : list_type, Xs1 : list_type, Zs2 : list_type, Ys2 : list_type, Us2 : list_type]: (((append_type @ Xs @ Xs1) = Zs2) => ((Ys2 = (append_type @ Xs1 @ Us2)) => ((append_type @ Xs @ Ys2) = (append_type @ Zs2 @ Us2))))))). % append_eq_appendI
thf(fact_108_type__induct, axiom,
    ((![P : type > $o, T : type]: ((![T3 : type]: ((![T12 : type, T23 : type]: ((T3 = (fun @ T12 @ T23)) => (P @ T12))) => ((![T12 : type, T23 : type]: ((T3 = (fun @ T12 @ T23)) => (P @ T23))) => (P @ T3)))) => (P @ T))))). % type_induct
thf(fact_109_append__Cons, axiom,
    ((![X : dB, Xs : list_dB, Ys2 : list_dB]: ((append_dB @ (cons_dB @ X @ Xs) @ Ys2) = (cons_dB @ X @ (append_dB @ Xs @ Ys2)))))). % append_Cons
thf(fact_110_append__Cons, axiom,
    ((![X : type, Xs : list_type, Ys2 : list_type]: ((append_type @ (cons_type @ X @ Xs) @ Ys2) = (cons_type @ X @ (append_type @ Xs @ Ys2)))))). % append_Cons
thf(fact_111_Cons__eq__appendI, axiom,
    ((![X : dB, Xs1 : list_dB, Ys2 : list_dB, Xs : list_dB, Zs2 : list_dB]: (((cons_dB @ X @ Xs1) = Ys2) => ((Xs = (append_dB @ Xs1 @ Zs2)) => ((cons_dB @ X @ Xs) = (append_dB @ Ys2 @ Zs2))))))). % Cons_eq_appendI
thf(fact_112_Cons__eq__appendI, axiom,
    ((![X : type, Xs1 : list_type, Ys2 : list_type, Xs : list_type, Zs2 : list_type]: (((cons_type @ X @ Xs1) = Ys2) => ((Xs = (append_type @ Xs1 @ Zs2)) => ((cons_type @ X @ Xs) = (append_type @ Ys2 @ Zs2))))))). % Cons_eq_appendI
thf(fact_113_append_Oleft__neutral, axiom,
    ((![A : list_dB]: ((append_dB @ nil_dB @ A) = A)))). % append.left_neutral
thf(fact_114_append_Oleft__neutral, axiom,
    ((![A : list_type]: ((append_type @ nil_type @ A) = A)))). % append.left_neutral
thf(fact_115_append__Nil, axiom,
    ((![Ys2 : list_dB]: ((append_dB @ nil_dB @ Ys2) = Ys2)))). % append_Nil
thf(fact_116_append__Nil, axiom,
    ((![Ys2 : list_type]: ((append_type @ nil_type @ 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_eq__Nil__appendI, axiom,
    ((![Xs : list_type, Ys2 : list_type]: ((Xs = Ys2) => (Xs = (append_type @ nil_type @ Ys2)))))). % eq_Nil_appendI
thf(fact_119_dB_Odistinct_I5_J, axiom,
    ((![X21 : dB, X22 : dB, X32 : dB]: (~ (((app @ X21 @ X22) = (abs @ X32))))))). % dB.distinct(5)
thf(fact_120_dB_Odistinct_I3_J, axiom,
    ((![X1 : nat, X32 : dB]: (~ (((var @ X1) = (abs @ X32))))))). % dB.distinct(3)
thf(fact_121_subst__App, axiom,
    ((![T2 : dB, U2 : dB, S : dB, K : nat]: ((subst @ (app @ T2 @ U2) @ S @ K) = (app @ (subst @ T2 @ S @ K) @ (subst @ U2 @ S @ K)))))). % subst_App
thf(fact_122_rev__exhaust2, axiom,
    ((![Xs : list_dB]: ((~ ((Xs = nil_dB))) => (~ ((![Ys3 : list_dB, Y3 : dB]: (~ ((Xs = (append_dB @ Ys3 @ (cons_dB @ Y3 @ nil_dB)))))))))))). % rev_exhaust2
thf(fact_123_rev__exhaust2, axiom,
    ((![Xs : list_type]: ((~ ((Xs = nil_type))) => (~ ((![Ys3 : list_type, Y3 : type]: (~ ((Xs = (append_type @ Ys3 @ (cons_type @ Y3 @ nil_type)))))))))))). % rev_exhaust2
thf(fact_124_rev__induct, axiom,
    ((![P : list_dB > $o, Xs : list_dB]: ((P @ nil_dB) => ((![X3 : dB, Xs2 : list_dB]: ((P @ Xs2) => (P @ (append_dB @ Xs2 @ (cons_dB @ X3 @ nil_dB))))) => (P @ Xs)))))). % rev_induct
thf(fact_125_rev__induct, axiom,
    ((![P : list_type > $o, Xs : list_type]: ((P @ nil_type) => ((![X3 : type, Xs2 : list_type]: ((P @ Xs2) => (P @ (append_type @ Xs2 @ (cons_type @ X3 @ nil_type))))) => (P @ Xs)))))). % rev_induct
thf(fact_126_rev__exhaust, axiom,
    ((![Xs : list_dB]: ((~ ((Xs = nil_dB))) => (~ ((![Ys3 : list_dB, Y3 : dB]: (~ ((Xs = (append_dB @ Ys3 @ (cons_dB @ Y3 @ nil_dB)))))))))))). % rev_exhaust
thf(fact_127_rev__exhaust, axiom,
    ((![Xs : list_type]: ((~ ((Xs = nil_type))) => (~ ((![Ys3 : list_type, Y3 : type]: (~ ((Xs = (append_type @ Ys3 @ (cons_type @ Y3 @ nil_type)))))))))))). % rev_exhaust
thf(fact_128_Cons__eq__append__conv, axiom,
    ((![X : dB, Xs : list_dB, Ys2 : list_dB, Zs2 : list_dB]: (((cons_dB @ X @ Xs) = (append_dB @ Ys2 @ Zs2)) = (((((Ys2 = nil_dB)) & (((cons_dB @ X @ Xs) = Zs2)))) | ((?[Ys4 : list_dB]: ((((cons_dB @ X @ Ys4) = Ys2)) & ((Xs = (append_dB @ Ys4 @ Zs2))))))))))). % Cons_eq_append_conv
thf(fact_129_Cons__eq__append__conv, axiom,
    ((![X : type, Xs : list_type, Ys2 : list_type, Zs2 : list_type]: (((cons_type @ X @ Xs) = (append_type @ Ys2 @ Zs2)) = (((((Ys2 = nil_type)) & (((cons_type @ X @ Xs) = Zs2)))) | ((?[Ys4 : list_type]: ((((cons_type @ X @ Ys4) = Ys2)) & ((Xs = (append_type @ Ys4 @ Zs2))))))))))). % Cons_eq_append_conv
thf(fact_130_append__eq__Cons__conv, axiom,
    ((![Ys2 : list_dB, Zs2 : list_dB, X : dB, Xs : list_dB]: (((append_dB @ Ys2 @ Zs2) = (cons_dB @ X @ Xs)) = (((((Ys2 = nil_dB)) & ((Zs2 = (cons_dB @ X @ Xs))))) | ((?[Ys4 : list_dB]: (((Ys2 = (cons_dB @ X @ Ys4))) & (((append_dB @ Ys4 @ Zs2) = Xs)))))))))). % append_eq_Cons_conv
thf(fact_131_append__eq__Cons__conv, axiom,
    ((![Ys2 : list_type, Zs2 : list_type, X : type, Xs : list_type]: (((append_type @ Ys2 @ Zs2) = (cons_type @ X @ Xs)) = (((((Ys2 = nil_type)) & ((Zs2 = (cons_type @ X @ Xs))))) | ((?[Ys4 : list_type]: (((Ys2 = (cons_type @ X @ Ys4))) & (((append_type @ Ys4 @ Zs2) = Xs)))))))))). % append_eq_Cons_conv
thf(fact_132_rev__nonempty__induct, axiom,
    ((![Xs : list_dB, P : list_dB > $o]: ((~ ((Xs = nil_dB))) => ((![X3 : dB]: (P @ (cons_dB @ X3 @ nil_dB))) => ((![X3 : dB, Xs2 : list_dB]: ((~ ((Xs2 = nil_dB))) => ((P @ Xs2) => (P @ (append_dB @ Xs2 @ (cons_dB @ X3 @ nil_dB)))))) => (P @ Xs))))))). % rev_nonempty_induct
thf(fact_133_rev__nonempty__induct, axiom,
    ((![Xs : list_type, P : list_type > $o]: ((~ ((Xs = nil_type))) => ((![X3 : type]: (P @ (cons_type @ X3 @ nil_type))) => ((![X3 : type, Xs2 : list_type]: ((~ ((Xs2 = nil_type))) => ((P @ Xs2) => (P @ (append_type @ Xs2 @ (cons_type @ X3 @ nil_type)))))) => (P @ Xs))))))). % rev_nonempty_induct
thf(fact_134_dB_Oexhaust, axiom,
    ((![Y : dB]: ((![X12 : nat]: (~ ((Y = (var @ X12))))) => ((![X212 : dB, X222 : dB]: (~ ((Y = (app @ X212 @ X222))))) => (~ ((![X33 : dB]: (~ ((Y = (abs @ X33)))))))))))). % dB.exhaust
thf(fact_135_dB_Oinduct, axiom,
    ((![P : dB > $o, DB : dB]: ((![X3 : nat]: (P @ (var @ X3))) => ((![X1a : dB, X2 : dB]: ((P @ X1a) => ((P @ X2) => (P @ (app @ X1a @ X2))))) => ((![X3 : dB]: ((P @ X3) => (P @ (abs @ X3)))) => (P @ DB))))))). % dB.induct
thf(fact_136_subst__Var__IT, axiom,
    ((![R : dB, I : nat, J : nat]: ((it @ R) => (it @ (subst @ R @ (var @ I) @ J)))))). % subst_Var_IT
thf(fact_137_subst__lemma, axiom,
    ((![E : nat > type, T2 : dB, T : type, E2 : nat > type, U2 : dB, U : type, I : nat]: ((typing @ E @ T2 @ T) => ((typing @ E2 @ U2 @ U) => ((E = (shift_type @ E2 @ I @ U)) => (typing @ E2 @ (subst @ T2 @ U2 @ I) @ T))))))). % subst_lemma
thf(fact_138_App, axiom,
    ((![Env : nat > type, S : dB, T : type, U : type, T2 : dB]: ((typing @ Env @ S @ (fun @ T @ U)) => ((typing @ Env @ T2 @ T) => (typing @ Env @ (app @ S @ T2) @ U)))))). % App
thf(fact_139_typing__elims_I2_J, axiom,
    ((![E : nat > type, T2 : dB, U2 : dB, T : type]: ((typing @ E @ (app @ T2 @ U2) @ T) => (~ ((![T3 : type]: ((typing @ E @ T2 @ (fun @ T3 @ T)) => (~ ((typing @ E @ U2 @ T3))))))))))). % typing_elims(2)
thf(fact_140_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_141_Abs__App__neq__Var__apps, axiom,
    ((![S : dB, T2 : dB, N : nat, Ss : list_dB]: (~ (((app @ (abs @ S) @ T2) = (foldl_dB_dB @ app @ (var @ N) @ Ss))))))). % Abs_App_neq_Var_apps
thf(fact_142_ex__head__tail, axiom,
    ((![T2 : dB]: (?[Ts2 : list_dB, H : dB]: ((T2 = (foldl_dB_dB @ app @ H @ Ts2)) & ((?[N2 : nat]: (H = (var @ N2))) | (?[U3 : dB]: (H = (abs @ U3))))))))). % ex_head_tail
thf(fact_143_Lambda, axiom,
    ((![R : dB]: ((it @ R) => (it @ (abs @ R)))))). % Lambda
thf(fact_144_typing_Oinducts, axiom,
    ((![X1 : nat > type, X23 : dB, X32 : type, P : (nat > type) > dB > type > $o]: ((typing @ X1 @ X23 @ X32) => ((![Env2 : nat > type, X3 : nat, T3 : type]: (((Env2 @ X3) = T3) => (P @ Env2 @ (var @ X3) @ T3))) => ((![Env2 : nat > type, T3 : type, T4 : dB, U4 : type]: ((typing @ (shift_type @ Env2 @ zero_zero_nat @ T3) @ T4 @ U4) => ((P @ (shift_type @ Env2 @ zero_zero_nat @ T3) @ T4 @ U4) => (P @ Env2 @ (abs @ T4) @ (fun @ T3 @ U4))))) => ((![Env2 : nat > type, S2 : dB, T3 : type, U4 : type, T4 : dB]: ((typing @ Env2 @ S2 @ (fun @ T3 @ U4)) => ((P @ Env2 @ S2 @ (fun @ T3 @ U4)) => ((typing @ Env2 @ T4 @ T3) => ((P @ Env2 @ T4 @ T3) => (P @ Env2 @ (app @ S2 @ T4) @ U4)))))) => (P @ X1 @ X23 @ X32)))))))). % typing.inducts
thf(fact_145_typing_Osimps, axiom,
    ((typing = (^[A12 : nat > type]: (^[A22 : dB]: (^[A32 : type]: (((?[Env3 : nat > type]: (?[X4 : nat]: (?[T5 : type]: (((A12 = Env3)) & ((((A22 = (var @ X4))) & ((((A32 = T5)) & (((Env3 @ X4) = T5))))))))))) | ((((?[Env3 : nat > type]: (?[T5 : type]: (?[T6 : dB]: (?[U5 : type]: (((A12 = Env3)) & ((((A22 = (abs @ T6))) & ((((A32 = (fun @ T5 @ U5))) & ((typing @ (shift_type @ Env3 @ zero_zero_nat @ T5) @ T6 @ U5)))))))))))) | ((?[Env3 : nat > type]: (?[S3 : dB]: (?[T5 : type]: (?[U5 : type]: (?[T6 : dB]: (((A12 = Env3)) & ((((A22 = (app @ S3 @ T6))) & ((((A32 = U5)) & ((((typing @ Env3 @ S3 @ (fun @ T5 @ U5))) & ((typing @ Env3 @ T6 @ T5)))))))))))))))))))))))). % typing.simps
thf(fact_146_typing_Ocases, axiom,
    ((![A1 : nat > type, A2 : dB, A33 : type]: ((typing @ A1 @ A2 @ A33) => ((![X3 : nat]: ((A2 = (var @ X3)) => (~ (((A1 @ X3) = A33))))) => ((![T3 : type, T4 : dB]: ((A2 = (abs @ T4)) => (![U4 : type]: ((A33 = (fun @ T3 @ U4)) => (~ ((typing @ (shift_type @ A1 @ zero_zero_nat @ T3) @ T4 @ U4))))))) => (~ ((![S2 : dB, T3 : type, U4 : type, T4 : dB]: ((A2 = (app @ S2 @ T4)) => ((A33 = U4) => ((typing @ A1 @ S2 @ (fun @ T3 @ U4)) => (~ ((typing @ A1 @ T4 @ T3))))))))))))))). % typing.cases
thf(fact_147_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_148_product__lists_Osimps_I1_J, axiom,
    (((product_lists_type @ nil_list_type) = (cons_list_type @ nil_type @ nil_list_type)))). % product_lists.simps(1)
thf(fact_149_abs__typeE, axiom,
    ((![E : nat > type, T2 : dB, T : type]: ((typing @ E @ (abs @ T2) @ T) => (~ ((![U4 : type, V : type]: (~ ((typing @ (shift_type @ E @ zero_zero_nat @ U4) @ T2 @ V)))))))))). % abs_typeE
thf(fact_150_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_151_typing__elims_I3_J, axiom,
    ((![E : nat > type, T2 : dB, T : type]: ((typing @ E @ (abs @ T2) @ T) => (~ ((![T3 : type, U4 : type]: ((T = (fun @ T3 @ U4)) => (~ ((typing @ (shift_type @ E @ zero_zero_nat @ T3) @ T2 @ U4))))))))))). % typing_elims(3)
thf(fact_152_Abs, axiom,
    ((![Env : nat > type, T : type, T2 : dB, U : type]: ((typing @ (shift_type @ Env @ zero_zero_nat @ T) @ T2 @ U) => (typing @ Env @ (abs @ T2) @ (fun @ T @ U)))))). % Abs
thf(fact_153_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_154_n__lists__Nil, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((n_lists_type @ N @ nil_type) = (cons_list_type @ nil_type @ nil_list_type))) & ((~ ((N = zero_zero_nat))) => ((n_lists_type @ N @ nil_type) = nil_list_type)))))). % n_lists_Nil
thf(fact_155_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_156_n__lists_Osimps_I1_J, axiom,
    ((![Xs : list_type]: ((n_lists_type @ zero_zero_nat @ Xs) = (cons_list_type @ nil_type @ nil_list_type))))). % n_lists.simps(1)
thf(fact_157_IT_Oinducts, axiom,
    ((![X : dB, P : dB > $o]: ((it @ X) => ((![Rs2 : list_dB, N2 : nat]: ((listsp_dB @ (^[X4 : dB]: (((it @ X4)) & ((P @ X4)))) @ Rs2) => (P @ (foldl_dB_dB @ app @ (var @ N2) @ Rs2)))) => ((![R2 : dB]: ((it @ R2) => ((P @ R2) => (P @ (abs @ R2))))) => ((![R2 : dB, S2 : dB, Ss3 : list_dB]: ((it @ (foldl_dB_dB @ app @ (subst @ R2 @ S2 @ zero_zero_nat) @ Ss3)) => ((P @ (foldl_dB_dB @ app @ (subst @ R2 @ S2 @ zero_zero_nat) @ Ss3)) => ((it @ S2) => ((P @ S2) => (P @ (foldl_dB_dB @ app @ (app @ (abs @ R2) @ S2) @ Ss3))))))) => (P @ X)))))))). % IT.inducts
thf(fact_158_listsp__conj__eq, axiom,
    ((![A4 : dB > $o, B2 : dB > $o]: ((listsp_dB @ (^[X4 : dB]: (((A4 @ X4)) & ((B2 @ X4))))) = (^[X4 : list_dB]: (((listsp_dB @ A4 @ X4)) & ((listsp_dB @ B2 @ X4)))))))). % listsp_conj_eq
thf(fact_159_listsp__simps_I1_J, axiom,
    ((![A4 : dB > $o]: (listsp_dB @ A4 @ nil_dB)))). % listsp_simps(1)
thf(fact_160_listsp__simps_I1_J, axiom,
    ((![A4 : type > $o]: (listsp_type @ A4 @ nil_type)))). % listsp_simps(1)
thf(fact_161_append__in__listsp__conv, axiom,
    ((![A4 : type > $o, Xs : list_type, Ys2 : list_type]: ((listsp_type @ A4 @ (append_type @ Xs @ Ys2)) = (((listsp_type @ A4 @ Xs)) & ((listsp_type @ A4 @ Ys2))))))). % append_in_listsp_conv
thf(fact_162_append__in__listsp__conv, axiom,
    ((![A4 : dB > $o, Xs : list_dB, Ys2 : list_dB]: ((listsp_dB @ A4 @ (append_dB @ Xs @ Ys2)) = (((listsp_dB @ A4 @ Xs)) & ((listsp_dB @ A4 @ Ys2))))))). % append_in_listsp_conv
thf(fact_163_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_164_listsp_OCons, axiom,
    ((![A4 : type > $o, A : type, L : list_type]: ((A4 @ A) => ((listsp_type @ A4 @ L) => (listsp_type @ A4 @ (cons_type @ A @ L))))))). % listsp.Cons
thf(fact_165_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_166_listspE, axiom,
    ((![A4 : type > $o, X : type, L : list_type]: ((listsp_type @ A4 @ (cons_type @ X @ L)) => (~ (((A4 @ X) => (~ ((listsp_type @ A4 @ L)))))))))). % listspE
thf(fact_167_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_168_listsp__simps_I2_J, axiom,
    ((![A4 : type > $o, X : type, Xs : list_type]: ((listsp_type @ A4 @ (cons_type @ X @ Xs)) = (((A4 @ X)) & ((listsp_type @ A4 @ Xs))))))). % listsp_simps(2)
thf(fact_169_listsp_ONil, axiom,
    ((![A4 : dB > $o]: (listsp_dB @ A4 @ nil_dB)))). % listsp.Nil
thf(fact_170_listsp_ONil, axiom,
    ((![A4 : type > $o]: (listsp_type @ A4 @ nil_type)))). % listsp.Nil
thf(fact_171_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_172_listsp_Ocases, axiom,
    ((![A4 : type > $o, A : list_type]: ((listsp_type @ A4 @ A) => ((~ ((A = nil_type))) => (~ ((![A3 : type, L2 : list_type]: ((A = (cons_type @ A3 @ L2)) => ((A4 @ A3) => (~ ((listsp_type @ A4 @ L2))))))))))))). % listsp.cases
thf(fact_173_listsp_Osimps, axiom,
    ((listsp_dB = (^[A5 : dB > $o]: (^[A6 : list_dB]: (((A6 = nil_dB)) | ((?[B3 : dB]: (?[L3 : list_dB]: (((A6 = (cons_dB @ B3 @ L3))) & ((((A5 @ B3)) & ((listsp_dB @ A5 @ L3)))))))))))))). % listsp.simps
thf(fact_174_listsp_Osimps, axiom,
    ((listsp_type = (^[A5 : type > $o]: (^[A6 : list_type]: (((A6 = nil_type)) | ((?[B3 : type]: (?[L3 : list_type]: (((A6 = (cons_type @ B3 @ L3))) & ((((A5 @ B3)) & ((listsp_type @ A5 @ L3)))))))))))))). % listsp.simps
thf(fact_175_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_176_listsp_Oinducts, axiom,
    ((![A4 : type > $o, X : list_type, P : list_type > $o]: ((listsp_type @ A4 @ X) => ((P @ nil_type) => ((![A3 : type, L2 : list_type]: ((A4 @ A3) => ((listsp_type @ A4 @ L2) => ((P @ L2) => (P @ (cons_type @ A3 @ L2)))))) => (P @ X))))))). % listsp.inducts
thf(fact_177_IT_OVar, axiom,
    ((![Rs : list_dB, N : nat]: ((listsp_dB @ it @ Rs) => (it @ (foldl_dB_dB @ app @ (var @ N) @ Rs)))))). % IT.Var
thf(fact_178_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_179_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_180_types__snoc, axiom,
    ((![E : nat > type, Ts : list_dB, Ts3 : list_type, T2 : dB, T : type]: ((typings @ E @ Ts @ Ts3) => ((typing @ E @ T2 @ T) => (typings @ E @ (append_dB @ Ts @ (cons_dB @ T2 @ nil_dB)) @ (append_type @ Ts3 @ (cons_type @ T @ nil_type)))))))). % types_snoc
thf(fact_181_typings_Osimps_I1_J, axiom,
    ((![E : nat > type, Ts3 : list_type]: ((typings @ E @ nil_dB @ Ts3) = (Ts3 = nil_type))))). % typings.simps(1)
thf(fact_182_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_183_types__snoc__eq, axiom,
    ((![E : nat > type, Ts : list_dB, T2 : dB, Ts3 : list_type, T : type]: ((typings @ E @ (append_dB @ Ts @ (cons_dB @ T2 @ nil_dB)) @ (append_type @ Ts3 @ (cons_type @ T @ nil_type))) = (((typings @ E @ Ts @ Ts3)) & ((typing @ E @ T2 @ T))))))). % types_snoc_eq
thf(fact_184_types__snocE, axiom,
    ((![E : nat > type, Ts : list_dB, T2 : dB, Ts3 : list_type]: ((typings @ E @ (append_dB @ Ts @ (cons_dB @ T2 @ nil_dB)) @ Ts3) => (~ ((![Us3 : list_type, U4 : type]: ((Ts3 = (append_type @ Us3 @ (cons_type @ U4 @ nil_type))) => ((typings @ E @ Ts @ Us3) => (~ ((typing @ E @ T2 @ U4)))))))))))). % types_snocE
thf(fact_185_var__app__types, axiom,
    ((![E : nat > type, I : nat, Ts : list_dB, Us2 : list_dB, T : type, Ts3 : list_type, U : type]: ((typing @ E @ (foldl_dB_dB @ app @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ Us2) @ T) => ((typings @ E @ Ts @ Ts3) => ((typing @ E @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ U) => (?[Us3 : list_type]: ((U = (foldr_type_type @ fun @ Us3 @ T)) & (typings @ E @ Us2 @ Us3))))))))). % var_app_types
thf(fact_186_var__app__typesE, axiom,
    ((![E : nat > type, I : nat, Ts : list_dB, T : type]: ((typing @ E @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ T) => (~ ((![Ts4 : list_type]: ((typing @ E @ (var @ I) @ (foldr_type_type @ fun @ Ts4 @ T)) => (~ ((typings @ E @ Ts @ Ts4))))))))))). % var_app_typesE
thf(fact_187_foldr__append, axiom,
    ((![F : type > type > type, Xs : list_type, Ys2 : list_type, A : type]: ((foldr_type_type @ F @ (append_type @ Xs @ Ys2) @ A) = (foldr_type_type @ F @ Xs @ (foldr_type_type @ F @ Ys2 @ A)))))). % foldr_append
thf(fact_188_list__app__typeI, axiom,
    ((![E : nat > type, T2 : dB, Ts3 : list_type, T : type, Ts : list_dB]: ((typing @ E @ T2 @ (foldr_type_type @ fun @ Ts3 @ T)) => ((typings @ E @ Ts @ Ts3) => (typing @ E @ (foldl_dB_dB @ app @ T2 @ Ts) @ T)))))). % list_app_typeI
thf(fact_189_list__app__typeE, axiom,
    ((![E : nat > type, T2 : dB, Ts : list_dB, T : type]: ((typing @ E @ (foldl_dB_dB @ app @ T2 @ Ts) @ T) => (~ ((![Ts4 : list_type]: ((typing @ E @ T2 @ (foldr_type_type @ fun @ Ts4 @ T)) => (~ ((typings @ E @ Ts @ Ts4))))))))))). % list_app_typeE
thf(fact_190_list__app__typeD, axiom,
    ((![E : nat > type, T2 : dB, Ts : list_dB, T : type]: ((typing @ E @ (foldl_dB_dB @ app @ T2 @ Ts) @ T) => (?[Ts4 : list_type]: ((typing @ E @ T2 @ (foldr_type_type @ fun @ Ts4 @ T)) & (typings @ E @ Ts @ Ts4))))))). % list_app_typeD
thf(fact_191_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_192_bind__simps_I2_J, axiom,
    ((![X : dB, Xs : list_dB, F : dB > list_type]: ((bind_dB_type @ (cons_dB @ X @ Xs) @ F) = (append_type @ (F @ X) @ (bind_dB_type @ Xs @ F)))))). % bind_simps(2)
thf(fact_193_bind__simps_I2_J, axiom,
    ((![X : type, Xs : list_type, F : type > list_dB]: ((bind_type_dB @ (cons_type @ X @ Xs) @ F) = (append_dB @ (F @ X) @ (bind_type_dB @ Xs @ F)))))). % bind_simps(2)
thf(fact_194_bind__simps_I2_J, axiom,
    ((![X : type, Xs : list_type, F : type > list_type]: ((bind_type_type @ (cons_type @ X @ Xs) @ F) = (append_type @ (F @ X) @ (bind_type_type @ Xs @ F)))))). % bind_simps(2)
thf(fact_195_typings_Osimps_I2_J, axiom,
    ((![E : nat > type, T2 : dB, Ts : list_dB, Ts3 : list_type]: ((typings @ E @ (cons_dB @ T2 @ Ts) @ Ts3) = (case_list_o_type @ $false @ (^[T5 : type]: (^[Ts5 : list_type]: (((typing @ E @ T2 @ T5)) & ((typings @ E @ Ts @ Ts5))))) @ Ts3))))). % typings.simps(2)
thf(fact_196_bind__simps_I1_J, axiom,
    ((![F : dB > list_dB]: ((bind_dB_dB @ nil_dB @ F) = nil_dB)))). % bind_simps(1)
thf(fact_197_bind__simps_I1_J, axiom,
    ((![F : dB > list_type]: ((bind_dB_type @ nil_dB @ F) = nil_type)))). % bind_simps(1)
thf(fact_198_bind__simps_I1_J, axiom,
    ((![F : type > list_dB]: ((bind_type_dB @ nil_type @ F) = nil_dB)))). % bind_simps(1)
thf(fact_199_bind__simps_I1_J, axiom,
    ((![F : type > list_type]: ((bind_type_type @ nil_type @ F) = nil_type)))). % bind_simps(1)
thf(fact_200_list_Ocase__distrib, axiom,
    ((![H2 : $o > $o, F1 : $o, F22 : type > list_type > $o, List : list_type]: ((H2 @ (case_list_o_type @ F1 @ F22 @ List)) = (case_list_o_type @ (H2 @ F1) @ (^[X13 : type]: (^[X24 : list_type]: (H2 @ (F22 @ X13 @ X24)))) @ List))))). % list.case_distrib
thf(fact_201_list_Osimps_I5_J, axiom,
    ((![F1 : $o, F22 : type > list_type > $o, X21 : type, X22 : list_type]: ((case_list_o_type @ F1 @ F22 @ (cons_type @ X21 @ X22)) = (F22 @ X21 @ X22))))). % list.simps(5)
thf(fact_202_list_Osimps_I4_J, axiom,
    ((![F1 : $o, F22 : type > list_type > $o]: ((case_list_o_type @ F1 @ F22 @ nil_type) = F1)))). % list.simps(4)
thf(fact_203_list_Odisc__eq__case_I1_J, axiom,
    ((![List : list_dB]: ((List = nil_dB) = (case_list_o_dB @ $true @ (^[Uu : dB]: (^[Uv : list_dB]: $false)) @ List))))). % list.disc_eq_case(1)
thf(fact_204_list_Odisc__eq__case_I1_J, axiom,
    ((![List : list_type]: ((List = nil_type) = (case_list_o_type @ $true @ (^[Uu : type]: (^[Uv : list_type]: $false)) @ List))))). % list.disc_eq_case(1)
thf(fact_205_list_Odisc__eq__case_I2_J, axiom,
    ((![List : list_dB]: ((~ ((List = nil_dB))) = (case_list_o_dB @ $false @ (^[Uu : dB]: (^[Uv : list_dB]: $true)) @ List))))). % list.disc_eq_case(2)
thf(fact_206_list_Odisc__eq__case_I2_J, axiom,
    ((![List : list_type]: ((~ ((List = nil_type))) = (case_list_o_type @ $false @ (^[Uu : type]: (^[Uv : list_type]: $true)) @ List))))). % list.disc_eq_case(2)
thf(fact_207_subseqs_Osimps_I1_J, axiom,
    (((subseqs_dB @ nil_dB) = (cons_list_dB @ nil_dB @ nil_list_dB)))). % subseqs.simps(1)
thf(fact_208_subseqs_Osimps_I1_J, axiom,
    (((subseqs_type @ nil_type) = (cons_list_type @ nil_type @ nil_list_type)))). % subseqs.simps(1)
thf(fact_209_dB_Osize__gen_I1_J, axiom,
    ((![X1 : nat]: ((size_dB @ (var @ X1)) = zero_zero_nat)))). % dB.size_gen(1)
thf(fact_210_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_211_maps__simps_I1_J, axiom,
    ((![F : dB > list_type, X : dB, Xs : list_dB]: ((maps_dB_type @ F @ (cons_dB @ X @ Xs)) = (append_type @ (F @ X) @ (maps_dB_type @ F @ Xs)))))). % maps_simps(1)
thf(fact_212_maps__simps_I1_J, axiom,
    ((![F : type > list_dB, X : type, Xs : list_type]: ((maps_type_dB @ F @ (cons_type @ X @ Xs)) = (append_dB @ (F @ X) @ (maps_type_dB @ F @ Xs)))))). % maps_simps(1)
thf(fact_213_maps__simps_I1_J, axiom,
    ((![F : type > list_type, X : type, Xs : list_type]: ((maps_type_type @ F @ (cons_type @ X @ Xs)) = (append_type @ (F @ X) @ (maps_type_type @ F @ Xs)))))). % maps_simps(1)
thf(fact_214_concat__eq__append__conv, axiom,
    ((![Xss2 : list_list_dB, Ys2 : list_dB, Zs2 : list_dB]: (((concat_dB @ Xss2) = (append_dB @ Ys2 @ Zs2)) = (((((Xss2 = nil_list_dB)) => ((((Ys2 = nil_dB)) & ((Zs2 = 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)))) & ((((Ys2 = (append_dB @ (concat_dB @ Xss1) @ Xs3))) & ((Zs2 = (append_dB @ Xs4 @ (concat_dB @ Xss22))))))))))))))))))). % concat_eq_append_conv
thf(fact_215_concat__eq__append__conv, axiom,
    ((![Xss2 : list_list_type, Ys2 : list_type, Zs2 : list_type]: (((concat_type @ Xss2) = (append_type @ Ys2 @ Zs2)) = (((((Xss2 = nil_list_type)) => ((((Ys2 = nil_type)) & ((Zs2 = nil_type)))))) & ((((~ ((Xss2 = nil_list_type)))) => ((?[Xss1 : list_list_type]: (?[Xs3 : list_type]: (?[Xs4 : list_type]: (?[Xss22 : list_list_type]: (((Xss2 = (append_list_type @ Xss1 @ (cons_list_type @ (append_type @ Xs3 @ Xs4) @ Xss22)))) & ((((Ys2 = (append_type @ (concat_type @ Xss1) @ Xs3))) & ((Zs2 = (append_type @ Xs4 @ (concat_type @ Xss22))))))))))))))))))). % concat_eq_append_conv
thf(fact_216_concat__append, axiom,
    ((![Xs : list_list_dB, Ys2 : list_list_dB]: ((concat_dB @ (append_list_dB @ Xs @ Ys2)) = (append_dB @ (concat_dB @ Xs) @ (concat_dB @ Ys2)))))). % concat_append
thf(fact_217_concat__append, axiom,
    ((![Xs : list_list_type, Ys2 : list_list_type]: ((concat_type @ (append_list_type @ Xs @ Ys2)) = (append_type @ (concat_type @ Xs) @ (concat_type @ Ys2)))))). % concat_append

% Conjectures (1)
thf(conj_0, conjecture,
    ((typing @ (shift_type @ e @ i @ t2) @ (foldl_dB_dB @ app @ (app @ (var @ n) @ a) @ as) @ t))).
