% 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_210__5211574_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:38:26.462

% Could-be-implicit typings (7)
thf(ty_n_t__List__Olist_It__LambdaType__Otype_J, type,
    list_type : $tType).
thf(ty_n_t__Set__Oset_It__LambdaType__Otype_J, type,
    set_type : $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__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 (53)
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_Obeta, type,
    beta : dB > 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_Oliftn, type,
    liftn : nat > dB > nat > dB).
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_ListOrder_Ostep1_001t__LambdaType__Otype, type,
    step1_type : (type > type > $o) > list_type > list_type > $o).
thf(sy_c_ListOrder_Ostep1_001t__Lambda__OdB, type,
    step1_dB : (dB > dB > $o) > list_dB > list_dB > $o).
thf(sy_c_List_Ocan__select_001t__Lambda__OdB, type,
    can_select_dB : (dB > $o) > set_dB > $o).
thf(sy_c_List_Ocount__list_001t__LambdaType__Otype, type,
    count_list_type : list_type > type > nat).
thf(sy_c_List_Ocount__list_001t__Lambda__OdB, type,
    count_list_dB : list_dB > dB > nat).
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_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_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_Omap_001t__LambdaType__Otype_001t__LambdaType__Otype, type,
    map_type_type : (type > type) > list_type > list_type).
thf(sy_c_List_Olist_Omap_001t__LambdaType__Otype_001t__Lambda__OdB, type,
    map_type_dB : (type > dB) > list_type > list_dB).
thf(sy_c_List_Olist_Omap_001t__Lambda__OdB_001t__LambdaType__Otype, type,
    map_dB_type : (dB > type) > list_dB > list_type).
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_Oset_001t__LambdaType__Otype, type,
    set_type2 : list_type > set_type).
thf(sy_c_List_Olist_Oset_001t__Lambda__OdB, type,
    set_dB2 : list_dB > set_dB).
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_Omap__tailrec_001t__Lambda__OdB_001t__Lambda__OdB, type,
    map_tailrec_dB_dB : (dB > dB) > list_dB > list_dB).
thf(sy_c_Nat_Osize__class_Osize_001t__Lambda__OdB, type,
    size_size_dB : dB > nat).
thf(sy_c_member_001t__LambdaType__Otype, type,
    member_type : type > set_type > $o).
thf(sy_c_member_001t__Lambda__OdB, type,
    member_dB : dB > set_dB > $o).
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_Ts____, type,
    ts : list_type).
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 (162)
thf(fact_0__092_060open_062IT_At_092_060close_062, axiom,
    ((it @ t3))). % \<open>IT t\<close>
thf(fact_1_Var_Oprems_I2_J, axiom,
    ((it @ u1))). % Var.prems(2)
thf(fact_2_False, axiom,
    ((~ ((n = i))))). % False
thf(fact_3_uIT, axiom,
    ((it @ u))). % uIT
thf(fact_4_Var__IT, axiom,
    ((![N : nat]: (it @ (var @ N))))). % Var_IT
thf(fact_5_app__Var__IT, axiom,
    ((![T : dB, I : nat]: ((it @ T) => (it @ (app @ T @ (var @ I))))))). % app_Var_IT
thf(fact_6_subst__Var__IT, axiom,
    ((![R : dB, I : nat, J : nat]: ((it @ R) => (it @ (subst @ R @ (var @ I) @ J)))))). % subst_Var_IT
thf(fact_7_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_8_subst__eq, axiom,
    ((![K : nat, U : dB]: ((subst @ (var @ K) @ U @ K) = U)))). % subst_eq
thf(fact_9_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_10_dB_Oinject_I1_J, axiom,
    ((![X1 : nat, Y1 : nat]: (((var @ X1) = (var @ Y1)) = (X1 = Y1))))). % dB.inject(1)
thf(fact_11_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_12__092_060open_062listsp_AIT_A_Imap_A_I_092_060lambda_062x_O_Ax_091u_Pi_093_J_Ars_J_092_060close_062, axiom,
    ((listsp_dB @ it @ (map_dB_dB @ (^[X : dB]: (subst @ X @ u @ i)) @ rs)))). % \<open>listsp IT (map (\<lambda>x. x[u/i]) rs)\<close>
thf(fact_13_IT_OVar, axiom,
    ((![Rs : list_dB, N : nat]: ((listsp_dB @ it @ Rs) => (it @ (foldl_dB_dB @ app @ (var @ N) @ Rs)))))). % IT.Var
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_nT, axiom,
    ((typing @ (shift_type @ e @ i @ t2) @ (foldl_dB_dB @ app @ (var @ n) @ rs) @ t))). % nT
thf(fact_17_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_18_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_19_uT, axiom,
    ((typing @ e @ u @ t2))). % uT
thf(fact_20_dB_Oinject_I3_J, axiom,
    ((![X3 : dB, Y3 : dB]: (((abs @ X3) = (abs @ Y3)) = (X3 = Y3))))). % dB.inject(3)
thf(fact_21_Var_Oprems_I3_J, axiom,
    ((typing @ e1 @ u1 @ t2))). % Var.prems(3)
thf(fact_22_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_23_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_24_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_25_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_26_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_27__092_060open_062e_092_060langle_062i_058T_092_060rangle_062_A_092_060tturnstile_062_Ars_A_058_ATs_092_060close_062, axiom,
    ((typings @ (shift_type @ e @ i @ t2) @ rs @ ts))). % \<open>e\<langle>i:T\<rangle> \<tturnstile> rs : Ts\<close>
thf(fact_28_dB_Odistinct_I5_J, axiom,
    ((![X21 : dB, X22 : dB, X3 : dB]: (~ (((app @ X21 @ X22) = (abs @ X3))))))). % dB.distinct(5)
thf(fact_29_dB_Odistinct_I3_J, axiom,
    ((![X1 : nat, X3 : dB]: (~ (((var @ X1) = (abs @ X3))))))). % dB.distinct(3)
thf(fact_30_Lambda, axiom,
    ((![R : dB]: ((it @ R) => (it @ (abs @ R)))))). % Lambda
thf(fact_31_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_32_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_33_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_34_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_35__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062Ts_O_Ae_092_060langle_062i_058T_092_060rangle_062_A_092_060tturnstile_062_Ars_A_058_ATs_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom,
    ((~ ((![Ts3 : list_type]: (~ ((typings @ (shift_type @ e @ i @ t2) @ rs @ Ts3)))))))). % \<open>\<And>thesis. (\<And>Ts. e\<langle>i:T\<rangle> \<tturnstile> rs : Ts \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_36_MI2, axiom,
    ((![T1 : type, T22 : type, T : dB, E : nat > type, I : nat, T3 : type, U : dB]: ((t2 = (fun @ T1 @ T22)) => ((it @ T) => ((typing @ (shift_type @ E @ I @ T22) @ T @ T3) => ((it @ U) => ((typing @ E @ U @ T22) => (it @ (subst @ T @ U @ I)))))))))). % MI2
thf(fact_37_MI1, axiom,
    ((![T1 : type, T22 : type, T : dB, E : nat > type, I : nat, T3 : type, U : dB]: ((t2 = (fun @ T1 @ T22)) => ((it @ T) => ((typing @ (shift_type @ E @ I @ T1) @ T @ T3) => ((it @ U) => ((typing @ E @ U @ T1) => (it @ (subst @ T @ U @ I)))))))))). % MI1
thf(fact_38__092_060open_062listsp_A_I_092_060lambda_062t_O_A_092_060exists_062T_H_O_Ae_092_060langle_062i_058T_092_060rangle_062_A_092_060turnstile_062_At_A_058_AT_H_J_Ars_092_060close_062, axiom,
    ((listsp_dB @ (^[T2 : dB]: (?[X4 : type]: (typing @ (shift_type @ e @ i @ t2) @ T2 @ X4))) @ rs))). % \<open>listsp (\<lambda>t. \<exists>T'. e\<langle>i:T\<rangle> \<turnstile> t : T') rs\<close>
thf(fact_39_listsp__simps_I1_J, axiom,
    ((![A : dB > $o]: (listsp_dB @ A @ nil_dB)))). % listsp_simps(1)
thf(fact_40_listsp__simps_I1_J, axiom,
    ((![A : type > $o]: (listsp_type @ A @ nil_type)))). % listsp_simps(1)
thf(fact_41_Nil__is__map__conv, axiom,
    ((![F : type > dB, Xs : list_type]: ((nil_dB = (map_type_dB @ F @ Xs)) = (Xs = nil_type))))). % Nil_is_map_conv
thf(fact_42_Nil__is__map__conv, axiom,
    ((![F : dB > type, Xs : list_dB]: ((nil_type = (map_dB_type @ F @ Xs)) = (Xs = nil_dB))))). % Nil_is_map_conv
thf(fact_43_Nil__is__map__conv, axiom,
    ((![F : type > type, Xs : list_type]: ((nil_type = (map_type_type @ F @ Xs)) = (Xs = nil_type))))). % Nil_is_map_conv
thf(fact_44_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_45_map__is__Nil__conv, axiom,
    ((![F : type > dB, Xs : list_type]: (((map_type_dB @ F @ Xs) = nil_dB) = (Xs = nil_type))))). % map_is_Nil_conv
thf(fact_46_map__is__Nil__conv, axiom,
    ((![F : dB > type, Xs : list_dB]: (((map_dB_type @ F @ Xs) = nil_type) = (Xs = nil_dB))))). % map_is_Nil_conv
thf(fact_47_map__is__Nil__conv, axiom,
    ((![F : type > type, Xs : list_type]: (((map_type_type @ F @ Xs) = nil_type) = (Xs = nil_type))))). % map_is_Nil_conv
thf(fact_48_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_49_list_Omap__disc__iff, axiom,
    ((![F : type > dB, A2 : list_type]: (((map_type_dB @ F @ A2) = nil_dB) = (A2 = nil_type))))). % list.map_disc_iff
thf(fact_50_list_Omap__disc__iff, axiom,
    ((![F : dB > type, A2 : list_dB]: (((map_dB_type @ F @ A2) = nil_type) = (A2 = nil_dB))))). % list.map_disc_iff
thf(fact_51_list_Omap__disc__iff, axiom,
    ((![F : type > type, A2 : list_type]: (((map_type_type @ F @ A2) = nil_type) = (A2 = nil_type))))). % list.map_disc_iff
thf(fact_52_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_53_var__app__type__eq, axiom,
    ((![E : nat > type, I : nat, Ts : list_dB, T3 : type, U3 : type]: ((typing @ E @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ T3) => ((typing @ E @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ U3) => (T3 = U3)))))). % var_app_type_eq
thf(fact_54_shift__eq, axiom,
    ((![I : nat, J : nat, E : nat > type, T3 : type]: ((I = J) => ((shift_type @ E @ I @ T3 @ J) = T3))))). % shift_eq
thf(fact_55_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_56_map__ident, axiom,
    (((map_dB_dB @ (^[X : dB]: X)) = (^[Xs2 : list_dB]: Xs2)))). % map_ident
thf(fact_57_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_58_typings_Osimps_I1_J, axiom,
    ((![E : nat > type, Ts4 : list_type]: ((typings @ E @ nil_dB @ Ts4) = (Ts4 = nil_type))))). % typings.simps(1)
thf(fact_59_type__induct, axiom,
    ((![P : type > $o, T3 : type]: ((![T4 : type]: ((![T12 : type, T23 : type]: ((T4 = (fun @ T12 @ T23)) => (P @ T12))) => ((![T12 : type, T23 : type]: ((T4 = (fun @ T12 @ T23)) => (P @ T23))) => (P @ T4)))) => (P @ T3))))). % type_induct
thf(fact_60_lists__typings, axiom,
    ((![E : nat > type, Ts : list_dB, Ts4 : list_type]: ((typings @ E @ Ts @ Ts4) => (listsp_dB @ (^[T2 : dB]: (?[X4 : type]: (typing @ E @ T2 @ X4))) @ Ts))))). % lists_typings
thf(fact_61_App, axiom,
    ((![Env : nat > type, S : dB, T3 : type, U3 : type, T : dB]: ((typing @ Env @ S @ (fun @ T3 @ U3)) => ((typing @ Env @ T @ T3) => (typing @ Env @ (app @ S @ T) @ U3)))))). % App
thf(fact_62_typing__elims_I2_J, axiom,
    ((![E : nat > type, T : dB, U : dB, T3 : type]: ((typing @ E @ (app @ T @ U) @ T3) => (~ ((![T4 : type]: ((typing @ E @ T @ (fun @ T4 @ T3)) => (~ ((typing @ E @ U @ T4))))))))))). % typing_elims(2)
thf(fact_63_list_Omap__ident, axiom,
    ((![T : list_dB]: ((map_dB_dB @ (^[X : dB]: X) @ T) = T)))). % list.map_ident
thf(fact_64_substs__lemma, axiom,
    ((![E : nat > type, U : dB, T3 : type, I : nat, Ts : list_dB, Ts4 : list_type]: ((typing @ E @ U @ T3) => ((typings @ (shift_type @ E @ I @ T3) @ Ts @ Ts4) => (typings @ E @ (map_dB_dB @ (^[T2 : dB]: (subst @ T2 @ U @ I)) @ Ts) @ Ts4)))))). % substs_lemma
thf(fact_65_list_Osimps_I8_J, axiom,
    ((![F : dB > type]: ((map_dB_type @ F @ nil_dB) = nil_type)))). % list.simps(8)
thf(fact_66_list_Osimps_I8_J, axiom,
    ((![F : type > dB]: ((map_type_dB @ F @ nil_type) = nil_dB)))). % list.simps(8)
thf(fact_67_list_Osimps_I8_J, axiom,
    ((![F : type > type]: ((map_type_type @ F @ nil_type) = nil_type)))). % list.simps(8)
thf(fact_68_list_Osimps_I8_J, axiom,
    ((![F : dB > dB]: ((map_dB_dB @ F @ nil_dB) = nil_dB)))). % list.simps(8)
thf(fact_69_foldl__Nil, axiom,
    ((![F : dB > dB > dB, A2 : dB]: ((foldl_dB_dB @ F @ A2 @ nil_dB) = A2)))). % foldl_Nil
thf(fact_70_listsp_ONil, axiom,
    ((![A : dB > $o]: (listsp_dB @ A @ nil_dB)))). % listsp.Nil
thf(fact_71_listsp_ONil, axiom,
    ((![A : type > $o]: (listsp_type @ A @ nil_type)))). % listsp.Nil
thf(fact_72_typing__elims_I1_J, axiom,
    ((![E : nat > type, I : nat, T3 : type]: ((typing @ E @ (var @ I) @ T3) => ((E @ I) = T3))))). % typing_elims(1)
thf(fact_73_typing_OVar, axiom,
    ((![Env : nat > type, X5 : nat, T3 : type]: (((Env @ X5) = T3) => (typing @ Env @ (var @ X5) @ T3))))). % typing.Var
thf(fact_74_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 @ (^[A3 : dB]: (^[X : dB]: (G @ A3 @ (F @ X)))) @ A2 @ Xs))))). % foldl_map
thf(fact_75_subst__lemma, axiom,
    ((![E : nat > type, T : dB, T3 : type, E2 : nat > type, U : dB, U3 : type, I : nat]: ((typing @ E @ T @ T3) => ((typing @ E2 @ U @ U3) => ((E = (shift_type @ E2 @ I @ U3)) => (typing @ E2 @ (subst @ T @ U @ I) @ T3))))))). % subst_lemma
thf(fact_76_IT_Oinducts, axiom,
    ((![X5 : dB, P : dB > $o]: ((it @ X5) => ((![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 @ X5)))))))). % IT.inducts
thf(fact_77_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_78_IT_Osimps, axiom,
    ((it = (^[A3 : dB]: (((?[Rs3 : list_dB]: (?[N3 : nat]: (((A3 = (foldl_dB_dB @ app @ (var @ N3) @ Rs3))) & ((listsp_dB @ it @ Rs3)))))) | ((((?[R3 : dB]: (((A3 = (abs @ R3))) & ((it @ R3))))) | ((?[R3 : dB]: (?[S3 : dB]: (?[Ss3 : list_dB]: (((A3 = (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_79_var__app__types, axiom,
    ((![E : nat > type, I : nat, Ts : list_dB, Us : list_dB, T3 : type, Ts4 : list_type, U3 : type]: ((typing @ E @ (foldl_dB_dB @ app @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ Us) @ T3) => ((typings @ E @ Ts @ Ts4) => ((typing @ E @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ U3) => (?[Us2 : list_type]: ((U3 = (foldr_type_type @ fun @ Us2 @ T3)) & (typings @ E @ Us @ Us2))))))))). % var_app_types
thf(fact_80_var__app__typesE, axiom,
    ((![E : nat > type, I : nat, Ts : list_dB, T3 : type]: ((typing @ E @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ T3) => (~ ((![Ts3 : list_type]: ((typing @ E @ (var @ I) @ (foldr_type_type @ fun @ Ts3 @ T3)) => (~ ((typings @ E @ Ts @ Ts3))))))))))). % var_app_typesE
thf(fact_81_typing_Oinducts, axiom,
    ((![X1 : nat > type, X24 : dB, X3 : type, P : (nat > type) > dB > type > $o]: ((typing @ X1 @ X24 @ X3) => ((![Env2 : nat > type, X2 : nat, T4 : type]: (((Env2 @ X2) = T4) => (P @ Env2 @ (var @ X2) @ T4))) => ((![Env2 : nat > type, T4 : type, T5 : dB, U4 : type]: ((typing @ (shift_type @ Env2 @ zero_zero_nat @ T4) @ T5 @ U4) => ((P @ (shift_type @ Env2 @ zero_zero_nat @ T4) @ T5 @ U4) => (P @ Env2 @ (abs @ T5) @ (fun @ T4 @ U4))))) => ((![Env2 : nat > type, S2 : dB, T4 : type, U4 : type, T5 : dB]: ((typing @ Env2 @ S2 @ (fun @ T4 @ U4)) => ((P @ Env2 @ S2 @ (fun @ T4 @ U4)) => ((typing @ Env2 @ T5 @ T4) => ((P @ Env2 @ T5 @ T4) => (P @ Env2 @ (app @ S2 @ T5) @ U4)))))) => (P @ X1 @ X24 @ X3)))))))). % typing.inducts
thf(fact_82_abs__typeE, axiom,
    ((![E : nat > type, T : dB, T3 : type]: ((typing @ E @ (abs @ T) @ T3) => (~ ((![U4 : type, V : type]: (~ ((typing @ (shift_type @ E @ zero_zero_nat @ U4) @ T @ V)))))))))). % abs_typeE
thf(fact_83_typing__elims_I3_J, axiom,
    ((![E : nat > type, T : dB, T3 : type]: ((typing @ E @ (abs @ T) @ T3) => (~ ((![T4 : type, U4 : type]: ((T3 = (fun @ T4 @ U4)) => (~ ((typing @ (shift_type @ E @ zero_zero_nat @ T4) @ T @ U4))))))))))). % typing_elims(3)
thf(fact_84_Abs, axiom,
    ((![Env : nat > type, T3 : type, T : dB, U3 : type]: ((typing @ (shift_type @ Env @ zero_zero_nat @ T3) @ T @ U3) => (typing @ Env @ (abs @ T) @ (fun @ T3 @ U3)))))). % Abs
thf(fact_85_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_86_list__app__typeI, axiom,
    ((![E : nat > type, T : dB, Ts4 : list_type, T3 : type, Ts : list_dB]: ((typing @ E @ T @ (foldr_type_type @ fun @ Ts4 @ T3)) => ((typings @ E @ Ts @ Ts4) => (typing @ E @ (foldl_dB_dB @ app @ T @ Ts) @ T3)))))). % list_app_typeI
thf(fact_87_list__app__typeE, axiom,
    ((![E : nat > type, T : dB, Ts : list_dB, T3 : type]: ((typing @ E @ (foldl_dB_dB @ app @ T @ Ts) @ T3) => (~ ((![Ts3 : list_type]: ((typing @ E @ T @ (foldr_type_type @ fun @ Ts3 @ T3)) => (~ ((typings @ E @ Ts @ Ts3))))))))))). % list_app_typeE
thf(fact_88_list__app__typeD, axiom,
    ((![E : nat > type, T : dB, Ts : list_dB, T3 : type]: ((typing @ E @ (foldl_dB_dB @ app @ T @ Ts) @ T3) => (?[Ts3 : list_type]: ((typing @ E @ T @ (foldr_type_type @ fun @ Ts3 @ T3)) & (typings @ E @ Ts @ Ts3))))))). % list_app_typeD
thf(fact_89_typing_Ocases, axiom,
    ((![A1 : nat > type, A22 : dB, A32 : type]: ((typing @ A1 @ A22 @ A32) => ((![X2 : nat]: ((A22 = (var @ X2)) => (~ (((A1 @ X2) = A32))))) => ((![T4 : type, T5 : dB]: ((A22 = (abs @ T5)) => (![U4 : type]: ((A32 = (fun @ T4 @ U4)) => (~ ((typing @ (shift_type @ A1 @ zero_zero_nat @ T4) @ T5 @ U4))))))) => (~ ((![S2 : dB, T4 : type, U4 : type, T5 : dB]: ((A22 = (app @ S2 @ T5)) => ((A32 = U4) => ((typing @ A1 @ S2 @ (fun @ T4 @ U4)) => (~ ((typing @ A1 @ T5 @ T4))))))))))))))). % typing.cases
thf(fact_90_typing_Osimps, axiom,
    ((typing = (^[A12 : nat > type]: (^[A23 : dB]: (^[A33 : type]: (((?[Env3 : nat > type]: (?[X : nat]: (?[T6 : type]: (((A12 = Env3)) & ((((A23 = (var @ X))) & ((((A33 = T6)) & (((Env3 @ X) = T6))))))))))) | ((((?[Env3 : nat > type]: (?[T6 : type]: (?[T2 : dB]: (?[U5 : type]: (((A12 = Env3)) & ((((A23 = (abs @ T2))) & ((((A33 = (fun @ T6 @ U5))) & ((typing @ (shift_type @ Env3 @ zero_zero_nat @ T6) @ T2 @ U5)))))))))))) | ((?[Env3 : nat > type]: (?[S3 : dB]: (?[T6 : type]: (?[U5 : type]: (?[T2 : dB]: (((A12 = Env3)) & ((((A23 = (app @ S3 @ T2))) & ((((A33 = U5)) & ((((typing @ Env3 @ S3 @ (fun @ T6 @ U5))) & ((typing @ Env3 @ T2 @ T6)))))))))))))))))))))))). % typing.simps
thf(fact_91_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_92_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_93_lift__types, axiom,
    ((![E : nat > type, Ts : list_dB, Ts4 : list_type, I : nat, U3 : type]: ((typings @ E @ Ts @ Ts4) => (typings @ (shift_type @ E @ I @ U3) @ (map_dB_dB @ (^[T2 : dB]: (lift @ T2 @ I)) @ Ts) @ Ts4))))). % lift_types
thf(fact_94_subst__lift, axiom,
    ((![T : dB, K : nat, S : dB]: ((subst @ (lift @ T @ K) @ S @ K) = T)))). % subst_lift
thf(fact_95_lift__IT, axiom,
    ((![T : dB, I : nat]: ((it @ T) => (it @ (lift @ T @ I)))))). % lift_IT
thf(fact_96_lift__type, axiom,
    ((![E : nat > type, T : dB, T3 : type, I : nat, U3 : type]: ((typing @ E @ T @ T3) => (typing @ (shift_type @ E @ I @ U3) @ (lift @ T @ I) @ T3))))). % lift_type
thf(fact_97_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_98_zero__reorient, axiom,
    ((![X5 : nat]: ((zero_zero_nat = X5) = (X5 = zero_zero_nat))))). % zero_reorient
thf(fact_99_dB_Osize__gen_I1_J, axiom,
    ((![X1 : nat]: ((size_dB @ (var @ X1)) = zero_zero_nat)))). % dB.size_gen(1)
thf(fact_100_map__eq__map__tailrec, axiom,
    ((map_dB_dB = map_tailrec_dB_dB))). % map_eq_map_tailrec
thf(fact_101_list__ex1__simps_I1_J, axiom,
    ((![P : dB > $o]: (~ ((list_ex1_dB @ P @ nil_dB)))))). % list_ex1_simps(1)
thf(fact_102_list__ex1__simps_I1_J, axiom,
    ((![P : type > $o]: (~ ((list_ex1_type @ P @ nil_type)))))). % list_ex1_simps(1)
thf(fact_103_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_104_Apps__dB__induct, axiom,
    ((![P : dB > $o, T : dB]: ((![N2 : nat, Ts2 : list_dB]: ((![X6 : dB]: ((member_dB @ X6 @ (set_dB2 @ Ts2)) => (P @ X6))) => (P @ (foldl_dB_dB @ app @ (var @ N2) @ Ts2)))) => ((![U2 : dB]: ((P @ U2) => (![Ts2 : list_dB]: ((![X6 : dB]: ((member_dB @ X6 @ (set_dB2 @ Ts2)) => (P @ X6))) => (P @ (foldl_dB_dB @ app @ (abs @ U2) @ Ts2)))))) => (P @ T)))))). % Apps_dB_induct
thf(fact_105_beta_Oinducts, axiom,
    ((![X1 : dB, X24 : dB, P : dB > dB > $o]: ((beta @ X1 @ X24) => ((![S2 : dB, T5 : dB]: (P @ (app @ (abs @ S2) @ T5) @ (subst @ S2 @ T5 @ zero_zero_nat))) => ((![S2 : dB, T5 : dB, U2 : dB]: ((beta @ S2 @ T5) => ((P @ S2 @ T5) => (P @ (app @ S2 @ U2) @ (app @ T5 @ U2))))) => ((![S2 : dB, T5 : dB, U2 : dB]: ((beta @ S2 @ T5) => ((P @ S2 @ T5) => (P @ (app @ U2 @ S2) @ (app @ U2 @ T5))))) => ((![S2 : dB, T5 : dB]: ((beta @ S2 @ T5) => ((P @ S2 @ T5) => (P @ (abs @ S2) @ (abs @ T5))))) => (P @ X1 @ X24))))))))). % beta.inducts
thf(fact_106_map__eq__conv, axiom,
    ((![F : dB > dB, Xs : list_dB, G : dB > dB]: (((map_dB_dB @ F @ Xs) = (map_dB_dB @ G @ Xs)) = (![X : dB]: (((member_dB @ X @ (set_dB2 @ Xs))) => (((F @ X) = (G @ X))))))))). % map_eq_conv
thf(fact_107_in__listspI, axiom,
    ((![Xs : list_dB, A : dB > $o]: ((![X2 : dB]: ((member_dB @ X2 @ (set_dB2 @ Xs)) => (A @ X2))) => (listsp_dB @ A @ Xs))))). % in_listspI
thf(fact_108_list__ex1__iff, axiom,
    ((list_ex1_dB = (^[P2 : dB > $o]: (^[Xs2 : list_dB]: (?[X : dB]: (((((member_dB @ X @ (set_dB2 @ Xs2))) & ((P2 @ X)))) & ((![Y2 : dB]: (((((member_dB @ Y2 @ (set_dB2 @ Xs2))) & ((P2 @ Y2)))) => ((Y2 = X)))))))))))). % list_ex1_iff
thf(fact_109_foldr__cong, axiom,
    ((![A2 : type, B2 : type, L : list_type, K : list_type, F : type > type > type, G : type > type > type]: ((A2 = B2) => ((L = K) => ((![A4 : type, X2 : type]: ((member_type @ X2 @ (set_type2 @ L)) => ((F @ X2 @ A4) = (G @ X2 @ A4)))) => ((foldr_type_type @ F @ L @ A2) = (foldr_type_type @ G @ K @ B2)))))))). % foldr_cong
thf(fact_110_list_Oinj__map__strong, axiom,
    ((![X5 : list_dB, Xa : list_dB, F : dB > dB, Fa : dB > dB]: ((![Z : dB, Za : dB]: ((member_dB @ Z @ (set_dB2 @ X5)) => ((member_dB @ Za @ (set_dB2 @ Xa)) => (((F @ Z) = (Fa @ Za)) => (Z = Za))))) => (((map_dB_dB @ F @ X5) = (map_dB_dB @ Fa @ Xa)) => (X5 = Xa)))))). % list.inj_map_strong
thf(fact_111_list_Omap__cong0, axiom,
    ((![X5 : list_dB, F : dB > dB, G : dB > dB]: ((![Z : dB]: ((member_dB @ Z @ (set_dB2 @ X5)) => ((F @ Z) = (G @ Z)))) => ((map_dB_dB @ F @ X5) = (map_dB_dB @ G @ X5)))))). % list.map_cong0
thf(fact_112_list_Omap__cong, axiom,
    ((![X5 : list_dB, Ya : list_dB, F : dB > dB, G : dB > dB]: ((X5 = Ya) => ((![Z : dB]: ((member_dB @ Z @ (set_dB2 @ Ya)) => ((F @ Z) = (G @ Z)))) => ((map_dB_dB @ F @ X5) = (map_dB_dB @ G @ Ya))))))). % list.map_cong
thf(fact_113_ex__map__conv, axiom,
    ((![Ys : list_dB, F : dB > dB]: ((?[Xs2 : list_dB]: (Ys = (map_dB_dB @ F @ Xs2))) = (![X : dB]: (((member_dB @ X @ (set_dB2 @ Ys))) => ((?[Y2 : dB]: (X = (F @ Y2)))))))))). % ex_map_conv
thf(fact_114_map__cong, axiom,
    ((![Xs : list_dB, Ys : list_dB, F : dB > dB, G : dB > dB]: ((Xs = Ys) => ((![X2 : dB]: ((member_dB @ X2 @ (set_dB2 @ Ys)) => ((F @ X2) = (G @ X2)))) => ((map_dB_dB @ F @ Xs) = (map_dB_dB @ G @ Ys))))))). % map_cong
thf(fact_115_map__idI, axiom,
    ((![Xs : list_dB, F : dB > dB]: ((![X2 : dB]: ((member_dB @ X2 @ (set_dB2 @ Xs)) => ((F @ X2) = X2))) => ((map_dB_dB @ F @ Xs) = Xs))))). % map_idI
thf(fact_116_map__ext, axiom,
    ((![Xs : list_dB, F : dB > dB, G : dB > dB]: ((![X2 : dB]: ((member_dB @ X2 @ (set_dB2 @ Xs)) => ((F @ X2) = (G @ X2)))) => ((map_dB_dB @ F @ Xs) = (map_dB_dB @ G @ Xs)))))). % map_ext
thf(fact_117_appR, axiom,
    ((![S : dB, T : dB, U : dB]: ((beta @ S @ T) => (beta @ (app @ U @ S) @ (app @ U @ T)))))). % appR
thf(fact_118_appL, axiom,
    ((![S : dB, T : dB, U : dB]: ((beta @ S @ T) => (beta @ (app @ S @ U) @ (app @ T @ U)))))). % appL
thf(fact_119_foldl__cong, axiom,
    ((![A2 : dB, B2 : dB, L : list_dB, K : list_dB, F : dB > dB > dB, G : dB > dB > dB]: ((A2 = B2) => ((L = K) => ((![A4 : dB, X2 : dB]: ((member_dB @ X2 @ (set_dB2 @ L)) => ((F @ A4 @ X2) = (G @ A4 @ X2)))) => ((foldl_dB_dB @ F @ A2 @ L) = (foldl_dB_dB @ G @ B2 @ K)))))))). % foldl_cong
thf(fact_120_subject__reduction, axiom,
    ((![E : nat > type, T : dB, T3 : type, T7 : dB]: ((typing @ E @ T @ T3) => ((beta @ T @ T7) => (typing @ E @ T7 @ T3)))))). % subject_reduction
thf(fact_121_in__listspD, axiom,
    ((![A : dB > $o, Xs : list_dB]: ((listsp_dB @ A @ Xs) => (![X6 : dB]: ((member_dB @ X6 @ (set_dB2 @ Xs)) => (A @ X6))))))). % in_listspD
thf(fact_122_in__listsp__conv__set, axiom,
    ((listsp_dB = (^[A5 : dB > $o]: (^[Xs2 : list_dB]: (![X : dB]: (((member_dB @ X @ (set_dB2 @ Xs2))) => ((A5 @ X))))))))). % in_listsp_conv_set
thf(fact_123_beta__cases_I2_J, axiom,
    ((![R : dB, S : dB]: ((beta @ (abs @ R) @ S) => (~ ((![T5 : dB]: ((S = (abs @ T5)) => (~ ((beta @ R @ T5))))))))))). % beta_cases(2)
thf(fact_124_abs, axiom,
    ((![S : dB, T : dB]: ((beta @ S @ T) => (beta @ (abs @ S) @ (abs @ T)))))). % abs
thf(fact_125_beta__cases_I1_J, axiom,
    ((![I : nat, T : dB]: (~ ((beta @ (var @ I) @ T)))))). % beta_cases(1)
thf(fact_126_subst__preserves__beta, axiom,
    ((![R : dB, S : dB, T : dB, I : nat]: ((beta @ R @ S) => (beta @ (subst @ R @ T @ I) @ (subst @ S @ T @ I)))))). % subst_preserves_beta
thf(fact_127_lift__preserves__beta, axiom,
    ((![R : dB, S : dB, I : nat]: ((beta @ R @ S) => (beta @ (lift @ R @ I) @ (lift @ S @ I)))))). % lift_preserves_beta
thf(fact_128_substn_Osimps_I2_J, axiom,
    ((![T : dB, U : dB, S : dB, K : nat]: ((substn @ (app @ T @ U) @ S @ K) = (app @ (substn @ T @ S @ K) @ (substn @ U @ S @ K)))))). % substn.simps(2)
thf(fact_129_beta__cases_I3_J, axiom,
    ((![S : dB, T : dB, U : dB]: ((beta @ (app @ S @ T) @ U) => ((![S2 : dB]: ((S = (abs @ S2)) => (~ ((U = (subst @ S2 @ T @ zero_zero_nat)))))) => ((![T5 : dB]: ((U = (app @ T5 @ T)) => (~ ((beta @ S @ T5))))) => (~ ((![T5 : dB]: ((U = (app @ S @ T5)) => (~ ((beta @ T @ T5))))))))))))). % beta_cases(3)
thf(fact_130_beta, axiom,
    ((![S : dB, T : dB]: (beta @ (app @ (abs @ S) @ T) @ (subst @ S @ T @ zero_zero_nat))))). % beta
thf(fact_131_beta_Ocases, axiom,
    ((![A1 : dB, A22 : dB]: ((beta @ A1 @ A22) => ((![S2 : dB, T5 : dB]: ((A1 = (app @ (abs @ S2) @ T5)) => (~ ((A22 = (subst @ S2 @ T5 @ zero_zero_nat)))))) => ((![S2 : dB, T5 : dB, U2 : dB]: ((A1 = (app @ S2 @ U2)) => ((A22 = (app @ T5 @ U2)) => (~ ((beta @ S2 @ T5)))))) => ((![S2 : dB, T5 : dB, U2 : dB]: ((A1 = (app @ U2 @ S2)) => ((A22 = (app @ U2 @ T5)) => (~ ((beta @ S2 @ T5)))))) => (~ ((![S2 : dB]: ((A1 = (abs @ S2)) => (![T5 : dB]: ((A22 = (abs @ T5)) => (~ ((beta @ S2 @ T5)))))))))))))))). % beta.cases
thf(fact_132_beta_Osimps, axiom,
    ((beta = (^[A12 : dB]: (^[A23 : dB]: (((?[S3 : dB]: (?[T2 : dB]: (((A12 = (app @ (abs @ S3) @ T2))) & ((A23 = (subst @ S3 @ T2 @ zero_zero_nat))))))) | ((((?[S3 : dB]: (?[T2 : dB]: (?[U6 : dB]: (((A12 = (app @ S3 @ U6))) & ((((A23 = (app @ T2 @ U6))) & ((beta @ S3 @ T2))))))))) | ((((?[S3 : dB]: (?[T2 : dB]: (?[U6 : dB]: (((A12 = (app @ U6 @ S3))) & ((((A23 = (app @ U6 @ T2))) & ((beta @ S3 @ T2))))))))) | ((?[S3 : dB]: (?[T2 : dB]: (((A12 = (abs @ S3))) & ((((A23 = (abs @ T2))) & ((beta @ S3 @ T2)))))))))))))))))). % beta.simps
thf(fact_133_apps__preserves__beta, axiom,
    ((![R : dB, S : dB, Ss : list_dB]: ((beta @ R @ S) => (beta @ (foldl_dB_dB @ app @ R @ Ss) @ (foldl_dB_dB @ app @ S @ Ss)))))). % apps_preserves_beta
thf(fact_134_can__select__set__list__ex1, axiom,
    ((![P : dB > $o, A : list_dB]: ((can_select_dB @ P @ (set_dB2 @ A)) = (list_ex1_dB @ P @ A))))). % can_select_set_list_ex1
thf(fact_135_lem, axiom,
    ((![P : dB > $o, T : dB, N : nat]: ((![N2 : nat, Ts2 : list_dB]: ((![X6 : dB]: ((member_dB @ X6 @ (set_dB2 @ Ts2)) => (P @ X6))) => (P @ (foldl_dB_dB @ app @ (var @ N2) @ Ts2)))) => ((![U2 : dB]: ((P @ U2) => (![Ts2 : list_dB]: ((![X6 : dB]: ((member_dB @ X6 @ (set_dB2 @ Ts2)) => (P @ X6))) => (P @ (foldl_dB_dB @ app @ (abs @ U2) @ Ts2)))))) => (((size_size_dB @ T) = N) => (P @ T))))))). % lem
thf(fact_136_dB_Osize_I4_J, axiom,
    ((![X1 : nat]: ((size_size_dB @ (var @ X1)) = zero_zero_nat)))). % dB.size(4)
thf(fact_137_substn__subst__n, axiom,
    ((substn = (^[T2 : dB]: (^[S3 : dB]: (^[N3 : nat]: (subst @ T2 @ (liftn @ N3 @ S3 @ zero_zero_nat) @ N3))))))). % substn_subst_n
thf(fact_138_count__notin, axiom,
    ((![X5 : dB, Xs : list_dB]: ((~ ((member_dB @ X5 @ (set_dB2 @ Xs)))) => ((count_list_dB @ Xs @ X5) = zero_zero_nat))))). % count_notin
thf(fact_139_liftn__0, axiom,
    ((![T : dB, K : nat]: ((liftn @ zero_zero_nat @ T @ K) = T)))). % liftn_0
thf(fact_140_liftn_Osimps_I2_J, axiom,
    ((![N : nat, S : dB, T : dB, K : nat]: ((liftn @ N @ (app @ S @ T) @ K) = (app @ (liftn @ N @ S @ K) @ (liftn @ N @ T @ K)))))). % liftn.simps(2)
thf(fact_141_count__list_Osimps_I1_J, axiom,
    ((![Y : dB]: ((count_list_dB @ nil_dB @ Y) = zero_zero_nat)))). % count_list.simps(1)
thf(fact_142_count__list_Osimps_I1_J, axiom,
    ((![Y : type]: ((count_list_type @ nil_type @ Y) = zero_zero_nat)))). % count_list.simps(1)
thf(fact_143_head__Var__reduction, axiom,
    ((![N : nat, Rs : list_dB, V2 : dB]: ((beta @ (foldl_dB_dB @ app @ (var @ N) @ Rs) @ V2) => (?[Ss2 : list_dB]: ((step1_dB @ beta @ Rs @ Ss2) & (V2 = (foldl_dB_dB @ app @ (var @ N) @ Ss2)))))))). % head_Var_reduction
thf(fact_144__092_060open_062listsp_A_I_092_060lambda_062t_O_A_I_092_060forall_062e_AT_H_Au_Ai_O_Ae_092_060langle_062i_058T_092_060rangle_062_A_092_060turnstile_062_At_A_058_AT_H_A_092_060longrightarrow_062_AIT_Au_A_092_060longrightarrow_062_Ae_A_092_060turnstile_062_Au_A_058_AT_A_092_060longrightarrow_062_AIT_A_It_091u_Pi_093_J_J_A_092_060and_062_A_I_092_060exists_062T_H_O_Ae_092_060langle_062i_058T_092_060rangle_062_A_092_060turnstile_062_At_A_058_AT_H_J_J_Ars_092_060close_062, axiom,
    ((listsp_dB @ (^[T2 : dB]: (((![E3 : nat > type]: (![T8 : type]: (![U6 : dB]: (![I2 : nat]: (((typing @ (shift_type @ E3 @ I2 @ t2) @ T2 @ T8)) => ((((it @ U6)) => ((((typing @ E3 @ U6 @ t2)) => ((it @ (subst @ T2 @ U6 @ I2))))))))))))) & ((?[X4 : type]: (typing @ (shift_type @ e @ i @ t2) @ T2 @ X4))))) @ rs))). % \<open>listsp (\<lambda>t. (\<forall>e T' u i. e\<langle>i:T\<rangle> \<turnstile> t : T' \<longrightarrow> IT u \<longrightarrow> e \<turnstile> u : T \<longrightarrow> IT (t[u/i])) \<and> (\<exists>T'. e\<langle>i:T\<rangle> \<turnstile> t : T')) rs\<close>
thf(fact_145_Var_Ohyps, axiom,
    ((listsp_dB @ (^[X : dB]: (((it @ X)) & ((![Y2 : nat > type]: (![Z2 : type]: (![Aa : dB]: (![Ab : nat]: (((typing @ (shift_type @ Y2 @ Ab @ t2) @ X @ Z2)) => ((((it @ Aa)) => ((((typing @ Y2 @ Aa @ t2)) => ((it @ (subst @ X @ Aa @ Ab))))))))))))))) @ rs))). % Var.hyps
thf(fact_146__092_060open_062listsp_A_I_092_060lambda_062t_O_A_092_060forall_062e_AT_H_Au_Ai_O_Ae_092_060langle_062i_058T_092_060rangle_062_A_092_060turnstile_062_At_A_058_AT_H_A_092_060longrightarrow_062_AIT_Au_A_092_060longrightarrow_062_Ae_A_092_060turnstile_062_Au_A_058_AT_A_092_060longrightarrow_062_AIT_A_It_091u_Pi_093_J_J_Ars_092_060close_062, axiom,
    ((listsp_dB @ (^[T2 : dB]: (![E3 : nat > type]: (![T8 : type]: (![U6 : dB]: (![I2 : nat]: (((typing @ (shift_type @ E3 @ I2 @ t2) @ T2 @ T8)) => ((((it @ U6)) => ((((typing @ E3 @ U6 @ t2)) => ((it @ (subst @ T2 @ U6 @ I2))))))))))))) @ rs))). % \<open>listsp (\<lambda>t. \<forall>e T' u i. e\<langle>i:T\<rangle> \<turnstile> t : T' \<longrightarrow> IT u \<longrightarrow> e \<turnstile> u : T \<longrightarrow> IT (t[u/i])) rs\<close>
thf(fact_147_apps__preserves__betas, axiom,
    ((![Rs : list_dB, Ss : list_dB, R : dB]: ((step1_dB @ beta @ Rs @ Ss) => (beta @ (foldl_dB_dB @ app @ R @ Rs) @ (foldl_dB_dB @ app @ R @ Ss)))))). % apps_preserves_betas
thf(fact_148_apps__betasE, axiom,
    ((![R : dB, Rs : list_dB, S : dB]: ((beta @ (foldl_dB_dB @ app @ R @ Rs) @ S) => ((![R4 : dB]: ((beta @ R @ R4) => (~ ((S = (foldl_dB_dB @ app @ R4 @ Rs)))))) => ((![Rs4 : list_dB]: ((step1_dB @ beta @ Rs @ Rs4) => (~ ((S = (foldl_dB_dB @ app @ R @ Rs4)))))) => (~ ((![T5 : dB]: ((R = (abs @ T5)) => (![U2 : dB, Us3 : list_dB]: ((Rs = (cons_dB @ U2 @ Us3)) => (~ ((S = (foldl_dB_dB @ app @ (subst @ T5 @ U2 @ zero_zero_nat) @ Us3)))))))))))))))). % apps_betasE
thf(fact_149_not__Nil__step1, axiom,
    ((![R : type > type > $o, Xs : list_type]: (~ ((step1_type @ R @ nil_type @ Xs)))))). % not_Nil_step1
thf(fact_150_not__Nil__step1, axiom,
    ((![R : dB > dB > $o, Xs : list_dB]: (~ ((step1_dB @ R @ nil_dB @ Xs)))))). % not_Nil_step1
thf(fact_151_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_152_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_153_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_154_set__ConsD, axiom,
    ((![Y : dB, X5 : dB, Xs : list_dB]: ((member_dB @ Y @ (set_dB2 @ (cons_dB @ X5 @ Xs))) => ((Y = X5) | (member_dB @ Y @ (set_dB2 @ Xs))))))). % set_ConsD
thf(fact_155_list_Oset__cases, axiom,
    ((![E : dB, A2 : list_dB]: ((member_dB @ E @ (set_dB2 @ A2)) => ((![Z22 : list_dB]: (~ ((A2 = (cons_dB @ E @ Z22))))) => (~ ((![Z1 : dB, Z22 : list_dB]: ((A2 = (cons_dB @ Z1 @ Z22)) => (~ ((member_dB @ E @ (set_dB2 @ Z22))))))))))))). % list.set_cases
thf(fact_156_listsp_Ocases, axiom,
    ((![A : type > $o, A2 : list_type]: ((listsp_type @ A @ A2) => ((~ ((A2 = nil_type))) => (~ ((![A4 : type, L2 : list_type]: ((A2 = (cons_type @ A4 @ L2)) => ((A @ A4) => (~ ((listsp_type @ A @ L2))))))))))))). % listsp.cases
thf(fact_157_listsp_Ocases, axiom,
    ((![A : dB > $o, A2 : list_dB]: ((listsp_dB @ A @ A2) => ((~ ((A2 = nil_dB))) => (~ ((![A4 : dB, L2 : list_dB]: ((A2 = (cons_dB @ A4 @ L2)) => ((A @ A4) => (~ ((listsp_dB @ A @ L2))))))))))))). % listsp.cases
thf(fact_158_listsp_Osimps, axiom,
    ((listsp_type = (^[A5 : type > $o]: (^[A3 : list_type]: (((A3 = nil_type)) | ((?[B3 : type]: (?[L3 : list_type]: (((A3 = (cons_type @ B3 @ L3))) & ((((A5 @ B3)) & ((listsp_type @ A5 @ L3)))))))))))))). % listsp.simps
thf(fact_159_listsp_Osimps, axiom,
    ((listsp_dB = (^[A5 : dB > $o]: (^[A3 : list_dB]: (((A3 = nil_dB)) | ((?[B3 : dB]: (?[L3 : list_dB]: (((A3 = (cons_dB @ B3 @ L3))) & ((((A5 @ B3)) & ((listsp_dB @ A5 @ L3)))))))))))))). % listsp.simps
thf(fact_160_listsp_Oinducts, axiom,
    ((![A : type > $o, X5 : list_type, P : list_type > $o]: ((listsp_type @ A @ X5) => ((P @ nil_type) => ((![A4 : type, L2 : list_type]: ((A @ A4) => ((listsp_type @ A @ L2) => ((P @ L2) => (P @ (cons_type @ A4 @ L2)))))) => (P @ X5))))))). % listsp.inducts
thf(fact_161_listsp_Oinducts, axiom,
    ((![A : dB > $o, X5 : list_dB, P : list_dB > $o]: ((listsp_dB @ A @ X5) => ((P @ nil_dB) => ((![A4 : dB, L2 : list_dB]: ((A @ A4) => ((listsp_dB @ A @ L2) => ((P @ L2) => (P @ (cons_dB @ A4 @ L2)))))) => (P @ X5))))))). % listsp.inducts

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