% 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_208__5211556_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:38:25.439

% 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 (57)
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_Olift, type,
    lift : dB > nat > dB).
thf(sy_c_Lambda_Osubst, type,
    subst : dB > dB > nat > 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_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_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_Orec__list_001t__List__Olist_It__Lambda__OdB_J_001t__Lambda__OdB, type,
    rec_list_list_dB_dB : list_dB > (dB > list_dB > list_dB > list_dB) > list_dB > list_dB).
thf(sy_c_List_Olist__ex_001t__LambdaType__Otype, type,
    list_ex_type : (type > $o) > list_type > $o).
thf(sy_c_List_Olist__ex_001t__Lambda__OdB, type,
    list_ex_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_List_Omember_001t__LambdaType__Otype, type,
    member_type : list_type > type > $o).
thf(sy_c_List_Omember_001t__Lambda__OdB, type,
    member_dB : list_dB > dB > $o).
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_Onull_001t__LambdaType__Otype, type,
    null_type : list_type > $o).
thf(sy_c_List_Onull_001t__Lambda__OdB, type,
    null_dB : list_dB > $o).
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_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_Ua____, type,
    ua : 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 (181)
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_uIT, axiom,
    ((it @ u))). % uIT
thf(fact_3_False, axiom,
    ((~ ((n = i))))). % False
thf(fact_4__092_060open_062listsp_AIT_A_Ia_091u_Pi_093_A_D_Amap_A_I_092_060lambda_062t_O_At_091u_Pi_093_J_Aas_J_092_060close_062, axiom,
    ((listsp_dB @ it @ (cons_dB @ (subst @ a @ u @ i) @ (map_dB_dB @ (^[T : dB]: (subst @ T @ u @ i)) @ as))))). % \<open>listsp IT (a[u/i] # map (\<lambda>t. t[u/i]) as)\<close>
thf(fact_5_Cons_Ohyps_I3_J, axiom,
    ((listsp_dB @ it @ (map_dB_dB @ (^[X : dB]: (subst @ X @ u @ i)) @ as)))). % Cons.hyps(3)
thf(fact_6__092_060open_062IT_A_Ia_091u_Pi_093_J_092_060close_062, axiom,
    ((it @ (subst @ a @ u @ i)))). % \<open>IT (a[u/i])\<close>
thf(fact_7_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_8_map__ident, axiom,
    (((map_dB_dB @ (^[X : dB]: X)) = (^[Xs : list_dB]: Xs)))). % map_ident
thf(fact_9_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_10_listsp__simps_I2_J, axiom,
    ((![A : dB > $o, X2 : dB, Xs2 : list_dB]: ((listsp_dB @ A @ (cons_dB @ X2 @ Xs2)) = (((A @ X2)) & ((listsp_dB @ A @ Xs2))))))). % listsp_simps(2)
thf(fact_11_listspE, axiom,
    ((![A : dB > $o, X2 : dB, L : list_dB]: ((listsp_dB @ A @ (cons_dB @ X2 @ L)) => (~ (((A @ X2) => (~ ((listsp_dB @ A @ L)))))))))). % listspE
thf(fact_12_listsp_OCons, axiom,
    ((![A : dB > $o, A2 : dB, L : list_dB]: ((A @ A2) => ((listsp_dB @ A @ L) => (listsp_dB @ A @ (cons_dB @ A2 @ L))))))). % listsp.Cons
thf(fact_13_list_Osimps_I9_J, axiom,
    ((![F : dB > dB, X21 : dB, X22 : list_dB]: ((map_dB_dB @ F @ (cons_dB @ X21 @ X22)) = (cons_dB @ (F @ X21) @ (map_dB_dB @ F @ X22)))))). % list.simps(9)
thf(fact_14_Cons__eq__map__D, axiom,
    ((![X2 : dB, Xs2 : list_dB, F : dB > dB, Ys : list_dB]: (((cons_dB @ X2 @ Xs2) = (map_dB_dB @ F @ Ys)) => (?[Z : dB, Zs : list_dB]: ((Ys = (cons_dB @ Z @ Zs)) & ((X2 = (F @ Z)) & (Xs2 = (map_dB_dB @ F @ Zs))))))))). % Cons_eq_map_D
thf(fact_15_map__eq__Cons__D, axiom,
    ((![F : dB > dB, Xs2 : list_dB, Y : dB, Ys : list_dB]: (((map_dB_dB @ F @ Xs2) = (cons_dB @ Y @ Ys)) => (?[Z : dB, Zs : list_dB]: ((Xs2 = (cons_dB @ Z @ Zs)) & (((F @ Z) = Y) & ((map_dB_dB @ F @ Zs) = Ys)))))))). % map_eq_Cons_D
thf(fact_16_Cons__eq__map__conv, axiom,
    ((![X2 : dB, Xs2 : list_dB, F : dB > dB, Ys : list_dB]: (((cons_dB @ X2 @ Xs2) = (map_dB_dB @ F @ Ys)) = (?[Z2 : dB]: (?[Zs2 : list_dB]: (((Ys = (cons_dB @ Z2 @ Zs2))) & ((((X2 = (F @ Z2))) & ((Xs2 = (map_dB_dB @ F @ Zs2)))))))))))). % Cons_eq_map_conv
thf(fact_17_map__eq__Cons__conv, axiom,
    ((![F : dB > dB, Xs2 : list_dB, Y : dB, Ys : list_dB]: (((map_dB_dB @ F @ Xs2) = (cons_dB @ Y @ Ys)) = (?[Z2 : dB]: (?[Zs2 : list_dB]: (((Xs2 = (cons_dB @ Z2 @ Zs2))) & (((((F @ Z2) = Y)) & (((map_dB_dB @ F @ Zs2) = Ys))))))))))). % map_eq_Cons_conv
thf(fact_18_list_Omap__ident, axiom,
    ((![T2 : list_dB]: ((map_dB_dB @ (^[X : dB]: X) @ T2) = T2)))). % list.map_ident
thf(fact_19_not__Cons__self2, axiom,
    ((![X2 : dB, Xs2 : list_dB]: (~ (((cons_dB @ X2 @ Xs2) = Xs2)))))). % not_Cons_self2
thf(fact_20_map__eq__map__tailrec, axiom,
    ((map_dB_dB = map_tailrec_dB_dB))). % map_eq_map_tailrec
thf(fact_21_member__rec_I1_J, axiom,
    ((![X2 : dB, Xs2 : list_dB, Y : dB]: ((member_dB @ (cons_dB @ X2 @ Xs2) @ Y) = (((X2 = Y)) | ((member_dB @ Xs2 @ Y))))))). % member_rec(1)
thf(fact_22_null__rec_I1_J, axiom,
    ((![X2 : dB, Xs2 : list_dB]: (~ ((null_dB @ (cons_dB @ X2 @ Xs2))))))). % null_rec(1)
thf(fact_23_Var_Oprems_I3_J, axiom,
    ((typing @ e1 @ u1 @ t2))). % Var.prems(3)
thf(fact_24_subst__Var__IT, axiom,
    ((![R : dB, I : nat, J : nat]: ((it @ R) => (it @ (subst @ R @ (var @ I) @ J)))))). % subst_Var_IT
thf(fact_25_lifts__IT, axiom,
    ((![Ts : list_dB]: ((listsp_dB @ it @ Ts) => (listsp_dB @ it @ (map_dB_dB @ (^[T : dB]: (lift @ T @ zero_zero_nat)) @ Ts)))))). % lifts_IT
thf(fact_26_list__ex__simps_I1_J, axiom,
    ((![P : dB > $o, X2 : dB, Xs2 : list_dB]: ((list_ex_dB @ P @ (cons_dB @ X2 @ Xs2)) = (((P @ X2)) | ((list_ex_dB @ P @ Xs2))))))). % list_ex_simps(1)
thf(fact_27_listsp_Oinducts, axiom,
    ((![A : type > $o, X2 : list_type, P : list_type > $o]: ((listsp_type @ A @ X2) => ((P @ nil_type) => ((![A3 : type, L2 : list_type]: ((A @ A3) => ((listsp_type @ A @ L2) => ((P @ L2) => (P @ (cons_type @ A3 @ L2)))))) => (P @ X2))))))). % listsp.inducts
thf(fact_28_listsp_Oinducts, axiom,
    ((![A : dB > $o, X2 : list_dB, P : list_dB > $o]: ((listsp_dB @ A @ X2) => ((P @ nil_dB) => ((![A3 : dB, L2 : list_dB]: ((A @ A3) => ((listsp_dB @ A @ L2) => ((P @ L2) => (P @ (cons_dB @ A3 @ L2)))))) => (P @ X2))))))). % listsp.inducts
thf(fact_29_listsp_Osimps, axiom,
    ((listsp_type = (^[A4 : type > $o]: (^[A5 : list_type]: (((A5 = nil_type)) | ((?[B2 : type]: (?[L3 : list_type]: (((A5 = (cons_type @ B2 @ L3))) & ((((A4 @ B2)) & ((listsp_type @ A4 @ L3)))))))))))))). % listsp.simps
thf(fact_30_listsp_Osimps, axiom,
    ((listsp_dB = (^[A4 : dB > $o]: (^[A5 : list_dB]: (((A5 = nil_dB)) | ((?[B2 : dB]: (?[L3 : list_dB]: (((A5 = (cons_dB @ B2 @ L3))) & ((((A4 @ B2)) & ((listsp_dB @ A4 @ L3)))))))))))))). % listsp.simps
thf(fact_31_Nil__is__map__conv, axiom,
    ((![F : type > dB, Xs2 : list_type]: ((nil_dB = (map_type_dB @ F @ Xs2)) = (Xs2 = nil_type))))). % Nil_is_map_conv
thf(fact_32_Nil__is__map__conv, axiom,
    ((![F : dB > type, Xs2 : list_dB]: ((nil_type = (map_dB_type @ F @ Xs2)) = (Xs2 = nil_dB))))). % Nil_is_map_conv
thf(fact_33_Nil__is__map__conv, axiom,
    ((![F : type > type, Xs2 : list_type]: ((nil_type = (map_type_type @ F @ Xs2)) = (Xs2 = nil_type))))). % Nil_is_map_conv
thf(fact_34_Nil__is__map__conv, axiom,
    ((![F : dB > dB, Xs2 : list_dB]: ((nil_dB = (map_dB_dB @ F @ Xs2)) = (Xs2 = nil_dB))))). % Nil_is_map_conv
thf(fact_35_map__is__Nil__conv, axiom,
    ((![F : type > dB, Xs2 : list_type]: (((map_type_dB @ F @ Xs2) = nil_dB) = (Xs2 = nil_type))))). % map_is_Nil_conv
thf(fact_36_map__is__Nil__conv, axiom,
    ((![F : dB > type, Xs2 : list_dB]: (((map_dB_type @ F @ Xs2) = nil_type) = (Xs2 = nil_dB))))). % map_is_Nil_conv
thf(fact_37_map__is__Nil__conv, axiom,
    ((![F : type > type, Xs2 : list_type]: (((map_type_type @ F @ Xs2) = nil_type) = (Xs2 = nil_type))))). % map_is_Nil_conv
thf(fact_38_map__is__Nil__conv, axiom,
    ((![F : dB > dB, Xs2 : list_dB]: (((map_dB_dB @ F @ Xs2) = nil_dB) = (Xs2 = nil_dB))))). % map_is_Nil_conv
thf(fact_39_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_40_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_41_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_42_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_43_listsp__simps_I1_J, axiom,
    ((![A : dB > $o]: (listsp_dB @ A @ nil_dB)))). % listsp_simps(1)
thf(fact_44_listsp__simps_I1_J, axiom,
    ((![A : type > $o]: (listsp_type @ A @ nil_type)))). % listsp_simps(1)
thf(fact_45_list__ex__simps_I2_J, axiom,
    ((![P : dB > $o]: (~ ((list_ex_dB @ P @ nil_dB)))))). % list_ex_simps(2)
thf(fact_46_list__ex__simps_I2_J, axiom,
    ((![P : type > $o]: (~ ((list_ex_type @ P @ nil_type)))))). % list_ex_simps(2)
thf(fact_47_lift__IT, axiom,
    ((![T2 : dB, I : nat]: ((it @ T2) => (it @ (lift @ T2 @ I)))))). % lift_IT
thf(fact_48_uT, axiom,
    ((typing @ e @ u @ t2))). % uT
thf(fact_49_null__rec_I2_J, axiom,
    ((null_dB @ nil_dB))). % null_rec(2)
thf(fact_50_null__rec_I2_J, axiom,
    ((null_type @ nil_type))). % null_rec(2)
thf(fact_51_member__rec_I2_J, axiom,
    ((![Y : dB]: (~ ((member_dB @ nil_dB @ Y)))))). % member_rec(2)
thf(fact_52_member__rec_I2_J, axiom,
    ((![Y : type]: (~ ((member_type @ nil_type @ Y)))))). % member_rec(2)
thf(fact_53_eq__Nil__null, axiom,
    ((![Xs2 : list_dB]: ((Xs2 = nil_dB) = (null_dB @ Xs2))))). % eq_Nil_null
thf(fact_54_eq__Nil__null, axiom,
    ((![Xs2 : list_type]: ((Xs2 = nil_type) = (null_type @ Xs2))))). % eq_Nil_null
thf(fact_55_transpose_Ocases, axiom,
    ((![X2 : list_list_type]: ((~ ((X2 = nil_list_type))) => ((![Xss : list_list_type]: (~ ((X2 = (cons_list_type @ nil_type @ Xss))))) => (~ ((![X3 : type, Xs3 : list_type, Xss : list_list_type]: (~ ((X2 = (cons_list_type @ (cons_type @ X3 @ Xs3) @ Xss)))))))))))). % transpose.cases
thf(fact_56_transpose_Ocases, axiom,
    ((![X2 : list_list_dB]: ((~ ((X2 = nil_list_dB))) => ((![Xss : list_list_dB]: (~ ((X2 = (cons_list_dB @ nil_dB @ Xss))))) => (~ ((![X3 : dB, Xs3 : list_dB, Xss : list_list_dB]: (~ ((X2 = (cons_list_dB @ (cons_dB @ X3 @ Xs3) @ Xss)))))))))))). % transpose.cases
thf(fact_57_list_Odistinct_I1_J, axiom,
    ((![X21 : type, X22 : list_type]: (~ ((nil_type = (cons_type @ X21 @ X22))))))). % list.distinct(1)
thf(fact_58_list_Odistinct_I1_J, axiom,
    ((![X21 : dB, X22 : list_dB]: (~ ((nil_dB = (cons_dB @ X21 @ X22))))))). % list.distinct(1)
thf(fact_59_list_OdiscI, axiom,
    ((![List : list_type, X21 : type, X22 : list_type]: ((List = (cons_type @ X21 @ X22)) => (~ ((List = nil_type))))))). % list.discI
thf(fact_60_list_OdiscI, axiom,
    ((![List : list_dB, X21 : dB, X22 : list_dB]: ((List = (cons_dB @ X21 @ X22)) => (~ ((List = nil_dB))))))). % list.discI
thf(fact_61_list_Oexhaust, axiom,
    ((![Y : list_type]: ((~ ((Y = nil_type))) => (~ ((![X212 : type, X222 : list_type]: (~ ((Y = (cons_type @ X212 @ X222))))))))))). % list.exhaust
thf(fact_62_list_Oexhaust, axiom,
    ((![Y : list_dB]: ((~ ((Y = nil_dB))) => (~ ((![X212 : dB, X222 : list_dB]: (~ ((Y = (cons_dB @ X212 @ X222))))))))))). % list.exhaust
thf(fact_63_list_Oinducts, axiom,
    ((![P : list_type > $o, List : list_type]: ((P @ nil_type) => ((![X1 : type, X23 : list_type]: ((P @ X23) => (P @ (cons_type @ X1 @ X23)))) => (P @ List)))))). % list.inducts
thf(fact_64_list_Oinducts, axiom,
    ((![P : list_dB > $o, List : list_dB]: ((P @ nil_dB) => ((![X1 : dB, X23 : list_dB]: ((P @ X23) => (P @ (cons_dB @ X1 @ X23)))) => (P @ List)))))). % list.inducts
thf(fact_65_neq__Nil__conv, axiom,
    ((![Xs2 : list_type]: ((~ ((Xs2 = nil_type))) = (?[Y2 : type]: (?[Ys2 : list_type]: (Xs2 = (cons_type @ Y2 @ Ys2)))))))). % neq_Nil_conv
thf(fact_66_neq__Nil__conv, axiom,
    ((![Xs2 : list_dB]: ((~ ((Xs2 = nil_dB))) = (?[Y2 : dB]: (?[Ys2 : list_dB]: (Xs2 = (cons_dB @ Y2 @ Ys2)))))))). % neq_Nil_conv
thf(fact_67_list__induct2_H, axiom,
    ((![P : list_type > list_type > $o, Xs2 : list_type, Ys : list_type]: ((P @ nil_type @ nil_type) => ((![X3 : type, Xs3 : list_type]: (P @ (cons_type @ X3 @ Xs3) @ nil_type)) => ((![Y3 : type, Ys3 : list_type]: (P @ nil_type @ (cons_type @ Y3 @ Ys3))) => ((![X3 : type, Xs3 : list_type, Y3 : type, Ys3 : list_type]: ((P @ Xs3 @ Ys3) => (P @ (cons_type @ X3 @ Xs3) @ (cons_type @ Y3 @ Ys3)))) => (P @ Xs2 @ Ys)))))))). % list_induct2'
thf(fact_68_list__induct2_H, axiom,
    ((![P : list_type > list_dB > $o, Xs2 : list_type, Ys : list_dB]: ((P @ nil_type @ nil_dB) => ((![X3 : type, Xs3 : list_type]: (P @ (cons_type @ X3 @ Xs3) @ nil_dB)) => ((![Y3 : dB, Ys3 : list_dB]: (P @ nil_type @ (cons_dB @ Y3 @ Ys3))) => ((![X3 : type, Xs3 : list_type, Y3 : dB, Ys3 : list_dB]: ((P @ Xs3 @ Ys3) => (P @ (cons_type @ X3 @ Xs3) @ (cons_dB @ Y3 @ Ys3)))) => (P @ Xs2 @ Ys)))))))). % list_induct2'
thf(fact_69_list__induct2_H, axiom,
    ((![P : list_dB > list_type > $o, Xs2 : list_dB, Ys : list_type]: ((P @ nil_dB @ nil_type) => ((![X3 : dB, Xs3 : list_dB]: (P @ (cons_dB @ X3 @ Xs3) @ nil_type)) => ((![Y3 : type, Ys3 : list_type]: (P @ nil_dB @ (cons_type @ Y3 @ Ys3))) => ((![X3 : dB, Xs3 : list_dB, Y3 : type, Ys3 : list_type]: ((P @ Xs3 @ Ys3) => (P @ (cons_dB @ X3 @ Xs3) @ (cons_type @ Y3 @ Ys3)))) => (P @ Xs2 @ Ys)))))))). % list_induct2'
thf(fact_70_list__induct2_H, axiom,
    ((![P : list_dB > list_dB > $o, Xs2 : list_dB, Ys : list_dB]: ((P @ nil_dB @ nil_dB) => ((![X3 : dB, Xs3 : list_dB]: (P @ (cons_dB @ X3 @ Xs3) @ nil_dB)) => ((![Y3 : dB, Ys3 : list_dB]: (P @ nil_dB @ (cons_dB @ Y3 @ Ys3))) => ((![X3 : dB, Xs3 : list_dB, Y3 : dB, Ys3 : list_dB]: ((P @ Xs3 @ Ys3) => (P @ (cons_dB @ X3 @ Xs3) @ (cons_dB @ Y3 @ Ys3)))) => (P @ Xs2 @ Ys)))))))). % list_induct2'
thf(fact_71_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, Xs3 : list_type, Ys3 : list_type]: ((P @ Ys3 @ Xs3) => (P @ (cons_type @ X3 @ Xs3) @ Ys3))) => (P @ A0 @ A1)))))). % splice.induct
thf(fact_72_splice_Oinduct, axiom,
    ((![P : list_dB > list_dB > $o, A0 : list_dB, A1 : list_dB]: ((![X_1 : list_dB]: (P @ nil_dB @ X_1)) => ((![X3 : dB, Xs3 : list_dB, Ys3 : list_dB]: ((P @ Ys3 @ Xs3) => (P @ (cons_dB @ X3 @ Xs3) @ Ys3))) => (P @ A0 @ A1)))))). % splice.induct
thf(fact_73_induct__list012, axiom,
    ((![P : list_type > $o, Xs2 : 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 @ Xs2))))))). % induct_list012
thf(fact_74_induct__list012, axiom,
    ((![P : list_dB > $o, Xs2 : 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 @ Xs2))))))). % induct_list012
thf(fact_75_shuffles_Oinduct, axiom,
    ((![P : list_type > list_type > $o, A0 : list_type, A1 : list_type]: ((![X_1 : list_type]: (P @ nil_type @ X_1)) => ((![Xs3 : list_type]: (P @ Xs3 @ nil_type)) => ((![X3 : type, Xs3 : list_type, Y3 : type, Ys3 : list_type]: ((P @ Xs3 @ (cons_type @ Y3 @ Ys3)) => ((P @ (cons_type @ X3 @ Xs3) @ Ys3) => (P @ (cons_type @ X3 @ Xs3) @ (cons_type @ Y3 @ Ys3))))) => (P @ A0 @ A1))))))). % shuffles.induct
thf(fact_76_shuffles_Oinduct, axiom,
    ((![P : list_dB > list_dB > $o, A0 : list_dB, A1 : list_dB]: ((![X_1 : list_dB]: (P @ nil_dB @ X_1)) => ((![Xs3 : list_dB]: (P @ Xs3 @ nil_dB)) => ((![X3 : dB, Xs3 : list_dB, Y3 : dB, Ys3 : list_dB]: ((P @ Xs3 @ (cons_dB @ Y3 @ Ys3)) => ((P @ (cons_dB @ X3 @ Xs3) @ Ys3) => (P @ (cons_dB @ X3 @ Xs3) @ (cons_dB @ Y3 @ Ys3))))) => (P @ A0 @ A1))))))). % shuffles.induct
thf(fact_77_remdups__adj_Ocases, axiom,
    ((![X2 : list_type]: ((~ ((X2 = nil_type))) => ((![X3 : type]: (~ ((X2 = (cons_type @ X3 @ nil_type))))) => (~ ((![X3 : type, Y3 : type, Xs3 : list_type]: (~ ((X2 = (cons_type @ X3 @ (cons_type @ Y3 @ Xs3))))))))))))). % remdups_adj.cases
thf(fact_78_remdups__adj_Ocases, axiom,
    ((![X2 : list_dB]: ((~ ((X2 = nil_dB))) => ((![X3 : dB]: (~ ((X2 = (cons_dB @ X3 @ nil_dB))))) => (~ ((![X3 : dB, Y3 : dB, Xs3 : list_dB]: (~ ((X2 = (cons_dB @ X3 @ (cons_dB @ Y3 @ Xs3))))))))))))). % remdups_adj.cases
thf(fact_79_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_80_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_81_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, Xs3 : list_type]: (((X3 = Y3) => (P @ (cons_type @ X3 @ Xs3))) => (((~ ((X3 = Y3))) => (P @ (cons_type @ Y3 @ Xs3))) => (P @ (cons_type @ X3 @ (cons_type @ Y3 @ Xs3)))))) => (P @ A0))))))). % remdups_adj.induct
thf(fact_82_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, Xs3 : list_dB]: (((X3 = Y3) => (P @ (cons_dB @ X3 @ Xs3))) => (((~ ((X3 = Y3))) => (P @ (cons_dB @ Y3 @ Xs3))) => (P @ (cons_dB @ X3 @ (cons_dB @ Y3 @ Xs3)))))) => (P @ A0))))))). % remdups_adj.induct
thf(fact_83_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, Xs3 : list_type]: ((P @ P2 @ (cons_type @ Y3 @ Xs3)) => (P @ P2 @ (cons_type @ X3 @ (cons_type @ Y3 @ Xs3))))) => (P @ A0 @ A1))))))). % successively.induct
thf(fact_84_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, Xs3 : list_dB]: ((P @ P2 @ (cons_dB @ Y3 @ Xs3)) => (P @ P2 @ (cons_dB @ X3 @ (cons_dB @ Y3 @ Xs3))))) => (P @ A0 @ A1))))))). % successively.induct
thf(fact_85_list__nonempty__induct, axiom,
    ((![Xs2 : list_type, P : list_type > $o]: ((~ ((Xs2 = nil_type))) => ((![X3 : type]: (P @ (cons_type @ X3 @ nil_type))) => ((![X3 : type, Xs3 : list_type]: ((~ ((Xs3 = nil_type))) => ((P @ Xs3) => (P @ (cons_type @ X3 @ Xs3))))) => (P @ Xs2))))))). % list_nonempty_induct
thf(fact_86_list__nonempty__induct, axiom,
    ((![Xs2 : list_dB, P : list_dB > $o]: ((~ ((Xs2 = nil_dB))) => ((![X3 : dB]: (P @ (cons_dB @ X3 @ nil_dB))) => ((![X3 : dB, Xs3 : list_dB]: ((~ ((Xs3 = nil_dB))) => ((P @ Xs3) => (P @ (cons_dB @ X3 @ Xs3))))) => (P @ Xs2))))))). % list_nonempty_induct
thf(fact_87_map__tailrec__rev_Oinduct, axiom,
    ((![P : (type > dB) > list_type > list_dB > $o, A0 : type > dB, A1 : list_type, A22 : 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 @ A22)))))). % map_tailrec_rev.induct
thf(fact_88_map__tailrec__rev_Oinduct, axiom,
    ((![P : (dB > dB) > list_dB > list_dB > $o, A0 : dB > dB, A1 : list_dB, A22 : list_dB]: ((![F2 : dB > dB, X_1 : list_dB]: (P @ F2 @ nil_dB @ X_1)) => ((![F2 : dB > dB, A3 : dB, As : list_dB, Bs : list_dB]: ((P @ F2 @ As @ (cons_dB @ (F2 @ A3) @ Bs)) => (P @ F2 @ (cons_dB @ A3 @ As) @ Bs))) => (P @ A0 @ A1 @ A22)))))). % map_tailrec_rev.induct
thf(fact_89_list_Osimps_I8_J, axiom,
    ((![F : dB > type]: ((map_dB_type @ F @ nil_dB) = nil_type)))). % list.simps(8)
thf(fact_90_list_Osimps_I8_J, axiom,
    ((![F : type > dB]: ((map_type_dB @ F @ nil_type) = nil_dB)))). % list.simps(8)
thf(fact_91_list_Osimps_I8_J, axiom,
    ((![F : type > type]: ((map_type_type @ F @ nil_type) = nil_type)))). % list.simps(8)
thf(fact_92_list_Osimps_I8_J, axiom,
    ((![F : dB > dB]: ((map_dB_dB @ F @ nil_dB) = nil_dB)))). % list.simps(8)
thf(fact_93_Var__IT, axiom,
    ((![N : nat]: (it @ (var @ N))))). % Var_IT
thf(fact_94_listsp_ONil, axiom,
    ((![A : dB > $o]: (listsp_dB @ A @ nil_dB)))). % listsp.Nil
thf(fact_95_listsp_ONil, axiom,
    ((![A : type > $o]: (listsp_type @ A @ nil_type)))). % listsp.Nil
thf(fact_96_listsp_Ocases, axiom,
    ((![A : type > $o, A2 : list_type]: ((listsp_type @ A @ A2) => ((~ ((A2 = nil_type))) => (~ ((![A3 : type, L2 : list_type]: ((A2 = (cons_type @ A3 @ L2)) => ((A @ A3) => (~ ((listsp_type @ A @ L2))))))))))))). % listsp.cases
thf(fact_97_listsp_Ocases, axiom,
    ((![A : dB > $o, A2 : list_dB]: ((listsp_dB @ A @ A2) => ((~ ((A2 = nil_dB))) => (~ ((![A3 : dB, L2 : list_dB]: ((A2 = (cons_dB @ A3 @ L2)) => ((A @ A3) => (~ ((listsp_dB @ A @ L2))))))))))))). % listsp.cases
thf(fact_98_I, axiom,
    ((![E : nat > type, T3 : type, U : dB, I2 : nat]: ((typing @ (shift_type @ E @ I2 @ t2) @ a @ T3) => ((it @ U) => ((typing @ E @ U @ t2) => (it @ (subst @ a @ U @ I2)))))))). % I
thf(fact_99_subst__lift, axiom,
    ((![T2 : dB, K : nat, S : dB]: ((subst @ (lift @ T2 @ K) @ S @ K) = T2)))). % subst_lift
thf(fact_100_subst__eq, axiom,
    ((![K : nat, U2 : dB]: ((subst @ (var @ K) @ U2 @ K) = U2)))). % subst_eq
thf(fact_101_Cons_Ohyps_I1_J, axiom,
    (((![E : nat > type, T3 : type, U : dB, I2 : nat]: ((typing @ (shift_type @ E @ I2 @ t2) @ a @ T3) => ((it @ U) => ((typing @ E @ U @ t2) => (it @ (subst @ a @ U @ I2)))))) & (?[X_1 : type]: (typing @ (shift_type @ e @ i @ t2) @ a @ X_1))))). % Cons.hyps(1)
thf(fact_102_dB_Oinject_I1_J, axiom,
    ((![X12 : nat, Y1 : nat]: (((var @ X12) = (var @ Y1)) = (X12 = Y1))))). % dB.inject(1)
thf(fact_103__092_060open_062e_092_060langle_062i_058T_092_060rangle_062_A_092_060turnstile_062_Aa_A_058_AU_092_060close_062, axiom,
    ((typing @ (shift_type @ e @ i @ t2) @ a @ ua))). % \<open>e\<langle>i:T\<rangle> \<turnstile> a : U\<close>
thf(fact_104_typing_OVar, axiom,
    ((![Env : nat > type, X2 : nat, T4 : type]: (((Env @ X2) = T4) => (typing @ Env @ (var @ X2) @ T4))))). % typing.Var
thf(fact_105_shift__eq, axiom,
    ((![I : nat, J : nat, E2 : nat > type, T4 : type]: ((I = J) => ((shift_type @ E2 @ I @ T4 @ J) = T4))))). % shift_eq
thf(fact_106__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062U_O_Ae_092_060langle_062i_058T_092_060rangle_062_A_092_060turnstile_062_Aa_A_058_AU_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom,
    ((~ ((![U3 : type]: (~ ((typing @ (shift_type @ e @ i @ t2) @ a @ U3)))))))). % \<open>\<And>thesis. (\<And>U. e\<langle>i:T\<rangle> \<turnstile> a : U \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_107_lift__type, axiom,
    ((![E2 : nat > type, T2 : dB, T4 : type, I : nat, U4 : type]: ((typing @ E2 @ T2 @ T4) => (typing @ (shift_type @ E2 @ I @ U4) @ (lift @ T2 @ I) @ T4))))). % lift_type
thf(fact_108_subst__lemma, axiom,
    ((![E2 : nat > type, T2 : dB, T4 : type, E3 : nat > type, U2 : dB, U4 : type, I : nat]: ((typing @ E2 @ T2 @ T4) => ((typing @ E3 @ U2 @ U4) => ((E2 = (shift_type @ E3 @ I @ U4)) => (typing @ E3 @ (subst @ T2 @ U2 @ I) @ T4))))))). % subst_lemma
thf(fact_109_zero__reorient, axiom,
    ((![X2 : nat]: ((zero_zero_nat = X2) = (X2 = zero_zero_nat))))). % zero_reorient
thf(fact_110_typing__elims_I1_J, axiom,
    ((![E2 : nat > type, I : nat, T4 : type]: ((typing @ E2 @ (var @ I) @ T4) => ((E2 @ I) = T4))))). % typing_elims(1)
thf(fact_111_MI2, axiom,
    ((![T1 : type, T22 : type, T2 : dB, E2 : nat > type, I : nat, T4 : type, U2 : dB]: ((t2 = (fun @ T1 @ T22)) => ((it @ T2) => ((typing @ (shift_type @ E2 @ I @ T22) @ T2 @ T4) => ((it @ U2) => ((typing @ E2 @ U2 @ T22) => (it @ (subst @ T2 @ U2 @ I)))))))))). % MI2
thf(fact_112_MI1, axiom,
    ((![T1 : type, T22 : type, T2 : dB, E2 : nat > type, I : nat, T4 : type, U2 : dB]: ((t2 = (fun @ T1 @ T22)) => ((it @ T2) => ((typing @ (shift_type @ E2 @ I @ T1) @ T2 @ T4) => ((it @ U2) => ((typing @ E2 @ U2 @ T1) => (it @ (subst @ T2 @ U2 @ I)))))))))). % MI1
thf(fact_113_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_114_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_115__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_116_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_117__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,
    ((~ ((![Ts2 : list_type]: (~ ((typings @ (shift_type @ e @ i @ t2) @ rs @ Ts2)))))))). % \<open>\<And>thesis. (\<And>Ts. e\<langle>i:T\<rangle> \<tturnstile> rs : Ts \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_118_typings_Osimps_I1_J, axiom,
    ((![E2 : nat > type, Ts3 : list_type]: ((typings @ E2 @ nil_dB @ Ts3) = (Ts3 = nil_type))))). % typings.simps(1)
thf(fact_119_type__induct, axiom,
    ((![P : type > $o, T4 : type]: ((![T5 : type]: ((![T12 : type, T23 : type]: ((T5 = (fun @ T12 @ T23)) => (P @ T12))) => ((![T12 : type, T23 : type]: ((T5 = (fun @ T12 @ T23)) => (P @ T23))) => (P @ T5)))) => (P @ T4))))). % type_induct
thf(fact_120_lift__types, axiom,
    ((![E2 : nat > type, Ts : list_dB, Ts3 : list_type, I : nat, U4 : type]: ((typings @ E2 @ Ts @ Ts3) => (typings @ (shift_type @ E2 @ I @ U4) @ (map_dB_dB @ (^[T : dB]: (lift @ T @ I)) @ Ts) @ Ts3))))). % lift_types
thf(fact_121_substs__lemma, axiom,
    ((![E2 : nat > type, U2 : dB, T4 : type, I : nat, Ts : list_dB, Ts3 : list_type]: ((typing @ E2 @ U2 @ T4) => ((typings @ (shift_type @ E2 @ I @ T4) @ Ts @ Ts3) => (typings @ E2 @ (map_dB_dB @ (^[T : dB]: (subst @ T @ U2 @ I)) @ Ts) @ Ts3)))))). % substs_lemma
thf(fact_122_n__lists_Osimps_I1_J, axiom,
    ((![Xs2 : list_dB]: ((n_lists_dB @ zero_zero_nat @ Xs2) = (cons_list_dB @ nil_dB @ nil_list_dB))))). % n_lists.simps(1)
thf(fact_123_n__lists_Osimps_I1_J, axiom,
    ((![Xs2 : list_type]: ((n_lists_type @ zero_zero_nat @ Xs2) = (cons_list_type @ nil_type @ nil_list_type))))). % n_lists.simps(1)
thf(fact_124__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 @ (^[T : dB]: (?[X4 : type]: (typing @ (shift_type @ e @ i @ t2) @ T @ X4))) @ rs))). % \<open>listsp (\<lambda>t. \<exists>T'. e\<langle>i:T\<rangle> \<turnstile> t : T') rs\<close>
thf(fact_125_nT, axiom,
    ((typing @ (shift_type @ e @ i @ t2) @ (foldl_dB_dB @ app @ (var @ n) @ rs) @ t))). % nT
thf(fact_126_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_127_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_128_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_129_lift_Osimps_I2_J, axiom,
    ((![S : dB, T2 : dB, K : nat]: ((lift @ (app @ S @ T2) @ K) = (app @ (lift @ S @ K) @ (lift @ T2 @ K)))))). % lift.simps(2)
thf(fact_130_dB_Odistinct_I1_J, axiom,
    ((![X12 : nat, X21 : dB, X22 : dB]: (~ (((var @ X12) = (app @ X21 @ X22))))))). % dB.distinct(1)
thf(fact_131_var__app__type__eq, axiom,
    ((![E2 : nat > type, I : nat, Ts : list_dB, T4 : type, U4 : type]: ((typing @ E2 @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ T4) => ((typing @ E2 @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ U4) => (T4 = U4)))))). % var_app_type_eq
thf(fact_132_foldl__Cons, axiom,
    ((![F : dB > dB > dB, A2 : dB, X2 : dB, Xs2 : list_dB]: ((foldl_dB_dB @ F @ A2 @ (cons_dB @ X2 @ Xs2)) = (foldl_dB_dB @ F @ (F @ A2 @ X2) @ Xs2))))). % foldl_Cons
thf(fact_133_foldl__Nil, axiom,
    ((![F : dB > dB > dB, A2 : dB]: ((foldl_dB_dB @ F @ A2 @ nil_dB) = A2)))). % foldl_Nil
thf(fact_134_app__Var__IT, axiom,
    ((![T2 : dB, I : nat]: ((it @ T2) => (it @ (app @ T2 @ (var @ I))))))). % app_Var_IT
thf(fact_135_typing__elims_I2_J, axiom,
    ((![E2 : nat > type, T2 : dB, U2 : dB, T4 : type]: ((typing @ E2 @ (app @ T2 @ U2) @ T4) => (~ ((![T5 : type]: ((typing @ E2 @ T2 @ (fun @ T5 @ T4)) => (~ ((typing @ E2 @ U2 @ T5))))))))))). % typing_elims(2)
thf(fact_136_App, axiom,
    ((![Env : nat > type, S : dB, T4 : type, U4 : type, T2 : dB]: ((typing @ Env @ S @ (fun @ T4 @ U4)) => ((typing @ Env @ T2 @ T4) => (typing @ Env @ (app @ S @ T2) @ U4)))))). % App
thf(fact_137_foldl__map, axiom,
    ((![G : dB > dB > dB, A2 : dB, F : dB > dB, Xs2 : list_dB]: ((foldl_dB_dB @ G @ A2 @ (map_dB_dB @ F @ Xs2)) = (foldl_dB_dB @ (^[A5 : dB]: (^[X : dB]: (G @ A5 @ (F @ X)))) @ A2 @ Xs2))))). % foldl_map
thf(fact_138_lists__typings, axiom,
    ((![E2 : nat > type, Ts : list_dB, Ts3 : list_type]: ((typings @ E2 @ Ts @ Ts3) => (listsp_dB @ (^[T : dB]: (?[X4 : type]: (typing @ E2 @ T @ X4))) @ Ts))))). % lists_typings
thf(fact_139_lift__map, axiom,
    ((![T2 : dB, Ts : list_dB, I : nat]: ((lift @ (foldl_dB_dB @ app @ T2 @ Ts) @ I) = (foldl_dB_dB @ app @ (lift @ T2 @ I) @ (map_dB_dB @ (^[T : dB]: (lift @ T @ I)) @ Ts)))))). % lift_map
thf(fact_140_subst__map, axiom,
    ((![T2 : dB, Ts : list_dB, U2 : dB, I : nat]: ((subst @ (foldl_dB_dB @ app @ T2 @ Ts) @ U2 @ I) = (foldl_dB_dB @ app @ (subst @ T2 @ U2 @ I) @ (map_dB_dB @ (^[T : dB]: (subst @ T @ U2 @ I)) @ Ts)))))). % subst_map
thf(fact_141_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_142_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_143_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_144_IT_OVar, axiom,
    ((![Rs : list_dB, N : nat]: ((listsp_dB @ it @ Rs) => (it @ (foldl_dB_dB @ app @ (var @ N) @ Rs)))))). % IT.Var
thf(fact_145_var__app__types, axiom,
    ((![E2 : nat > type, I : nat, Ts : list_dB, Us : list_dB, T4 : type, Ts3 : list_type, U4 : type]: ((typing @ E2 @ (foldl_dB_dB @ app @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ Us) @ T4) => ((typings @ E2 @ Ts @ Ts3) => ((typing @ E2 @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ U4) => (?[Us2 : list_type]: ((U4 = (foldr_type_type @ fun @ Us2 @ T4)) & (typings @ E2 @ Us @ Us2))))))))). % var_app_types
thf(fact_146_list__app__typeI, axiom,
    ((![E2 : nat > type, T2 : dB, Ts3 : list_type, T4 : type, Ts : list_dB]: ((typing @ E2 @ T2 @ (foldr_type_type @ fun @ Ts3 @ T4)) => ((typings @ E2 @ Ts @ Ts3) => (typing @ E2 @ (foldl_dB_dB @ app @ T2 @ Ts) @ T4)))))). % list_app_typeI
thf(fact_147_list__app__typeE, axiom,
    ((![E2 : nat > type, T2 : dB, Ts : list_dB, T4 : type]: ((typing @ E2 @ (foldl_dB_dB @ app @ T2 @ Ts) @ T4) => (~ ((![Ts2 : list_type]: ((typing @ E2 @ T2 @ (foldr_type_type @ fun @ Ts2 @ T4)) => (~ ((typings @ E2 @ Ts @ Ts2))))))))))). % list_app_typeE
thf(fact_148_list__app__typeD, axiom,
    ((![E2 : nat > type, T2 : dB, Ts : list_dB, T4 : type]: ((typing @ E2 @ (foldl_dB_dB @ app @ T2 @ Ts) @ T4) => (?[Ts2 : list_type]: ((typing @ E2 @ T2 @ (foldr_type_type @ fun @ Ts2 @ T4)) & (typings @ E2 @ Ts @ Ts2))))))). % list_app_typeD
thf(fact_149_var__app__typesE, axiom,
    ((![E2 : nat > type, I : nat, Ts : list_dB, T4 : type]: ((typing @ E2 @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ T4) => (~ ((![Ts2 : list_type]: ((typing @ E2 @ (var @ I) @ (foldr_type_type @ fun @ Ts2 @ T4)) => (~ ((typings @ E2 @ Ts @ Ts2))))))))))). % var_app_typesE
thf(fact_150_IT_Oinducts, axiom,
    ((![X2 : dB, P : dB > $o]: ((it @ X2) => ((![Rs2 : list_dB, N2 : nat]: ((listsp_dB @ (^[X : dB]: (((it @ X)) & ((P @ X)))) @ Rs2) => (P @ (foldl_dB_dB @ app @ (var @ N2) @ Rs2)))) => ((![R2 : dB]: ((it @ R2) => ((P @ R2) => (P @ (abs @ R2))))) => ((![R2 : dB, S2 : dB, Ss2 : list_dB]: ((it @ (foldl_dB_dB @ app @ (subst @ R2 @ S2 @ zero_zero_nat) @ Ss2)) => ((P @ (foldl_dB_dB @ app @ (subst @ R2 @ S2 @ zero_zero_nat) @ Ss2)) => ((it @ S2) => ((P @ S2) => (P @ (foldl_dB_dB @ app @ (app @ (abs @ R2) @ S2) @ Ss2))))))) => (P @ X2)))))))). % IT.inducts
thf(fact_151_IT_Osimps, axiom,
    ((it = (^[A5 : dB]: (((?[Rs3 : list_dB]: (?[N3 : nat]: (((A5 = (foldl_dB_dB @ app @ (var @ N3) @ Rs3))) & ((listsp_dB @ it @ Rs3)))))) | ((((?[R3 : dB]: (((A5 = (abs @ R3))) & ((it @ R3))))) | ((?[R3 : dB]: (?[S3 : dB]: (?[Ss3 : list_dB]: (((A5 = (foldl_dB_dB @ app @ (app @ (abs @ R3) @ S3) @ Ss3))) & ((((it @ (foldl_dB_dB @ app @ (subst @ R3 @ S3 @ zero_zero_nat) @ Ss3))) & ((it @ S3)))))))))))))))). % IT.simps
thf(fact_152_dB_Oinject_I3_J, axiom,
    ((![X32 : dB, Y32 : dB]: (((abs @ X32) = (abs @ Y32)) = (X32 = Y32))))). % dB.inject(3)
thf(fact_153_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_154_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_155_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_156_Lambda, axiom,
    ((![R : dB]: ((it @ R) => (it @ (abs @ R)))))). % Lambda
thf(fact_157_dB_Odistinct_I5_J, axiom,
    ((![X21 : dB, X22 : dB, X32 : dB]: (~ (((app @ X21 @ X22) = (abs @ X32))))))). % dB.distinct(5)
thf(fact_158_dB_Odistinct_I3_J, axiom,
    ((![X12 : nat, X32 : dB]: (~ (((var @ X12) = (abs @ X32))))))). % dB.distinct(3)
thf(fact_159_dB_Oexhaust, axiom,
    ((![Y : dB]: ((![X1 : nat]: (~ ((Y = (var @ X1))))) => ((![X212 : dB, X222 : dB]: (~ ((Y = (app @ X212 @ X222))))) => (~ ((![X33 : dB]: (~ ((Y = (abs @ X33)))))))))))). % dB.exhaust
thf(fact_160_dB_Oinduct, axiom,
    ((![P : dB > $o, DB : dB]: ((![X3 : nat]: (P @ (var @ X3))) => ((![X1a : dB, X23 : dB]: ((P @ X1a) => ((P @ X23) => (P @ (app @ X1a @ X23))))) => ((![X3 : dB]: ((P @ X3) => (P @ (abs @ X3)))) => (P @ DB))))))). % dB.induct
thf(fact_161_abs__typeE, axiom,
    ((![E2 : nat > type, T2 : dB, T4 : type]: ((typing @ E2 @ (abs @ T2) @ T4) => (~ ((![U3 : type, V : type]: (~ ((typing @ (shift_type @ E2 @ zero_zero_nat @ U3) @ T2 @ V)))))))))). % abs_typeE
thf(fact_162_ex__head__tail, axiom,
    ((![T2 : dB]: (?[Ts4 : list_dB, H : dB]: ((T2 = (foldl_dB_dB @ app @ H @ Ts4)) & ((?[N2 : nat]: (H = (var @ N2))) | (?[U5 : dB]: (H = (abs @ U5))))))))). % ex_head_tail
thf(fact_163_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_164_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_165_typing__elims_I3_J, axiom,
    ((![E2 : nat > type, T2 : dB, T4 : type]: ((typing @ E2 @ (abs @ T2) @ T4) => (~ ((![T5 : type, U3 : type]: ((T4 = (fun @ T5 @ U3)) => (~ ((typing @ (shift_type @ E2 @ zero_zero_nat @ T5) @ T2 @ U3))))))))))). % typing_elims(3)
thf(fact_166_Abs, axiom,
    ((![Env : nat > type, T4 : type, T2 : dB, U4 : type]: ((typing @ (shift_type @ Env @ zero_zero_nat @ T4) @ T2 @ U4) => (typing @ Env @ (abs @ T2) @ (fun @ T4 @ U4)))))). % Abs
thf(fact_167_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_168_typing_Ocases, axiom,
    ((![A1 : nat > type, A22 : dB, A32 : type]: ((typing @ A1 @ A22 @ A32) => ((![X3 : nat]: ((A22 = (var @ X3)) => (~ (((A1 @ X3) = A32))))) => ((![T5 : type, T6 : dB]: ((A22 = (abs @ T6)) => (![U3 : type]: ((A32 = (fun @ T5 @ U3)) => (~ ((typing @ (shift_type @ A1 @ zero_zero_nat @ T5) @ T6 @ U3))))))) => (~ ((![S2 : dB, T5 : type, U3 : type, T6 : dB]: ((A22 = (app @ S2 @ T6)) => ((A32 = U3) => ((typing @ A1 @ S2 @ (fun @ T5 @ U3)) => (~ ((typing @ A1 @ T6 @ T5))))))))))))))). % typing.cases
thf(fact_169_typing_Osimps, axiom,
    ((typing = (^[A12 : nat > type]: (^[A23 : dB]: (^[A33 : type]: (((?[Env2 : nat > type]: (?[X : nat]: (?[T7 : type]: (((A12 = Env2)) & ((((A23 = (var @ X))) & ((((A33 = T7)) & (((Env2 @ X) = T7))))))))))) | ((((?[Env2 : nat > type]: (?[T7 : type]: (?[T : dB]: (?[U6 : type]: (((A12 = Env2)) & ((((A23 = (abs @ T))) & ((((A33 = (fun @ T7 @ U6))) & ((typing @ (shift_type @ Env2 @ zero_zero_nat @ T7) @ T @ U6)))))))))))) | ((?[Env2 : nat > type]: (?[S3 : dB]: (?[T7 : type]: (?[U6 : type]: (?[T : dB]: (((A12 = Env2)) & ((((A23 = (app @ S3 @ T))) & ((((A33 = U6)) & ((((typing @ Env2 @ S3 @ (fun @ T7 @ U6))) & ((typing @ Env2 @ T @ T7)))))))))))))))))))))))). % typing.simps
thf(fact_170_typing_Oinducts, axiom,
    ((![X12 : nat > type, X24 : dB, X32 : type, P : (nat > type) > dB > type > $o]: ((typing @ X12 @ X24 @ X32) => ((![Env3 : nat > type, X3 : nat, T5 : type]: (((Env3 @ X3) = T5) => (P @ Env3 @ (var @ X3) @ T5))) => ((![Env3 : nat > type, T5 : type, T6 : dB, U3 : type]: ((typing @ (shift_type @ Env3 @ zero_zero_nat @ T5) @ T6 @ U3) => ((P @ (shift_type @ Env3 @ zero_zero_nat @ T5) @ T6 @ U3) => (P @ Env3 @ (abs @ T6) @ (fun @ T5 @ U3))))) => ((![Env3 : nat > type, S2 : dB, T5 : type, U3 : type, T6 : dB]: ((typing @ Env3 @ S2 @ (fun @ T5 @ U3)) => ((P @ Env3 @ S2 @ (fun @ T5 @ U3)) => ((typing @ Env3 @ T6 @ T5) => ((P @ Env3 @ T6 @ T5) => (P @ Env3 @ (app @ S2 @ T6) @ U3)))))) => (P @ X12 @ X24 @ X32)))))))). % typing.inducts
thf(fact_171_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_172_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_173_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_174_map__rec, axiom,
    ((map_dB_dB = (^[F3 : dB > dB]: (rec_list_list_dB_dB @ nil_dB @ (^[X : dB]: (^[Uu : list_dB]: (cons_dB @ (F3 @ X))))))))). % map_rec
thf(fact_175_typings_Osimps_I2_J, axiom,
    ((![E2 : nat > type, T2 : dB, Ts : list_dB, Ts3 : list_type]: ((typings @ E2 @ (cons_dB @ T2 @ Ts) @ Ts3) = (case_list_o_type @ $false @ (^[T7 : type]: (^[Ts5 : list_type]: (((typing @ E2 @ T2 @ T7)) & ((typings @ E2 @ Ts @ Ts5))))) @ Ts3))))). % typings.simps(2)
thf(fact_176_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_177_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_178_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_179_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_180_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]: (^[X25 : list_type]: (H2 @ (F22 @ X13 @ X25)))) @ List))))). % list.case_distrib

% Conjectures (1)
thf(conj_0, conjecture,
    ((listsp_dB @ it @ (map_dB_dB @ (^[X : dB]: (subst @ X @ u @ i)) @ (cons_dB @ a @ as))))).
