% 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_166__5211094_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:38:01.075

% 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 (48)
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__Lambda__OdB, type,
    step1_dB : (dB > dB > $o) > list_dB > list_dB > $o).
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__Lambda__OdB, type,
    cons_dB : dB > list_dB > list_dB).
thf(sy_c_List_Olist_ONil_001t__Lambda__OdB, type,
    nil_dB : list_dB).
thf(sy_c_List_Olist_Omap_001t__Lambda__OdB_001t__Lambda__OdB, type,
    map_dB_dB : (dB > dB) > list_dB > list_dB).
thf(sy_c_List_Olist_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_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_H____, type,
    t : type).
thf(sy_v_T_H____, type,
    t2 : type).
thf(sy_v_T____, type,
    t3 : 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_b____, type,
    b : dB).
thf(sy_v_bs____, type,
    bs : 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,
    t4 : dB).
thf(sy_v_u1____, type,
    u1 : dB).
thf(sy_v_u____, type,
    u : dB).

% Relevant facts (144)
thf(fact_0__092_060open_062IT_At_092_060close_062, axiom,
    ((it @ t4))). % \<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_uT, axiom,
    ((typing @ e @ u @ t3))). % uT
thf(fact_4_True, axiom,
    ((n = i))). % True
thf(fact_5_I, axiom,
    ((![E : nat > type, T : type, U : dB, I : nat]: ((typing @ (shift_type @ E @ I @ t3) @ b @ T) => ((it @ U) => ((typing @ E @ U @ t3) => (it @ (subst @ b @ U @ I)))))))). % I
thf(fact_6__092_060open_062e_092_060langle_062i_058T_092_060rangle_062_A_092_060turnstile_062_Ab_A_058_AU_092_060close_062, axiom,
    ((typing @ (shift_type @ e @ i @ t3) @ b @ ua))). % \<open>e\<langle>i:T\<rangle> \<turnstile> b : U\<close>
thf(fact_7_Cons_Ohyps_I1_J, axiom,
    (((![E : nat > type, T : type, U : dB, I : nat]: ((typing @ (shift_type @ E @ I @ t3) @ b @ T) => ((it @ U) => ((typing @ E @ U @ t3) => (it @ (subst @ b @ U @ I)))))) & (?[X_1 : type]: (typing @ (shift_type @ e @ i @ t3) @ b @ X_1))))). % Cons.hyps(1)
thf(fact_8__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_Ab_A_058_AU_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom,
    ((~ ((![U2 : type]: (~ ((typing @ (shift_type @ e @ i @ t3) @ b @ U2)))))))). % \<open>\<And>thesis. (\<And>U. e\<langle>i:T\<rangle> \<turnstile> b : U \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_9__092_060open_062IT_A_Iu_A_092_060degree_062_Aa_091u_Pi_093_J_092_060close_062, axiom,
    ((it @ (app @ u @ (subst @ a @ u @ i))))). % \<open>IT (u \<degree> a[u/i])\<close>
thf(fact_10_subst__Var__IT, axiom,
    ((![R : dB, I2 : nat, J : nat]: ((it @ R) => (it @ (subst @ R @ (var @ I2) @ J)))))). % subst_Var_IT
thf(fact_11_MI1, axiom,
    ((![T1 : type, T2 : type, T3 : dB, E2 : nat > type, I2 : nat, T4 : type, U3 : dB]: ((t3 = (fun @ T1 @ T2)) => ((it @ T3) => ((typing @ (shift_type @ E2 @ I2 @ T1) @ T3 @ T4) => ((it @ U3) => ((typing @ E2 @ U3 @ T1) => (it @ (subst @ T3 @ U3 @ I2)))))))))). % MI1
thf(fact_12_MI2, axiom,
    ((![T1 : type, T2 : type, T3 : dB, E2 : nat > type, I2 : nat, T4 : type, U3 : dB]: ((t3 = (fun @ T1 @ T2)) => ((it @ T3) => ((typing @ (shift_type @ E2 @ I2 @ T2) @ T3 @ T4) => ((it @ U3) => ((typing @ E2 @ U3 @ T2) => (it @ (subst @ T3 @ U3 @ I2)))))))))). % MI2
thf(fact_13_lift__IT, axiom,
    ((![T3 : dB, I2 : nat]: ((it @ T3) => (it @ (lift @ T3 @ I2)))))). % lift_IT
thf(fact_14_argT, axiom,
    ((typing @ (shift_type @ e @ i @ t3) @ a @ t))). % argT
thf(fact_15__092_060open_062IT_A_I_Ilift_Au_A0_A_092_060degree_062_AVar_A0_J_091a_091u_Pi_093_P0_093_J_092_060close_062, axiom,
    ((it @ (subst @ (app @ (lift @ u @ zero_zero_nat) @ (var @ zero_zero_nat)) @ (subst @ a @ u @ i) @ zero_zero_nat)))). % \<open>IT ((lift u 0 \<degree> Var 0)[a[u/i]/0])\<close>
thf(fact_16_Var__IT, axiom,
    ((![N : nat]: (it @ (var @ N))))). % Var_IT
thf(fact_17_subst__lemma, axiom,
    ((![E2 : nat > type, T3 : dB, T4 : type, E3 : nat > type, U3 : dB, U4 : type, I2 : nat]: ((typing @ E2 @ T3 @ T4) => ((typing @ E3 @ U3 @ U4) => ((E2 = (shift_type @ E3 @ I2 @ U4)) => (typing @ E3 @ (subst @ T3 @ U3 @ I2) @ T4))))))). % subst_lemma
thf(fact_18_argsT, axiom,
    ((typings @ (shift_type @ e @ i @ t3) @ as @ ts))). % argsT
thf(fact_19_Cons_Ohyps_I3_J, axiom,
    ((listsp_dB @ it @ (map_dB_dB @ (^[T5 : dB]: (lift @ T5 @ zero_zero_nat)) @ (map_dB_dB @ (^[T5 : dB]: (subst @ T5 @ u @ i)) @ bs))))). % Cons.hyps(3)
thf(fact_20_Var_Oprems_I3_J, axiom,
    ((typing @ e1 @ u1 @ t3))). % Var.prems(3)
thf(fact_21_shift__eq, axiom,
    ((![I2 : nat, J : nat, E2 : nat > type, T4 : type]: ((I2 = J) => ((shift_type @ E2 @ I2 @ T4 @ J) = T4))))). % shift_eq
thf(fact_22_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_23_lift__type, axiom,
    ((![E2 : nat > type, T3 : dB, T4 : type, I2 : nat, U4 : type]: ((typing @ E2 @ T3 @ T4) => (typing @ (shift_type @ E2 @ I2 @ U4) @ (lift @ T3 @ I2) @ T4))))). % lift_type
thf(fact_24_typing__elims_I2_J, axiom,
    ((![E2 : nat > type, T3 : dB, U3 : dB, T4 : type]: ((typing @ E2 @ (app @ T3 @ U3) @ T4) => (~ ((![T6 : type]: ((typing @ E2 @ T3 @ (fun @ T6 @ T4)) => (~ ((typing @ E2 @ U3 @ T6))))))))))). % typing_elims(2)
thf(fact_25_typing__elims_I1_J, axiom,
    ((![E2 : nat > type, I2 : nat, T4 : type]: ((typing @ E2 @ (var @ I2) @ T4) => ((E2 @ I2) = T4))))). % typing_elims(1)
thf(fact_26_lift__types, axiom,
    ((![E2 : nat > type, Ts : list_dB, Ts2 : list_type, I2 : nat, U4 : type]: ((typings @ E2 @ Ts @ Ts2) => (typings @ (shift_type @ E2 @ I2 @ U4) @ (map_dB_dB @ (^[T5 : dB]: (lift @ T5 @ I2)) @ Ts) @ Ts2))))). % lift_types
thf(fact_27_App, axiom,
    ((![Env : nat > type, S : dB, T4 : type, U4 : type, T3 : dB]: ((typing @ Env @ S @ (fun @ T4 @ U4)) => ((typing @ Env @ T3 @ T4) => (typing @ Env @ (app @ S @ T3) @ U4)))))). % App
thf(fact_28_typing_OVar, axiom,
    ((![Env : nat > type, X : nat, T4 : type]: (((Env @ X) = T4) => (typing @ Env @ (var @ X) @ T4))))). % typing.Var
thf(fact_29_type__induct, axiom,
    ((![P : type > $o, T4 : type]: ((![T6 : type]: ((![T12 : type, T22 : type]: ((T6 = (fun @ T12 @ T22)) => (P @ T12))) => ((![T12 : type, T22 : type]: ((T6 = (fun @ T12 @ T22)) => (P @ T22))) => (P @ T6)))) => (P @ T4))))). % type_induct
thf(fact_30_substs__lemma, axiom,
    ((![E2 : nat > type, U3 : dB, T4 : type, I2 : nat, Ts : list_dB, Ts2 : list_type]: ((typing @ E2 @ U3 @ T4) => ((typings @ (shift_type @ E2 @ I2 @ T4) @ Ts @ Ts2) => (typings @ E2 @ (map_dB_dB @ (^[T5 : dB]: (subst @ T5 @ U3 @ I2)) @ Ts) @ Ts2)))))). % substs_lemma
thf(fact_31_lifts__IT, axiom,
    ((![Ts : list_dB]: ((listsp_dB @ it @ Ts) => (listsp_dB @ it @ (map_dB_dB @ (^[T5 : dB]: (lift @ T5 @ zero_zero_nat)) @ Ts)))))). % lifts_IT
thf(fact_32_app__Var__IT, axiom,
    ((![T3 : dB, I2 : nat]: ((it @ T3) => (it @ (app @ T3 @ (var @ I2))))))). % app_Var_IT
thf(fact_33_headT, axiom,
    ((typing @ (shift_type @ e @ i @ t3) @ (app @ (var @ n) @ a) @ (foldr_type_type @ fun @ ts @ t2)))). % headT
thf(fact_34_varT, axiom,
    ((typing @ (shift_type @ e @ i @ t3) @ (var @ n) @ (fun @ t @ (foldr_type_type @ fun @ ts @ t2))))). % varT
thf(fact_35__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_Aas_092_060close_062, axiom,
    ((listsp_dB @ (^[T5 : dB]: (?[X2 : type]: (typing @ (shift_type @ e @ i @ t3) @ T5 @ X2))) @ as))). % \<open>listsp (\<lambda>t. \<exists>T'. e\<langle>i:T\<rangle> \<turnstile> t : T') as\<close>
thf(fact_36_subst__lift, axiom,
    ((![T3 : dB, K : nat, S : dB]: ((subst @ (lift @ T3 @ K) @ S @ K) = T3)))). % subst_lift
thf(fact_37_subst__eq, axiom,
    ((![K : nat, U3 : dB]: ((subst @ (var @ K) @ U3 @ K) = U3)))). % subst_eq
thf(fact_38__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062Ts_O_A_092_060lbrakk_062e_092_060langle_062i_058T_092_060rangle_062_A_092_060turnstile_062_AVar_An_A_092_060degree_062_Aa_A_058_ATs_A_092_060Rrightarrow_062_AT_H_059_Ae_092_060langle_062i_058T_092_060rangle_062_A_092_060tturnstile_062_Aas_A_058_ATs_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom,
    ((~ ((![Ts3 : list_type]: ((typing @ (shift_type @ e @ i @ t3) @ (app @ (var @ n) @ a) @ (foldr_type_type @ fun @ Ts3 @ t2)) => (~ ((typings @ (shift_type @ e @ i @ t3) @ as @ Ts3))))))))). % \<open>\<And>thesis. (\<And>Ts. \<lbrakk>e\<langle>i:T\<rangle> \<turnstile> Var n \<degree> a : Ts \<Rrightarrow> T'; e\<langle>i:T\<rangle> \<tturnstile> as : Ts\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_39__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062T_H_H_O_A_092_060lbrakk_062e_092_060langle_062i_058T_092_060rangle_062_A_092_060turnstile_062_AVar_An_A_058_AT_H_H_A_092_060Rightarrow_062_ATs_A_092_060Rrightarrow_062_AT_H_059_Ae_092_060langle_062i_058T_092_060rangle_062_A_092_060turnstile_062_Aa_A_058_AT_H_H_092_060rbrakk_062_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062, axiom,
    ((~ ((![T7 : type]: ((typing @ (shift_type @ e @ i @ t3) @ (var @ n) @ (fun @ T7 @ (foldr_type_type @ fun @ ts @ t2))) => (~ ((typing @ (shift_type @ e @ i @ t3) @ a @ T7))))))))). % \<open>\<And>thesis. (\<And>T''. \<lbrakk>e\<langle>i:T\<rangle> \<turnstile> Var n : T'' \<Rightarrow> Ts \<Rrightarrow> T'; e\<langle>i:T\<rangle> \<turnstile> a : T''\<rbrakk> \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_40_listsp__conj__eq, axiom,
    ((![A : dB > $o, B : dB > $o]: ((listsp_dB @ (^[X3 : dB]: (((A @ X3)) & ((B @ X3))))) = (^[X3 : list_dB]: (((listsp_dB @ A @ X3)) & ((listsp_dB @ B @ X3)))))))). % listsp_conj_eq
thf(fact_41__092_060open_062e_092_060langle_062i_058T_092_060rangle_062_A_092_060turnstile_062_AVar_An_A_092_060degree_062_Aa_A_092_060degree_062_092_060degree_062_Aas_A_058_AT_H_092_060close_062, axiom,
    ((typing @ (shift_type @ e @ i @ t3) @ (foldl_dB_dB @ app @ (app @ (var @ n) @ a) @ as) @ t2))). % \<open>e\<langle>i:T\<rangle> \<turnstile> Var n \<degree> a \<degree>\<degree> as : T'\<close>
thf(fact_42_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_43_dB_Oinject_I1_J, axiom,
    ((![X1 : nat, Y1 : nat]: (((var @ X1) = (var @ Y1)) = (X1 = Y1))))). % dB.inject(1)
thf(fact_44_map__ident, axiom,
    (((map_dB_dB @ (^[X3 : dB]: X3)) = (^[Xs : list_dB]: Xs)))). % map_ident
thf(fact_45_T, axiom,
    ((t3 = (fun @ t @ (foldr_type_type @ fun @ ts @ t2))))). % T
thf(fact_46_nT, axiom,
    ((typing @ (shift_type @ e @ i @ t3) @ (foldl_dB_dB @ app @ (var @ n) @ rs) @ t2))). % nT
thf(fact_47_uT_H, axiom,
    ((typing @ e @ u @ (fun @ t @ (foldr_type_type @ fun @ ts @ t2))))). % uT'
thf(fact_48_Var_Oprems_I1_J, axiom,
    ((typing @ (shift_type @ e1 @ i1 @ t3) @ (foldl_dB_dB @ app @ (var @ n) @ rs) @ t_1))). % Var.prems(1)
thf(fact_49_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 @ (^[A3 : dB]: (^[X3 : dB]: (G @ A3 @ (F @ X3)))) @ A2 @ Xs2))))). % foldl_map
thf(fact_50_list__app__typeI, axiom,
    ((![E2 : nat > type, T3 : dB, Ts2 : list_type, T4 : type, Ts : list_dB]: ((typing @ E2 @ T3 @ (foldr_type_type @ fun @ Ts2 @ T4)) => ((typings @ E2 @ Ts @ Ts2) => (typing @ E2 @ (foldl_dB_dB @ app @ T3 @ Ts) @ T4)))))). % list_app_typeI
thf(fact_51_list__app__typeE, axiom,
    ((![E2 : nat > type, T3 : dB, Ts : list_dB, T4 : type]: ((typing @ E2 @ (foldl_dB_dB @ app @ T3 @ Ts) @ T4) => (~ ((![Ts3 : list_type]: ((typing @ E2 @ T3 @ (foldr_type_type @ fun @ Ts3 @ T4)) => (~ ((typings @ E2 @ Ts @ Ts3))))))))))). % list_app_typeE
thf(fact_52_list__app__typeD, axiom,
    ((![E2 : nat > type, T3 : dB, Ts : list_dB, T4 : type]: ((typing @ E2 @ (foldl_dB_dB @ app @ T3 @ Ts) @ T4) => (?[Ts3 : list_type]: ((typing @ E2 @ T3 @ (foldr_type_type @ fun @ Ts3 @ T4)) & (typings @ E2 @ Ts @ Ts3))))))). % list_app_typeD
thf(fact_53_var__app__typesE, axiom,
    ((![E2 : nat > type, I2 : nat, Ts : list_dB, T4 : type]: ((typing @ E2 @ (foldl_dB_dB @ app @ (var @ I2) @ Ts) @ T4) => (~ ((![Ts3 : list_type]: ((typing @ E2 @ (var @ I2) @ (foldr_type_type @ fun @ Ts3 @ T4)) => (~ ((typings @ E2 @ Ts @ Ts3))))))))))). % var_app_typesE
thf(fact_54_var__app__types, axiom,
    ((![E2 : nat > type, I2 : nat, Ts : list_dB, Us : list_dB, T4 : type, Ts2 : list_type, U4 : type]: ((typing @ E2 @ (foldl_dB_dB @ app @ (foldl_dB_dB @ app @ (var @ I2) @ Ts) @ Us) @ T4) => ((typings @ E2 @ Ts @ Ts2) => ((typing @ E2 @ (foldl_dB_dB @ app @ (var @ I2) @ Ts) @ U4) => (?[Us2 : list_type]: ((U4 = (foldr_type_type @ fun @ Us2 @ T4)) & (typings @ E2 @ Us @ Us2))))))))). % var_app_types
thf(fact_55_var__app__type__eq, axiom,
    ((![E2 : nat > type, I2 : nat, Ts : list_dB, T4 : type, U4 : type]: ((typing @ E2 @ (foldl_dB_dB @ app @ (var @ I2) @ Ts) @ T4) => ((typing @ E2 @ (foldl_dB_dB @ app @ (var @ I2) @ Ts) @ U4) => (T4 = U4)))))). % var_app_type_eq
thf(fact_56_lists__typings, axiom,
    ((![E2 : nat > type, Ts : list_dB, Ts2 : list_type]: ((typings @ E2 @ Ts @ Ts2) => (listsp_dB @ (^[T5 : dB]: (?[X2 : type]: (typing @ E2 @ T5 @ X2))) @ Ts))))). % lists_typings
thf(fact_57_list_Omap__ident, axiom,
    ((![T3 : list_dB]: ((map_dB_dB @ (^[X3 : dB]: X3) @ T3) = T3)))). % list.map_ident
thf(fact_58_dB_Odistinct_I1_J, axiom,
    ((![X1 : nat, X21 : dB, X22 : dB]: (~ (((var @ X1) = (app @ X21 @ X22))))))). % dB.distinct(1)
thf(fact_59_subst__App, axiom,
    ((![T3 : dB, U3 : dB, S : dB, K : nat]: ((subst @ (app @ T3 @ U3) @ S @ K) = (app @ (subst @ T3 @ S @ K) @ (subst @ U3 @ S @ K)))))). % subst_App
thf(fact_60_lift_Osimps_I2_J, axiom,
    ((![S : dB, T3 : dB, K : nat]: ((lift @ (app @ S @ T3) @ K) = (app @ (lift @ S @ K) @ (lift @ T3 @ K)))))). % lift.simps(2)
thf(fact_61_lift__map, axiom,
    ((![T3 : dB, Ts : list_dB, I2 : nat]: ((lift @ (foldl_dB_dB @ app @ T3 @ Ts) @ I2) = (foldl_dB_dB @ app @ (lift @ T3 @ I2) @ (map_dB_dB @ (^[T5 : dB]: (lift @ T5 @ I2)) @ Ts)))))). % lift_map
thf(fact_62_subst__map, axiom,
    ((![T3 : dB, Ts : list_dB, U3 : dB, I2 : nat]: ((subst @ (foldl_dB_dB @ app @ T3 @ Ts) @ U3 @ I2) = (foldl_dB_dB @ app @ (subst @ T3 @ U3 @ I2) @ (map_dB_dB @ (^[T5 : dB]: (subst @ T5 @ U3 @ I2)) @ Ts)))))). % subst_map
thf(fact_63_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_64_IT_OVar, axiom,
    ((![Rs : list_dB, N : nat]: ((listsp_dB @ it @ Rs) => (it @ (foldl_dB_dB @ app @ (var @ N) @ Rs)))))). % IT.Var
thf(fact_65_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_66_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_67_IT_Oinducts, axiom,
    ((![X : dB, P : dB > $o]: ((it @ X) => ((![Rs2 : list_dB, N2 : nat]: ((listsp_dB @ (^[X3 : dB]: (((it @ X3)) & ((P @ X3)))) @ 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 @ X)))))))). % IT.inducts
thf(fact_68_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_69_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_70_typing_Ocases, axiom,
    ((![A1 : nat > type, A22 : dB, A32 : type]: ((typing @ A1 @ A22 @ A32) => ((![X4 : nat]: ((A22 = (var @ X4)) => (~ (((A1 @ X4) = A32))))) => ((![T6 : type, T8 : dB]: ((A22 = (abs @ T8)) => (![U2 : type]: ((A32 = (fun @ T6 @ U2)) => (~ ((typing @ (shift_type @ A1 @ zero_zero_nat @ T6) @ T8 @ U2))))))) => (~ ((![S2 : dB, T6 : type, U2 : type, T8 : dB]: ((A22 = (app @ S2 @ T8)) => ((A32 = U2) => ((typing @ A1 @ S2 @ (fun @ T6 @ U2)) => (~ ((typing @ A1 @ T8 @ T6))))))))))))))). % typing.cases
thf(fact_71_dB_Oinject_I3_J, axiom,
    ((![X32 : dB, Y3 : dB]: (((abs @ X32) = (abs @ Y3)) = (X32 = Y3))))). % dB.inject(3)
thf(fact_72_Abs__apps__eq__Abs__apps__conv, axiom,
    ((![R : dB, Rs : list_dB, S : dB, Ss : list_dB]: (((foldl_dB_dB @ app @ (abs @ R) @ Rs) = (foldl_dB_dB @ app @ (abs @ S) @ Ss)) = (((R = S)) & ((Rs = Ss))))))). % Abs_apps_eq_Abs_apps_conv
thf(fact_73_Lambda, axiom,
    ((![R : dB]: ((it @ R) => (it @ (abs @ R)))))). % Lambda
thf(fact_74_dB_Odistinct_I5_J, axiom,
    ((![X21 : dB, X22 : dB, X32 : dB]: (~ (((app @ X21 @ X22) = (abs @ X32))))))). % dB.distinct(5)
thf(fact_75_dB_Odistinct_I3_J, axiom,
    ((![X1 : nat, X32 : dB]: (~ (((var @ X1) = (abs @ X32))))))). % dB.distinct(3)
thf(fact_76_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_77_dB_Oinduct, axiom,
    ((![P : dB > $o, DB : dB]: ((![X4 : nat]: (P @ (var @ X4))) => ((![X1a : dB, X23 : dB]: ((P @ X1a) => ((P @ X23) => (P @ (app @ X1a @ X23))))) => ((![X4 : dB]: ((P @ X4) => (P @ (abs @ X4)))) => (P @ DB))))))). % dB.induct
thf(fact_78_abs__typeE, axiom,
    ((![E2 : nat > type, T3 : dB, T4 : type]: ((typing @ E2 @ (abs @ T3) @ T4) => (~ ((![U2 : type, V : type]: (~ ((typing @ (shift_type @ E2 @ zero_zero_nat @ U2) @ T3 @ V)))))))))). % abs_typeE
thf(fact_79_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_80_Abs__App__neq__Var__apps, axiom,
    ((![S : dB, T3 : dB, N : nat, Ss : list_dB]: (~ (((app @ (abs @ S) @ T3) = (foldl_dB_dB @ app @ (var @ N) @ Ss))))))). % Abs_App_neq_Var_apps
thf(fact_81_ex__head__tail, axiom,
    ((![T3 : dB]: (?[Ts4 : list_dB, H : dB]: ((T3 = (foldl_dB_dB @ app @ H @ Ts4)) & ((?[N2 : nat]: (H = (var @ N2))) | (?[U5 : dB]: (H = (abs @ U5))))))))). % ex_head_tail
thf(fact_82_Abs, axiom,
    ((![Env : nat > type, T4 : type, T3 : dB, U4 : type]: ((typing @ (shift_type @ Env @ zero_zero_nat @ T4) @ T3 @ U4) => (typing @ Env @ (abs @ T3) @ (fun @ T4 @ U4)))))). % Abs
thf(fact_83_typing__elims_I3_J, axiom,
    ((![E2 : nat > type, T3 : dB, T4 : type]: ((typing @ E2 @ (abs @ T3) @ T4) => (~ ((![T6 : type, U2 : type]: ((T4 = (fun @ T6 @ U2)) => (~ ((typing @ (shift_type @ E2 @ zero_zero_nat @ T6) @ T3 @ U2))))))))))). % typing_elims(3)
thf(fact_84_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_85_typing_Oinducts, axiom,
    ((![X1 : nat > type, X24 : dB, X32 : type, P : (nat > type) > dB > type > $o]: ((typing @ X1 @ X24 @ X32) => ((![Env2 : nat > type, X4 : nat, T6 : type]: (((Env2 @ X4) = T6) => (P @ Env2 @ (var @ X4) @ T6))) => ((![Env2 : nat > type, T6 : type, T8 : dB, U2 : type]: ((typing @ (shift_type @ Env2 @ zero_zero_nat @ T6) @ T8 @ U2) => ((P @ (shift_type @ Env2 @ zero_zero_nat @ T6) @ T8 @ U2) => (P @ Env2 @ (abs @ T8) @ (fun @ T6 @ U2))))) => ((![Env2 : nat > type, S2 : dB, T6 : type, U2 : type, T8 : dB]: ((typing @ Env2 @ S2 @ (fun @ T6 @ U2)) => ((P @ Env2 @ S2 @ (fun @ T6 @ U2)) => ((typing @ Env2 @ T8 @ T6) => ((P @ Env2 @ T8 @ T6) => (P @ Env2 @ (app @ S2 @ T8) @ U2)))))) => (P @ X1 @ X24 @ X32)))))))). % typing.inducts
thf(fact_86_typing_Osimps, axiom,
    ((typing = (^[A12 : nat > type]: (^[A23 : dB]: (^[A33 : type]: (((?[Env3 : nat > type]: (?[X3 : nat]: (?[T9 : type]: (((A12 = Env3)) & ((((A23 = (var @ X3))) & ((((A33 = T9)) & (((Env3 @ X3) = T9))))))))))) | ((((?[Env3 : nat > type]: (?[T9 : type]: (?[T5 : dB]: (?[U6 : type]: (((A12 = Env3)) & ((((A23 = (abs @ T5))) & ((((A33 = (fun @ T9 @ U6))) & ((typing @ (shift_type @ Env3 @ zero_zero_nat @ T9) @ T5 @ U6)))))))))))) | ((?[Env3 : nat > type]: (?[S3 : dB]: (?[T9 : type]: (?[U6 : type]: (?[T5 : dB]: (((A12 = Env3)) & ((((A23 = (app @ S3 @ T5))) & ((((A33 = U6)) & ((((typing @ Env3 @ S3 @ (fun @ T9 @ U6))) & ((typing @ Env3 @ T5 @ T9)))))))))))))))))))))))). % typing.simps
thf(fact_87_Cons_Ohyps_I2_J, axiom,
    ((listsp_dB @ (^[T5 : dB]: (((![E4 : nat > type]: (![T10 : type]: (![U7 : dB]: (![I3 : nat]: (((typing @ (shift_type @ E4 @ I3 @ t3) @ T5 @ T10)) => ((((it @ U7)) => ((((typing @ E4 @ U7 @ t3)) => ((it @ (subst @ T5 @ U7 @ I3))))))))))))) & ((?[X2 : type]: (typing @ (shift_type @ e @ i @ t3) @ T5 @ X2))))) @ bs))). % Cons.hyps(2)
thf(fact_88_dB_Osize__gen_I1_J, axiom,
    ((![X1 : nat]: ((size_dB @ (var @ X1)) = zero_zero_nat)))). % dB.size_gen(1)
thf(fact_89_map__eq__map__tailrec, axiom,
    ((map_dB_dB = map_tailrec_dB_dB))). % map_eq_map_tailrec
thf(fact_90_Var_Ohyps, axiom,
    ((listsp_dB @ (^[X3 : dB]: (((it @ X3)) & ((![Y2 : nat > type]: (![Z : type]: (![Aa : dB]: (![Ab : nat]: (((typing @ (shift_type @ Y2 @ Ab @ t3) @ X3 @ Z)) => ((((it @ Aa)) => ((((typing @ Y2 @ Aa @ t3)) => ((it @ (subst @ X3 @ Aa @ Ab))))))))))))))) @ rs))). % Var.hyps
thf(fact_91__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_Aas_092_060close_062, axiom,
    ((listsp_dB @ (^[T5 : dB]: (![E4 : nat > type]: (![T10 : type]: (![U7 : dB]: (![I3 : nat]: (((typing @ (shift_type @ E4 @ I3 @ t3) @ T5 @ T10)) => ((((it @ U7)) => ((((typing @ E4 @ U7 @ t3)) => ((it @ (subst @ T5 @ U7 @ I3))))))))))))) @ as))). % \<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])) as\<close>
thf(fact_92__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_Aas_092_060close_062, axiom,
    ((listsp_dB @ (^[T5 : dB]: (((![E4 : nat > type]: (![T10 : type]: (![U7 : dB]: (![I3 : nat]: (((typing @ (shift_type @ E4 @ I3 @ t3) @ T5 @ T10)) => ((((it @ U7)) => ((((typing @ E4 @ U7 @ t3)) => ((it @ (subst @ T5 @ U7 @ I3))))))))))))) & ((?[X2 : type]: (typing @ (shift_type @ e @ i @ t3) @ T5 @ X2))))) @ as))). % \<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')) as\<close>
thf(fact_93_substn__subst__0, axiom,
    ((![T3 : dB, S : dB]: ((substn @ T3 @ S @ zero_zero_nat) = (subst @ T3 @ S @ zero_zero_nat))))). % substn_subst_0
thf(fact_94_Apps__dB__induct, axiom,
    ((![P : dB > $o, T3 : dB]: ((![N2 : nat, Ts4 : list_dB]: ((![X5 : dB]: ((member_dB @ X5 @ (set_dB2 @ Ts4)) => (P @ X5))) => (P @ (foldl_dB_dB @ app @ (var @ N2) @ Ts4)))) => ((![U5 : dB]: ((P @ U5) => (![Ts4 : list_dB]: ((![X5 : dB]: ((member_dB @ X5 @ (set_dB2 @ Ts4)) => (P @ X5))) => (P @ (foldl_dB_dB @ app @ (abs @ U5) @ Ts4)))))) => (P @ T3)))))). % Apps_dB_induct
thf(fact_95_beta__cases_I3_J, axiom,
    ((![S : dB, T3 : dB, U3 : dB]: ((beta @ (app @ S @ T3) @ U3) => ((![S2 : dB]: ((S = (abs @ S2)) => (~ ((U3 = (subst @ S2 @ T3 @ zero_zero_nat)))))) => ((![T8 : dB]: ((U3 = (app @ T8 @ T3)) => (~ ((beta @ S @ T8))))) => (~ ((![T8 : dB]: ((U3 = (app @ S @ T8)) => (~ ((beta @ T3 @ T8))))))))))))). % beta_cases(3)
thf(fact_96_map__eq__conv, axiom,
    ((![F : dB > dB, Xs2 : list_dB, G : dB > dB]: (((map_dB_dB @ F @ Xs2) = (map_dB_dB @ G @ Xs2)) = (![X3 : dB]: (((member_dB @ X3 @ (set_dB2 @ Xs2))) => (((F @ X3) = (G @ X3))))))))). % map_eq_conv
thf(fact_97_in__listspI, axiom,
    ((![Xs2 : list_dB, A : dB > $o]: ((![X4 : dB]: ((member_dB @ X4 @ (set_dB2 @ Xs2)) => (A @ X4))) => (listsp_dB @ A @ Xs2))))). % in_listspI
thf(fact_98_list_Oinj__map__strong, axiom,
    ((![X : list_dB, Xa : list_dB, F : dB > dB, Fa : dB > dB]: ((![Z2 : dB, Za : dB]: ((member_dB @ Z2 @ (set_dB2 @ X)) => ((member_dB @ Za @ (set_dB2 @ Xa)) => (((F @ Z2) = (Fa @ Za)) => (Z2 = Za))))) => (((map_dB_dB @ F @ X) = (map_dB_dB @ Fa @ Xa)) => (X = Xa)))))). % list.inj_map_strong
thf(fact_99_list_Omap__cong0, axiom,
    ((![X : list_dB, F : dB > dB, G : dB > dB]: ((![Z2 : dB]: ((member_dB @ Z2 @ (set_dB2 @ X)) => ((F @ Z2) = (G @ Z2)))) => ((map_dB_dB @ F @ X) = (map_dB_dB @ G @ X)))))). % list.map_cong0
thf(fact_100_list_Omap__cong, axiom,
    ((![X : list_dB, Ya : list_dB, F : dB > dB, G : dB > dB]: ((X = Ya) => ((![Z2 : dB]: ((member_dB @ Z2 @ (set_dB2 @ Ya)) => ((F @ Z2) = (G @ Z2)))) => ((map_dB_dB @ F @ X) = (map_dB_dB @ G @ Ya))))))). % list.map_cong
thf(fact_101_ex__map__conv, axiom,
    ((![Ys : list_dB, F : dB > dB]: ((?[Xs : list_dB]: (Ys = (map_dB_dB @ F @ Xs))) = (![X3 : dB]: (((member_dB @ X3 @ (set_dB2 @ Ys))) => ((?[Y2 : dB]: (X3 = (F @ Y2)))))))))). % ex_map_conv
thf(fact_102_map__cong, axiom,
    ((![Xs2 : list_dB, Ys : list_dB, F : dB > dB, G : dB > dB]: ((Xs2 = Ys) => ((![X4 : dB]: ((member_dB @ X4 @ (set_dB2 @ Ys)) => ((F @ X4) = (G @ X4)))) => ((map_dB_dB @ F @ Xs2) = (map_dB_dB @ G @ Ys))))))). % map_cong
thf(fact_103_map__idI, axiom,
    ((![Xs2 : list_dB, F : dB > dB]: ((![X4 : dB]: ((member_dB @ X4 @ (set_dB2 @ Xs2)) => ((F @ X4) = X4))) => ((map_dB_dB @ F @ Xs2) = Xs2))))). % map_idI
thf(fact_104_map__ext, axiom,
    ((![Xs2 : list_dB, F : dB > dB, G : dB > dB]: ((![X4 : dB]: ((member_dB @ X4 @ (set_dB2 @ Xs2)) => ((F @ X4) = (G @ X4)))) => ((map_dB_dB @ F @ Xs2) = (map_dB_dB @ G @ Xs2)))))). % map_ext
thf(fact_105_appL, axiom,
    ((![S : dB, T3 : dB, U3 : dB]: ((beta @ S @ T3) => (beta @ (app @ S @ U3) @ (app @ T3 @ U3)))))). % appL
thf(fact_106_appR, axiom,
    ((![S : dB, T3 : dB, U3 : dB]: ((beta @ S @ T3) => (beta @ (app @ U3 @ S) @ (app @ U3 @ T3)))))). % appR
thf(fact_107_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, X4 : dB]: ((member_dB @ X4 @ (set_dB2 @ L)) => ((F @ A4 @ X4) = (G @ A4 @ X4)))) => ((foldl_dB_dB @ F @ A2 @ L) = (foldl_dB_dB @ G @ B2 @ K)))))))). % foldl_cong
thf(fact_108_subject__reduction, axiom,
    ((![E2 : nat > type, T3 : dB, T4 : type, T11 : dB]: ((typing @ E2 @ T3 @ T4) => ((beta @ T3 @ T11) => (typing @ E2 @ T11 @ T4)))))). % subject_reduction
thf(fact_109_beta__cases_I1_J, axiom,
    ((![I2 : nat, T3 : dB]: (~ ((beta @ (var @ I2) @ T3)))))). % beta_cases(1)
thf(fact_110_in__listspD, axiom,
    ((![A : dB > $o, Xs2 : list_dB]: ((listsp_dB @ A @ Xs2) => (![X5 : dB]: ((member_dB @ X5 @ (set_dB2 @ Xs2)) => (A @ X5))))))). % in_listspD
thf(fact_111_in__listsp__conv__set, axiom,
    ((listsp_dB = (^[A5 : dB > $o]: (^[Xs : list_dB]: (![X3 : dB]: (((member_dB @ X3 @ (set_dB2 @ Xs))) => ((A5 @ X3))))))))). % in_listsp_conv_set
thf(fact_112_subst__preserves__beta, axiom,
    ((![R : dB, S : dB, T3 : dB, I2 : nat]: ((beta @ R @ S) => (beta @ (subst @ R @ T3 @ I2) @ (subst @ S @ T3 @ I2)))))). % subst_preserves_beta
thf(fact_113_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, X4 : type]: ((member_type @ X4 @ (set_type2 @ L)) => ((F @ X4 @ A4) = (G @ X4 @ A4)))) => ((foldr_type_type @ F @ L @ A2) = (foldr_type_type @ G @ K @ B2)))))))). % foldr_cong
thf(fact_114_lift__preserves__beta, axiom,
    ((![R : dB, S : dB, I2 : nat]: ((beta @ R @ S) => (beta @ (lift @ R @ I2) @ (lift @ S @ I2)))))). % lift_preserves_beta
thf(fact_115_abs, axiom,
    ((![S : dB, T3 : dB]: ((beta @ S @ T3) => (beta @ (abs @ S) @ (abs @ T3)))))). % abs
thf(fact_116_beta__cases_I2_J, axiom,
    ((![R : dB, S : dB]: ((beta @ (abs @ R) @ S) => (~ ((![T8 : dB]: ((S = (abs @ T8)) => (~ ((beta @ R @ T8))))))))))). % beta_cases(2)
thf(fact_117_substn_Osimps_I2_J, axiom,
    ((![T3 : dB, U3 : dB, S : dB, K : nat]: ((substn @ (app @ T3 @ U3) @ S @ K) = (app @ (substn @ T3 @ S @ K) @ (substn @ U3 @ S @ K)))))). % substn.simps(2)
thf(fact_118_beta_Oinducts, axiom,
    ((![X1 : dB, X24 : dB, P : dB > dB > $o]: ((beta @ X1 @ X24) => ((![S2 : dB, T8 : dB]: (P @ (app @ (abs @ S2) @ T8) @ (subst @ S2 @ T8 @ zero_zero_nat))) => ((![S2 : dB, T8 : dB, U5 : dB]: ((beta @ S2 @ T8) => ((P @ S2 @ T8) => (P @ (app @ S2 @ U5) @ (app @ T8 @ U5))))) => ((![S2 : dB, T8 : dB, U5 : dB]: ((beta @ S2 @ T8) => ((P @ S2 @ T8) => (P @ (app @ U5 @ S2) @ (app @ U5 @ T8))))) => ((![S2 : dB, T8 : dB]: ((beta @ S2 @ T8) => ((P @ S2 @ T8) => (P @ (abs @ S2) @ (abs @ T8))))) => (P @ X1 @ X24))))))))). % beta.inducts
thf(fact_119_beta_Osimps, axiom,
    ((beta = (^[A12 : dB]: (^[A23 : dB]: (((?[S3 : dB]: (?[T5 : dB]: (((A12 = (app @ (abs @ S3) @ T5))) & ((A23 = (subst @ S3 @ T5 @ zero_zero_nat))))))) | ((((?[S3 : dB]: (?[T5 : dB]: (?[U7 : dB]: (((A12 = (app @ S3 @ U7))) & ((((A23 = (app @ T5 @ U7))) & ((beta @ S3 @ T5))))))))) | ((((?[S3 : dB]: (?[T5 : dB]: (?[U7 : dB]: (((A12 = (app @ U7 @ S3))) & ((((A23 = (app @ U7 @ T5))) & ((beta @ S3 @ T5))))))))) | ((?[S3 : dB]: (?[T5 : dB]: (((A12 = (abs @ S3))) & ((((A23 = (abs @ T5))) & ((beta @ S3 @ T5)))))))))))))))))). % beta.simps
thf(fact_120_beta_Ocases, axiom,
    ((![A1 : dB, A22 : dB]: ((beta @ A1 @ A22) => ((![S2 : dB, T8 : dB]: ((A1 = (app @ (abs @ S2) @ T8)) => (~ ((A22 = (subst @ S2 @ T8 @ zero_zero_nat)))))) => ((![S2 : dB, T8 : dB, U5 : dB]: ((A1 = (app @ S2 @ U5)) => ((A22 = (app @ T8 @ U5)) => (~ ((beta @ S2 @ T8)))))) => ((![S2 : dB, T8 : dB, U5 : dB]: ((A1 = (app @ U5 @ S2)) => ((A22 = (app @ U5 @ T8)) => (~ ((beta @ S2 @ T8)))))) => (~ ((![S2 : dB]: ((A1 = (abs @ S2)) => (![T8 : dB]: ((A22 = (abs @ T8)) => (~ ((beta @ S2 @ T8)))))))))))))))). % beta.cases
thf(fact_121_beta, axiom,
    ((![S : dB, T3 : dB]: (beta @ (app @ (abs @ S) @ T3) @ (subst @ S @ T3 @ zero_zero_nat))))). % beta
thf(fact_122_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_123_lem, axiom,
    ((![P : dB > $o, T3 : dB, N : nat]: ((![N2 : nat, Ts4 : list_dB]: ((![X5 : dB]: ((member_dB @ X5 @ (set_dB2 @ Ts4)) => (P @ X5))) => (P @ (foldl_dB_dB @ app @ (var @ N2) @ Ts4)))) => ((![U5 : dB]: ((P @ U5) => (![Ts4 : list_dB]: ((![X5 : dB]: ((member_dB @ X5 @ (set_dB2 @ Ts4)) => (P @ X5))) => (P @ (foldl_dB_dB @ app @ (abs @ U5) @ Ts4)))))) => (((size_size_dB @ T3) = N) => (P @ T3))))))). % lem
thf(fact_124_substn__subst__n, axiom,
    ((substn = (^[T5 : dB]: (^[S3 : dB]: (^[N3 : nat]: (subst @ T5 @ (liftn @ N3 @ S3 @ zero_zero_nat) @ N3))))))). % substn_subst_n
thf(fact_125_liftn__0, axiom,
    ((![T3 : dB, K : nat]: ((liftn @ zero_zero_nat @ T3 @ K) = T3)))). % liftn_0
thf(fact_126_liftn_Osimps_I2_J, axiom,
    ((![N : nat, S : dB, T3 : dB, K : nat]: ((liftn @ N @ (app @ S @ T3) @ K) = (app @ (liftn @ N @ S @ K) @ (liftn @ N @ T3 @ K)))))). % liftn.simps(2)
thf(fact_127_dB_Osize_I4_J, axiom,
    ((![X1 : nat]: ((size_size_dB @ (var @ X1)) = zero_zero_nat)))). % dB.size(4)
thf(fact_128_count__notin, axiom,
    ((![X : dB, Xs2 : list_dB]: ((~ ((member_dB @ X @ (set_dB2 @ Xs2)))) => ((count_list_dB @ Xs2 @ X) = zero_zero_nat))))). % count_notin
thf(fact_129_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_130_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_131_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)))))) => (~ ((![T8 : dB]: ((R = (abs @ T8)) => (![U5 : dB, Us3 : list_dB]: ((Rs = (cons_dB @ U5 @ Us3)) => (~ ((S = (foldl_dB_dB @ app @ (subst @ T8 @ U5 @ zero_zero_nat) @ Us3)))))))))))))))). % apps_betasE
thf(fact_132_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_133_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_134_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_135_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_136_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_137_listsp__simps_I1_J, axiom,
    ((![A : dB > $o]: (listsp_dB @ A @ nil_dB)))). % listsp_simps(1)
thf(fact_138_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_139_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_140_count__list_Osimps_I1_J, axiom,
    ((![Y : dB]: ((count_list_dB @ nil_dB @ Y) = zero_zero_nat)))). % count_list.simps(1)
thf(fact_141_foldl__Nil, axiom,
    ((![F : dB > dB > dB, A2 : dB]: ((foldl_dB_dB @ F @ A2 @ nil_dB) = A2)))). % foldl_Nil
thf(fact_142_listsp_ONil, axiom,
    ((![A : dB > $o]: (listsp_dB @ A @ nil_dB)))). % listsp.Nil
thf(fact_143_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

% Conjectures (1)
thf(conj_0, conjecture,
    ((it @ (subst @ b @ u @ i)))).
