% 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_135__5210612_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:37:41.637

% 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 (49)
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_If_001t__List__Olist_It__Lambda__OdB_J, type,
    if_list_dB : $o > list_dB > list_dB > list_dB).
thf(sy_c_InductTermi_OIT, type,
    it : dB > $o).
thf(sy_c_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_Oinsert_001t__Lambda__OdB, type,
    insert_dB : dB > list_dB > list_dB).
thf(sy_c_List_Olist_OCons_001t__LambdaType__Otype, type,
    cons_type : type > list_type > list_type).
thf(sy_c_List_Olist_OCons_001t__Lambda__OdB, type,
    cons_dB : dB > list_dB > list_dB).
thf(sy_c_List_Olist_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_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_Nat_Osize__class_Osize_001t__List__Olist_It__Lambda__OdB_J, type,
    size_size_list_dB : list_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_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,
    t4 : dB).
thf(sy_v_u1____, type,
    u1 : dB).
thf(sy_v_u____, type,
    u : dB).

% Relevant facts (143)
thf(fact_0_T, axiom,
    ((t3 = (fun @ t @ (foldr_type_type @ fun @ ts @ t2))))). % T
thf(fact_1__092_060open_062IT_At_092_060close_062, axiom,
    ((it @ t4))). % \<open>IT t\<close>
thf(fact_2_Var_Oprems_I2_J, axiom,
    ((it @ u1))). % Var.prems(2)
thf(fact_3_uIT, axiom,
    ((it @ u))). % uIT
thf(fact_4_True, axiom,
    ((n = i))). % True
thf(fact_5_lift__IT, axiom,
    ((![T : dB, I : nat]: ((it @ T) => (it @ (lift @ T @ I)))))). % lift_IT
thf(fact_6_Var__IT, axiom,
    ((![N : nat]: (it @ (var @ N))))). % Var_IT
thf(fact_7_app__Var__IT, axiom,
    ((![T : dB, I : nat]: ((it @ T) => (it @ (app @ T @ (var @ I))))))). % app_Var_IT
thf(fact_8_subst__Var__IT, axiom,
    ((![R : dB, I : nat, J : nat]: ((it @ R) => (it @ (subst @ R @ (var @ I) @ J)))))). % subst_Var_IT
thf(fact_9_subst__lift, axiom,
    ((![T : dB, K : nat, S : dB]: ((subst @ (lift @ T @ K) @ S @ K) = T)))). % subst_lift
thf(fact_10_subst__eq, axiom,
    ((![K : nat, U : dB]: ((subst @ (var @ K) @ U @ K) = U)))). % subst_eq
thf(fact_11_dB_Oinject_I1_J, axiom,
    ((![X1 : nat, Y1 : nat]: (((var @ X1) = (var @ Y1)) = (X1 = Y1))))). % dB.inject(1)
thf(fact_12_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_13_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_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_zero__natural_Orsp, axiom,
    ((zero_zero_nat = zero_zero_nat))). % zero_natural.rsp
thf(fact_17_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_18_uT, axiom,
    ((typing @ e @ u @ t3))). % uT
thf(fact_19_uT_H, axiom,
    ((typing @ e @ u @ (fun @ t @ (foldr_type_type @ fun @ ts @ t2))))). % uT'
thf(fact_20_argT, axiom,
    ((typing @ (shift_type @ e @ i @ t3) @ a @ t))). % argT
thf(fact_21_varT, axiom,
    ((typing @ (shift_type @ e @ i @ t3) @ (var @ n) @ (fun @ t @ (foldr_type_type @ fun @ ts @ t2))))). % varT
thf(fact_22_headT, axiom,
    ((typing @ (shift_type @ e @ i @ t3) @ (app @ (var @ n) @ a) @ (foldr_type_type @ fun @ ts @ t2)))). % headT
thf(fact_23__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,
    ((~ ((![T2 : type]: ((typing @ (shift_type @ e @ i @ t3) @ (var @ n) @ (fun @ T2 @ (foldr_type_type @ fun @ ts @ t2))) => (~ ((typing @ (shift_type @ e @ i @ t3) @ a @ T2))))))))). % \<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_24_MI2, axiom,
    ((![T1 : type, T22 : type, T : dB, E : nat > type, I : nat, T3 : type, U : dB]: ((t3 = (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_25_MI1, axiom,
    ((![T1 : type, T22 : type, T : dB, E : nat > type, I : nat, T3 : type, U : dB]: ((t3 = (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_26_App, axiom,
    ((![Env : nat > type, S : dB, T3 : type, U2 : type, T : dB]: ((typing @ Env @ S @ (fun @ T3 @ U2)) => ((typing @ Env @ T @ T3) => (typing @ Env @ (app @ S @ T) @ U2)))))). % App
thf(fact_27_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_28__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,
    ((~ ((![Ts : list_type]: ((typing @ (shift_type @ e @ i @ t3) @ (app @ (var @ n) @ a) @ (foldr_type_type @ fun @ Ts @ t2)) => (~ ((typings @ (shift_type @ e @ i @ t3) @ as @ Ts))))))))). % \<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_29_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_30_nT, axiom,
    ((typing @ (shift_type @ e @ i @ t3) @ (foldl_dB_dB @ app @ (var @ n) @ rs) @ t2))). % nT
thf(fact_31__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_32_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_33_argsT, axiom,
    ((typings @ (shift_type @ e @ i @ t3) @ as @ ts))). % argsT
thf(fact_34_lift__type, axiom,
    ((![E : nat > type, T : dB, T3 : type, I : nat, U2 : type]: ((typing @ E @ T @ T3) => (typing @ (shift_type @ E @ I @ U2) @ (lift @ T @ I) @ T3))))). % lift_type
thf(fact_35_local_OCons, axiom,
    ((rs = (cons_dB @ a @ as)))). % local.Cons
thf(fact_36_subst__lemma, axiom,
    ((![E : nat > type, T : dB, T3 : type, E2 : nat > type, U : dB, U2 : type, I : nat]: ((typing @ E @ T @ T3) => ((typing @ E2 @ U @ U2) => ((E = (shift_type @ E2 @ I @ U2)) => (typing @ E2 @ (subst @ T @ U @ I) @ T3))))))). % subst_lemma
thf(fact_37_list__app__typeI, axiom,
    ((![E : nat > type, T : dB, Ts2 : list_type, T3 : type, Ts3 : list_dB]: ((typing @ E @ T @ (foldr_type_type @ fun @ Ts2 @ T3)) => ((typings @ E @ Ts3 @ Ts2) => (typing @ E @ (foldl_dB_dB @ app @ T @ Ts3) @ T3)))))). % list_app_typeI
thf(fact_38_list__app__typeE, axiom,
    ((![E : nat > type, T : dB, Ts3 : list_dB, T3 : type]: ((typing @ E @ (foldl_dB_dB @ app @ T @ Ts3) @ T3) => (~ ((![Ts : list_type]: ((typing @ E @ T @ (foldr_type_type @ fun @ Ts @ T3)) => (~ ((typings @ E @ Ts3 @ Ts))))))))))). % list_app_typeE
thf(fact_39_list__app__typeD, axiom,
    ((![E : nat > type, T : dB, Ts3 : list_dB, T3 : type]: ((typing @ E @ (foldl_dB_dB @ app @ T @ Ts3) @ T3) => (?[Ts : list_type]: ((typing @ E @ T @ (foldr_type_type @ fun @ Ts @ T3)) & (typings @ E @ Ts3 @ Ts))))))). % list_app_typeD
thf(fact_40_var__app__type__eq, axiom,
    ((![E : nat > type, I : nat, Ts3 : list_dB, T3 : type, U2 : type]: ((typing @ E @ (foldl_dB_dB @ app @ (var @ I) @ Ts3) @ T3) => ((typing @ E @ (foldl_dB_dB @ app @ (var @ I) @ Ts3) @ U2) => (T3 = U2)))))). % var_app_type_eq
thf(fact_41_var__app__types, axiom,
    ((![E : nat > type, I : nat, Ts3 : list_dB, Us : list_dB, T3 : type, Ts2 : list_type, U2 : type]: ((typing @ E @ (foldl_dB_dB @ app @ (foldl_dB_dB @ app @ (var @ I) @ Ts3) @ Us) @ T3) => ((typings @ E @ Ts3 @ Ts2) => ((typing @ E @ (foldl_dB_dB @ app @ (var @ I) @ Ts3) @ U2) => (?[Us2 : list_type]: ((U2 = (foldr_type_type @ fun @ Us2 @ T3)) & (typings @ E @ Us @ Us2))))))))). % var_app_types
thf(fact_42_var__app__typesE, axiom,
    ((![E : nat > type, I : nat, Ts3 : list_dB, T3 : type]: ((typing @ E @ (foldl_dB_dB @ app @ (var @ I) @ Ts3) @ T3) => (~ ((![Ts : list_type]: ((typing @ E @ (var @ I) @ (foldr_type_type @ fun @ Ts @ T3)) => (~ ((typings @ E @ Ts3 @ Ts))))))))))). % var_app_typesE
thf(fact_43_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_44_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_45_typing_OVar, axiom,
    ((![Env : nat > type, X : nat, T3 : type]: (((Env @ X) = T3) => (typing @ Env @ (var @ X) @ T3))))). % typing.Var
thf(fact_46_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_47_apps__eq__tail__conv, axiom,
    ((![R : dB, Ts3 : list_dB, S : dB]: (((foldl_dB_dB @ app @ R @ Ts3) = (foldl_dB_dB @ app @ S @ Ts3)) = (R = S))))). % apps_eq_tail_conv
thf(fact_48_typing_Ocases, axiom,
    ((![A1 : nat > type, A2 : dB, A3 : type]: ((typing @ A1 @ A2 @ A3) => ((![X2 : nat]: ((A2 = (var @ X2)) => (~ (((A1 @ X2) = A3))))) => ((![T4 : type, T5 : dB]: ((A2 = (abs @ T5)) => (![U3 : type]: ((A3 = (fun @ T4 @ U3)) => (~ ((typing @ (shift_type @ A1 @ zero_zero_nat @ T4) @ T5 @ U3))))))) => (~ ((![S2 : dB, T4 : type, U3 : type, T5 : dB]: ((A2 = (app @ S2 @ T5)) => ((A3 = U3) => ((typing @ A1 @ S2 @ (fun @ T4 @ U3)) => (~ ((typing @ A1 @ T5 @ T4))))))))))))))). % typing.cases
thf(fact_49_typing_Osimps, axiom,
    ((typing = (^[A12 : nat > type]: (^[A22 : dB]: (^[A32 : type]: (((?[Env2 : nat > type]: (?[X3 : nat]: (?[T6 : type]: (((A12 = Env2)) & ((((A22 = (var @ X3))) & ((((A32 = T6)) & (((Env2 @ X3) = T6))))))))))) | ((((?[Env2 : nat > type]: (?[T6 : type]: (?[T7 : dB]: (?[U4 : type]: (((A12 = Env2)) & ((((A22 = (abs @ T7))) & ((((A32 = (fun @ T6 @ U4))) & ((typing @ (shift_type @ Env2 @ zero_zero_nat @ T6) @ T7 @ U4)))))))))))) | ((?[Env2 : nat > type]: (?[S3 : dB]: (?[T6 : type]: (?[U4 : type]: (?[T7 : dB]: (((A12 = Env2)) & ((((A22 = (app @ S3 @ T7))) & ((((A32 = U4)) & ((((typing @ Env2 @ S3 @ (fun @ T6 @ U4))) & ((typing @ Env2 @ T7 @ T6)))))))))))))))))))))))). % typing.simps
thf(fact_50_typing_Oinducts, axiom,
    ((![X1 : nat > type, X23 : dB, X32 : type, P : (nat > type) > dB > type > $o]: ((typing @ X1 @ X23 @ X32) => ((![Env3 : nat > type, X2 : nat, T4 : type]: (((Env3 @ X2) = T4) => (P @ Env3 @ (var @ X2) @ T4))) => ((![Env3 : nat > type, T4 : type, T5 : dB, U3 : type]: ((typing @ (shift_type @ Env3 @ zero_zero_nat @ T4) @ T5 @ U3) => ((P @ (shift_type @ Env3 @ zero_zero_nat @ T4) @ T5 @ U3) => (P @ Env3 @ (abs @ T5) @ (fun @ T4 @ U3))))) => ((![Env3 : nat > type, S2 : dB, T4 : type, U3 : type, T5 : dB]: ((typing @ Env3 @ S2 @ (fun @ T4 @ U3)) => ((P @ Env3 @ S2 @ (fun @ T4 @ U3)) => ((typing @ Env3 @ T5 @ T4) => ((P @ Env3 @ T5 @ T4) => (P @ Env3 @ (app @ S2 @ T5) @ U3)))))) => (P @ X1 @ X23 @ X32)))))))). % typing.inducts
thf(fact_51_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_52_dB_Osize__gen_I1_J, axiom,
    ((![X1 : nat]: ((size_dB @ (var @ X1)) = zero_zero_nat)))). % dB.size_gen(1)
thf(fact_53_Var_Oprems_I3_J, axiom,
    ((typing @ e1 @ u1 @ t3))). % Var.prems(3)
thf(fact_54_dB_Oinject_I3_J, axiom,
    ((![X32 : dB, Y3 : dB]: (((abs @ X32) = (abs @ Y3)) = (X32 = Y3))))). % dB.inject(3)
thf(fact_55_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_56_dB_Odistinct_I5_J, axiom,
    ((![X21 : dB, X22 : dB, X32 : dB]: (~ (((app @ X21 @ X22) = (abs @ X32))))))). % dB.distinct(5)
thf(fact_57_dB_Odistinct_I3_J, axiom,
    ((![X1 : nat, X32 : dB]: (~ (((var @ X1) = (abs @ X32))))))). % dB.distinct(3)
thf(fact_58_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_59_dB_Oinduct, axiom,
    ((![P : dB > $o, DB : dB]: ((![X2 : nat]: (P @ (var @ X2))) => ((![X1a : dB, X24 : dB]: ((P @ X1a) => ((P @ X24) => (P @ (app @ X1a @ X24))))) => ((![X2 : dB]: ((P @ X2) => (P @ (abs @ X2)))) => (P @ DB))))))). % dB.induct
thf(fact_60_Var__apps__neq__Abs__apps, axiom,
    ((![N : nat, Ts3 : list_dB, R : dB, Ss : list_dB]: (~ (((foldl_dB_dB @ app @ (var @ N) @ Ts3) = (foldl_dB_dB @ app @ (abs @ R) @ Ss))))))). % Var_apps_neq_Abs_apps
thf(fact_61_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_62_ex__head__tail, axiom,
    ((![T : dB]: (?[Ts4 : list_dB, H : dB]: ((T = (foldl_dB_dB @ app @ H @ Ts4)) & ((?[N2 : nat]: (H = (var @ N2))) | (?[U5 : dB]: (H = (abs @ U5))))))))). % ex_head_tail
thf(fact_63_abs__typeE, axiom,
    ((![E : nat > type, T : dB, T3 : type]: ((typing @ E @ (abs @ T) @ T3) => (~ ((![U3 : type, V : type]: (~ ((typing @ (shift_type @ E @ zero_zero_nat @ U3) @ T @ V)))))))))). % abs_typeE
thf(fact_64_typing__elims_I3_J, axiom,
    ((![E : nat > type, T : dB, T3 : type]: ((typing @ E @ (abs @ T) @ T3) => (~ ((![T4 : type, U3 : type]: ((T3 = (fun @ T4 @ U3)) => (~ ((typing @ (shift_type @ E @ zero_zero_nat @ T4) @ T @ U3))))))))))). % typing_elims(3)
thf(fact_65_Abs, axiom,
    ((![Env : nat > type, T3 : type, T : dB, U2 : type]: ((typing @ (shift_type @ Env @ zero_zero_nat @ T3) @ T @ U2) => (typing @ Env @ (abs @ T) @ (fun @ T3 @ U2)))))). % Abs
thf(fact_66_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_67_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_68_Lambda, axiom,
    ((![R : dB]: ((it @ R) => (it @ (abs @ R)))))). % Lambda
thf(fact_69_foldl__Cons, axiom,
    ((![F : dB > dB > dB, A : dB, X : dB, Xs : list_dB]: ((foldl_dB_dB @ F @ A @ (cons_dB @ X @ Xs)) = (foldl_dB_dB @ F @ (F @ A @ X) @ Xs))))). % foldl_Cons
thf(fact_70_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_71_listsp__conj__eq, axiom,
    ((![A4 : dB > $o, B : dB > $o]: ((listsp_dB @ (^[X3 : dB]: (((A4 @ X3)) & ((B @ X3))))) = (^[X3 : list_dB]: (((listsp_dB @ A4 @ X3)) & ((listsp_dB @ B @ X3)))))))). % listsp_conj_eq
thf(fact_72_listsp__simps_I2_J, axiom,
    ((![A4 : dB > $o, X : dB, Xs : list_dB]: ((listsp_dB @ A4 @ (cons_dB @ X @ Xs)) = (((A4 @ X)) & ((listsp_dB @ A4 @ Xs))))))). % listsp_simps(2)
thf(fact_73_listspE, axiom,
    ((![A4 : dB > $o, X : dB, L : list_dB]: ((listsp_dB @ A4 @ (cons_dB @ X @ L)) => (~ (((A4 @ X) => (~ ((listsp_dB @ A4 @ L)))))))))). % listspE
thf(fact_74_listsp_OCons, axiom,
    ((![A4 : dB > $o, A : dB, L : list_dB]: ((A4 @ A) => ((listsp_dB @ A4 @ L) => (listsp_dB @ A4 @ (cons_dB @ A @ L))))))). % listsp.Cons
thf(fact_75_not__Cons__self2, axiom,
    ((![X : dB, Xs : list_dB]: (~ (((cons_dB @ X @ Xs) = Xs)))))). % not_Cons_self2
thf(fact_76_IT_OVar, axiom,
    ((![Rs : list_dB, N : nat]: ((listsp_dB @ it @ Rs) => (it @ (foldl_dB_dB @ app @ (var @ N) @ Rs)))))). % IT.Var
thf(fact_77_IT_Ocases, axiom,
    ((![A : dB]: ((it @ A) => ((![Rs2 : list_dB]: ((?[N2 : nat]: (A = (foldl_dB_dB @ app @ (var @ N2) @ Rs2))) => (~ ((listsp_dB @ it @ Rs2))))) => ((![R2 : dB]: ((A = (abs @ R2)) => (~ ((it @ R2))))) => (~ ((![R2 : dB, S2 : dB, Ss2 : list_dB]: ((A = (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 = (^[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_79_typings_Osimps_I2_J, axiom,
    ((![E : nat > type, T : dB, Ts3 : list_dB, Ts2 : list_type]: ((typings @ E @ (cons_dB @ T @ Ts3) @ Ts2) = (case_list_o_type @ $false @ (^[T6 : type]: (^[Ts5 : list_type]: (((typing @ E @ T @ T6)) & ((typings @ E @ Ts3 @ Ts5))))) @ Ts2))))). % typings.simps(2)
thf(fact_80_lifts__IT, axiom,
    ((![Ts3 : list_dB]: ((listsp_dB @ it @ Ts3) => (listsp_dB @ it @ (map_dB_dB @ (^[T7 : dB]: (lift @ T7 @ zero_zero_nat)) @ Ts3)))))). % lifts_IT
thf(fact_81_substs__lemma, axiom,
    ((![E : nat > type, U : dB, T3 : type, I : nat, Ts3 : list_dB, Ts2 : list_type]: ((typing @ E @ U @ T3) => ((typings @ (shift_type @ E @ I @ T3) @ Ts3 @ Ts2) => (typings @ E @ (map_dB_dB @ (^[T7 : dB]: (subst @ T7 @ U @ I)) @ Ts3) @ Ts2)))))). % substs_lemma
thf(fact_82_map__ident, axiom,
    (((map_dB_dB @ (^[X3 : dB]: X3)) = (^[Xs2 : list_dB]: Xs2)))). % map_ident
thf(fact_83_subst__map, axiom,
    ((![T : dB, Ts3 : list_dB, U : dB, I : nat]: ((subst @ (foldl_dB_dB @ app @ T @ Ts3) @ U @ I) = (foldl_dB_dB @ app @ (subst @ T @ U @ I) @ (map_dB_dB @ (^[T7 : dB]: (subst @ T7 @ U @ I)) @ Ts3)))))). % subst_map
thf(fact_84_lift__map, axiom,
    ((![T : dB, Ts3 : list_dB, I : nat]: ((lift @ (foldl_dB_dB @ app @ T @ Ts3) @ I) = (foldl_dB_dB @ app @ (lift @ T @ I) @ (map_dB_dB @ (^[T7 : dB]: (lift @ T7 @ I)) @ Ts3)))))). % lift_map
thf(fact_85_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_86_Cons__eq__map__D, axiom,
    ((![X : dB, Xs : list_dB, F : dB > dB, Ys : list_dB]: (((cons_dB @ X @ Xs) = (map_dB_dB @ F @ Ys)) => (?[Z : dB, Zs : list_dB]: ((Ys = (cons_dB @ Z @ Zs)) & ((X = (F @ Z)) & (Xs = (map_dB_dB @ F @ Zs))))))))). % Cons_eq_map_D
thf(fact_87_map__eq__Cons__D, axiom,
    ((![F : dB > dB, Xs : list_dB, Y : dB, Ys : list_dB]: (((map_dB_dB @ F @ Xs) = (cons_dB @ Y @ Ys)) => (?[Z : dB, Zs : list_dB]: ((Xs = (cons_dB @ Z @ Zs)) & (((F @ Z) = Y) & ((map_dB_dB @ F @ Zs) = Ys)))))))). % map_eq_Cons_D
thf(fact_88_Cons__eq__map__conv, axiom,
    ((![X : dB, Xs : list_dB, F : dB > dB, Ys : list_dB]: (((cons_dB @ X @ Xs) = (map_dB_dB @ F @ Ys)) = (?[Z2 : dB]: (?[Zs2 : list_dB]: (((Ys = (cons_dB @ Z2 @ Zs2))) & ((((X = (F @ Z2))) & ((Xs = (map_dB_dB @ F @ Zs2)))))))))))). % Cons_eq_map_conv
thf(fact_89_map__eq__Cons__conv, axiom,
    ((![F : dB > dB, Xs : list_dB, Y : dB, Ys : list_dB]: (((map_dB_dB @ F @ Xs) = (cons_dB @ Y @ Ys)) = (?[Z2 : dB]: (?[Zs2 : list_dB]: (((Xs = (cons_dB @ Z2 @ Zs2))) & (((((F @ Z2) = Y)) & (((map_dB_dB @ F @ Zs2) = Ys))))))))))). % map_eq_Cons_conv
thf(fact_90_list_Ocase__distrib, axiom,
    ((![H2 : $o > $o, F1 : $o, F2 : type > list_type > $o, List : list_type]: ((H2 @ (case_list_o_type @ F1 @ F2 @ List)) = (case_list_o_type @ (H2 @ F1) @ (^[X13 : type]: (^[X25 : list_type]: (H2 @ (F2 @ X13 @ X25)))) @ List))))). % list.case_distrib
thf(fact_91_list_Omap__ident, axiom,
    ((![T : list_dB]: ((map_dB_dB @ (^[X3 : dB]: X3) @ T) = T)))). % list.map_ident
thf(fact_92_foldl__map, axiom,
    ((![G : dB > dB > dB, A : dB, F : dB > dB, Xs : list_dB]: ((foldl_dB_dB @ G @ A @ (map_dB_dB @ F @ Xs)) = (foldl_dB_dB @ (^[A5 : dB]: (^[X3 : dB]: (G @ A5 @ (F @ X3)))) @ A @ Xs))))). % foldl_map
thf(fact_93_list_Osimps_I5_J, axiom,
    ((![F1 : $o, F2 : type > list_type > $o, X21 : type, X22 : list_type]: ((case_list_o_type @ F1 @ F2 @ (cons_type @ X21 @ X22)) = (F2 @ X21 @ X22))))). % list.simps(5)
thf(fact_94_lift__types, axiom,
    ((![E : nat > type, Ts3 : list_dB, Ts2 : list_type, I : nat, U2 : type]: ((typings @ E @ Ts3 @ Ts2) => (typings @ (shift_type @ E @ I @ U2) @ (map_dB_dB @ (^[T7 : dB]: (lift @ T7 @ I)) @ Ts3) @ Ts2))))). % lift_types
thf(fact_95_map__eq__map__tailrec, axiom,
    ((map_dB_dB = map_tailrec_dB_dB))). % map_eq_map_tailrec
thf(fact_96_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_97_Apps__dB__induct, axiom,
    ((![P : dB > $o, T : dB]: ((![N2 : nat, Ts4 : list_dB]: ((![X4 : dB]: ((member_dB @ X4 @ (set_dB2 @ Ts4)) => (P @ X4))) => (P @ (foldl_dB_dB @ app @ (var @ N2) @ Ts4)))) => ((![U5 : dB]: ((P @ U5) => (![Ts4 : list_dB]: ((![X4 : dB]: ((member_dB @ X4 @ (set_dB2 @ Ts4)) => (P @ X4))) => (P @ (foldl_dB_dB @ app @ (abs @ U5) @ Ts4)))))) => (P @ T)))))). % Apps_dB_induct
thf(fact_98_map__eq__conv, axiom,
    ((![F : dB > dB, Xs : list_dB, G : dB > dB]: (((map_dB_dB @ F @ Xs) = (map_dB_dB @ G @ Xs)) = (![X3 : dB]: (((member_dB @ X3 @ (set_dB2 @ Xs))) => (((F @ X3) = (G @ X3))))))))). % map_eq_conv
thf(fact_99_in__listspI, axiom,
    ((![Xs : list_dB, A4 : dB > $o]: ((![X2 : dB]: ((member_dB @ X2 @ (set_dB2 @ Xs)) => (A4 @ X2))) => (listsp_dB @ A4 @ Xs))))). % in_listspI
thf(fact_100_list_Oinj__map__strong, axiom,
    ((![X : list_dB, Xa : list_dB, F : dB > dB, Fa : dB > dB]: ((![Z : dB, Za : dB]: ((member_dB @ Z @ (set_dB2 @ X)) => ((member_dB @ Za @ (set_dB2 @ Xa)) => (((F @ Z) = (Fa @ Za)) => (Z = Za))))) => (((map_dB_dB @ F @ X) = (map_dB_dB @ Fa @ Xa)) => (X = Xa)))))). % list.inj_map_strong
thf(fact_101_list_Omap__cong0, axiom,
    ((![X : list_dB, F : dB > dB, G : dB > dB]: ((![Z : dB]: ((member_dB @ Z @ (set_dB2 @ X)) => ((F @ Z) = (G @ Z)))) => ((map_dB_dB @ F @ X) = (map_dB_dB @ G @ X)))))). % list.map_cong0
thf(fact_102_in__listsp__conv__set, axiom,
    ((listsp_dB = (^[A6 : dB > $o]: (^[Xs2 : list_dB]: (![X3 : dB]: (((member_dB @ X3 @ (set_dB2 @ Xs2))) => ((A6 @ X3))))))))). % in_listsp_conv_set
thf(fact_103_list_Omap__cong, axiom,
    ((![X : list_dB, Ya : list_dB, F : dB > dB, G : dB > dB]: ((X = Ya) => ((![Z : dB]: ((member_dB @ Z @ (set_dB2 @ Ya)) => ((F @ Z) = (G @ Z)))) => ((map_dB_dB @ F @ X) = (map_dB_dB @ G @ Ya))))))). % list.map_cong
thf(fact_104_ex__map__conv, axiom,
    ((![Ys : list_dB, F : dB > dB]: ((?[Xs2 : list_dB]: (Ys = (map_dB_dB @ F @ Xs2))) = (![X3 : dB]: (((member_dB @ X3 @ (set_dB2 @ Ys))) => ((?[Y2 : dB]: (X3 = (F @ Y2)))))))))). % ex_map_conv
thf(fact_105_in__listspD, axiom,
    ((![A4 : dB > $o, Xs : list_dB]: ((listsp_dB @ A4 @ Xs) => (![X4 : dB]: ((member_dB @ X4 @ (set_dB2 @ Xs)) => (A4 @ X4))))))). % in_listspD
thf(fact_106_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_107_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_108_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_109_foldl__cong, axiom,
    ((![A : dB, B2 : dB, L : list_dB, K : list_dB, F : dB > dB > dB, G : dB > dB > dB]: ((A = B2) => ((L = K) => ((![A7 : dB, X2 : dB]: ((member_dB @ X2 @ (set_dB2 @ L)) => ((F @ A7 @ X2) = (G @ A7 @ X2)))) => ((foldl_dB_dB @ F @ A @ L) = (foldl_dB_dB @ G @ B2 @ K)))))))). % foldl_cong
thf(fact_110_foldr__cong, axiom,
    ((![A : type, B2 : type, L : list_type, K : list_type, F : type > type > type, G : type > type > type]: ((A = B2) => ((L = K) => ((![A7 : type, X2 : type]: ((member_type @ X2 @ (set_type2 @ L)) => ((F @ X2 @ A7) = (G @ X2 @ A7)))) => ((foldr_type_type @ F @ L @ A) = (foldr_type_type @ G @ K @ B2)))))))). % foldr_cong
thf(fact_111_list_Oset__cases, axiom,
    ((![E : dB, A : list_dB]: ((member_dB @ E @ (set_dB2 @ A)) => ((![Z22 : list_dB]: (~ ((A = (cons_dB @ E @ Z22))))) => (~ ((![Z1 : dB, Z22 : list_dB]: ((A = (cons_dB @ Z1 @ Z22)) => (~ ((member_dB @ E @ (set_dB2 @ Z22))))))))))))). % list.set_cases
thf(fact_112_set__ConsD, axiom,
    ((![Y : dB, X : dB, Xs : list_dB]: ((member_dB @ Y @ (set_dB2 @ (cons_dB @ X @ Xs))) => ((Y = X) | (member_dB @ Y @ (set_dB2 @ Xs))))))). % set_ConsD
thf(fact_113_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_114_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_115_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_116_lem, axiom,
    ((![P : dB > $o, T : dB, N : nat]: ((![N2 : nat, Ts4 : list_dB]: ((![X4 : dB]: ((member_dB @ X4 @ (set_dB2 @ Ts4)) => (P @ X4))) => (P @ (foldl_dB_dB @ app @ (var @ N2) @ Ts4)))) => ((![U5 : dB]: ((P @ U5) => (![Ts4 : list_dB]: ((![X4 : dB]: ((member_dB @ X4 @ (set_dB2 @ Ts4)) => (P @ X4))) => (P @ (foldl_dB_dB @ app @ (abs @ U5) @ Ts4)))))) => (((size_size_dB @ T) = N) => (P @ T))))))). % lem
thf(fact_117_substn__subst__n, axiom,
    ((substn = (^[T7 : dB]: (^[S3 : dB]: (^[N3 : nat]: (subst @ T7 @ (liftn @ N3 @ S3 @ zero_zero_nat) @ N3))))))). % substn_subst_n
thf(fact_118_count__notin, axiom,
    ((![X : dB, Xs : list_dB]: ((~ ((member_dB @ X @ (set_dB2 @ Xs)))) => ((count_list_dB @ Xs @ X) = zero_zero_nat))))). % count_notin
thf(fact_119_liftn__0, axiom,
    ((![T : dB, K : nat]: ((liftn @ zero_zero_nat @ T @ K) = T)))). % liftn_0
thf(fact_120_size__neq__size__imp__neq, axiom,
    ((![X : dB, Y : dB]: ((~ (((size_size_dB @ X) = (size_size_dB @ Y)))) => (~ ((X = Y))))))). % size_neq_size_imp_neq
thf(fact_121_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_122_dB_Osize_I4_J, axiom,
    ((![X1 : nat]: ((size_size_dB @ (var @ X1)) = zero_zero_nat)))). % dB.size(4)
thf(fact_123_not__in__set__insert, axiom,
    ((![X : dB, Xs : list_dB]: ((~ ((member_dB @ X @ (set_dB2 @ Xs)))) => ((insert_dB @ X @ Xs) = (cons_dB @ X @ Xs)))))). % not_in_set_insert
thf(fact_124_beta_Oinducts, axiom,
    ((![X1 : dB, X23 : dB, P : dB > dB > $o]: ((beta @ X1 @ X23) => ((![S2 : dB, T5 : dB]: (P @ (app @ (abs @ S2) @ T5) @ (subst @ S2 @ T5 @ zero_zero_nat))) => ((![S2 : dB, T5 : dB, U5 : dB]: ((beta @ S2 @ T5) => ((P @ S2 @ T5) => (P @ (app @ S2 @ U5) @ (app @ T5 @ U5))))) => ((![S2 : dB, T5 : dB, U5 : dB]: ((beta @ S2 @ T5) => ((P @ S2 @ T5) => (P @ (app @ U5 @ S2) @ (app @ U5 @ T5))))) => ((![S2 : dB, T5 : dB]: ((beta @ S2 @ T5) => ((P @ S2 @ T5) => (P @ (abs @ S2) @ (abs @ T5))))) => (P @ X1 @ X23))))))))). % beta.inducts
thf(fact_125_length__map, axiom,
    ((![F : dB > dB, Xs : list_dB]: ((size_size_list_dB @ (map_dB_dB @ F @ Xs)) = (size_size_list_dB @ Xs))))). % length_map
thf(fact_126_in__set__insert, axiom,
    ((![X : dB, Xs : list_dB]: ((member_dB @ X @ (set_dB2 @ Xs)) => ((insert_dB @ X @ Xs) = Xs))))). % in_set_insert
thf(fact_127_map__eq__imp__length__eq, axiom,
    ((![F : dB > dB, Xs : list_dB, G : dB > dB, Ys : list_dB]: (((map_dB_dB @ F @ Xs) = (map_dB_dB @ G @ Ys)) => ((size_size_list_dB @ Xs) = (size_size_list_dB @ Ys)))))). % map_eq_imp_length_eq
thf(fact_128_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_129_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_130_abs, axiom,
    ((![S : dB, T : dB]: ((beta @ S @ T) => (beta @ (abs @ S) @ (abs @ T)))))). % abs
thf(fact_131_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_132_beta__cases_I1_J, axiom,
    ((![I : nat, T : dB]: (~ ((beta @ (var @ I) @ T)))))). % beta_cases(1)
thf(fact_133_subject__reduction, axiom,
    ((![E : nat > type, T : dB, T3 : type, T8 : dB]: ((typing @ E @ T @ T3) => ((beta @ T @ T8) => (typing @ E @ T8 @ T3)))))). % subject_reduction
thf(fact_134_appR, axiom,
    ((![S : dB, T : dB, U : dB]: ((beta @ S @ T) => (beta @ (app @ U @ S) @ (app @ U @ T)))))). % appR
thf(fact_135_appL, axiom,
    ((![S : dB, T : dB, U : dB]: ((beta @ S @ T) => (beta @ (app @ S @ U) @ (app @ T @ U)))))). % appL
thf(fact_136_List_Oinsert__def, axiom,
    ((insert_dB = (^[X3 : dB]: (^[Xs2 : list_dB]: (if_list_dB @ (member_dB @ X3 @ (set_dB2 @ Xs2)) @ Xs2 @ (cons_dB @ X3 @ Xs2))))))). % List.insert_def
thf(fact_137_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_138_beta, axiom,
    ((![S : dB, T : dB]: (beta @ (app @ (abs @ S) @ T) @ (subst @ S @ T @ zero_zero_nat))))). % beta
thf(fact_139_beta_Ocases, axiom,
    ((![A1 : dB, A2 : dB]: ((beta @ A1 @ A2) => ((![S2 : dB, T5 : dB]: ((A1 = (app @ (abs @ S2) @ T5)) => (~ ((A2 = (subst @ S2 @ T5 @ zero_zero_nat)))))) => ((![S2 : dB, T5 : dB, U5 : dB]: ((A1 = (app @ S2 @ U5)) => ((A2 = (app @ T5 @ U5)) => (~ ((beta @ S2 @ T5)))))) => ((![S2 : dB, T5 : dB, U5 : dB]: ((A1 = (app @ U5 @ S2)) => ((A2 = (app @ U5 @ T5)) => (~ ((beta @ S2 @ T5)))))) => (~ ((![S2 : dB]: ((A1 = (abs @ S2)) => (![T5 : dB]: ((A2 = (abs @ T5)) => (~ ((beta @ S2 @ T5)))))))))))))))). % beta.cases
thf(fact_140_beta_Osimps, axiom,
    ((beta = (^[A12 : dB]: (^[A22 : dB]: (((?[S3 : dB]: (?[T7 : dB]: (((A12 = (app @ (abs @ S3) @ T7))) & ((A22 = (subst @ S3 @ T7 @ zero_zero_nat))))))) | ((((?[S3 : dB]: (?[T7 : dB]: (?[U6 : dB]: (((A12 = (app @ S3 @ U6))) & ((((A22 = (app @ T7 @ U6))) & ((beta @ S3 @ T7))))))))) | ((((?[S3 : dB]: (?[T7 : dB]: (?[U6 : dB]: (((A12 = (app @ U6 @ S3))) & ((((A22 = (app @ U6 @ T7))) & ((beta @ S3 @ T7))))))))) | ((?[S3 : dB]: (?[T7 : dB]: (((A12 = (abs @ S3))) & ((((A22 = (abs @ T7))) & ((beta @ S3 @ T7)))))))))))))))))). % beta.simps
thf(fact_141_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_142_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)) => (![U5 : dB, Us3 : list_dB]: ((Rs = (cons_dB @ U5 @ Us3)) => (~ ((S = (foldl_dB_dB @ app @ (subst @ T5 @ U5 @ zero_zero_nat) @ Us3)))))))))))))))). % apps_betasE

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

% Conjectures (1)
thf(conj_0, conjecture,
    ((it @ (subst @ (app @ (lift @ u @ zero_zero_nat) @ (var @ zero_zero_nat)) @ (subst @ a @ u @ i) @ zero_zero_nat)))).
