% 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_238__5212030_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:38:36.568

% Could-be-implicit typings (6)
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__List__Olist_It__Nat__Onat_J, type,
    list_nat : $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 (45)
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat, type,
    plus_plus_nat : nat > nat > nat).
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_Otype_Osize__type, type,
    size_type : type > nat).
thf(sy_c_LambdaType_Otyping, type,
    typing : (nat > type) > dB > type > $o).
thf(sy_c_LambdaType_Otypings, type,
    typings : (nat > type) > list_dB > list_type > $o).
thf(sy_c_Lambda_OdB_OAbs, type,
    abs : dB > dB).
thf(sy_c_Lambda_OdB_OApp, type,
    app : dB > dB > dB).
thf(sy_c_Lambda_OdB_OVar, type,
    var : nat > dB).
thf(sy_c_Lambda_OdB_Osize__dB, type,
    size_dB : dB > nat).
thf(sy_c_Lambda_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_List_Ofoldl_001t__Lambda__OdB_001t__Lambda__OdB, type,
    foldl_dB_dB : (dB > dB > dB) > dB > list_dB > dB).
thf(sy_c_List_Ofoldl_001t__Lambda__OdB_001t__Nat__Onat, type,
    foldl_dB_nat : (dB > nat > dB) > dB > list_nat > dB).
thf(sy_c_List_Ofoldl_001t__Nat__Onat_001t__Lambda__OdB, type,
    foldl_nat_dB : (nat > dB > nat) > nat > list_dB > nat).
thf(sy_c_List_Ofoldl_001t__Nat__Onat_001t__Nat__Onat, type,
    foldl_nat_nat : (nat > nat > nat) > nat > list_nat > nat).
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_Omap_001t__Lambda__OdB_001t__Lambda__OdB, type,
    map_dB_dB : (dB > dB) > list_dB > list_dB).
thf(sy_c_List_Olist_Omap_001t__Lambda__OdB_001t__Nat__Onat, type,
    map_dB_nat : (dB > nat) > list_dB > list_nat).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat, type,
    map_nat_nat : (nat > nat) > list_nat > list_nat).
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_Omap__tailrec_001t__Lambda__OdB_001t__Nat__Onat, type,
    map_tailrec_dB_nat : (dB > nat) > list_dB > list_nat).
thf(sy_c_Nat_OSuc, type,
    suc : nat > nat).
thf(sy_c_Nat_Osize__class_Osize_001t__LambdaType__Otype, type,
    size_size_type : type > nat).
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_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J, type,
    size_size_list_nat : list_nat > nat).
thf(sy_v_T_H1____, type,
    t_1 : type).
thf(sy_v_T_H____, type,
    t : type).
thf(sy_v_T____, type,
    t2 : type).
thf(sy_v_a____, type,
    a : dB).
thf(sy_v_as____, type,
    as : list_dB).
thf(sy_v_e1____, type,
    e1 : nat > type).
thf(sy_v_e____, type,
    e : nat > type).
thf(sy_v_i1____, type,
    i1 : nat).
thf(sy_v_i____, type,
    i : nat).
thf(sy_v_r____, type,
    r : dB).
thf(sy_v_t____, type,
    t3 : dB).
thf(sy_v_u1____, type,
    u1 : dB).
thf(sy_v_u____, type,
    u : dB).

% Relevant facts (147)
thf(fact_0__092_060open_062IT_At_092_060close_062, axiom,
    ((it @ t3))). % \<open>IT t\<close>
thf(fact_1_Beta_Oprems_I2_J, axiom,
    ((it @ u1))). % Beta.prems(2)
thf(fact_2_Beta_Ohyps_I3_J, axiom,
    ((it @ a))). % Beta.hyps(3)
thf(fact_3_uIT, axiom,
    ((it @ u))). % uIT
thf(fact_4_Beta_Ohyps_I1_J, axiom,
    ((it @ (foldl_dB_dB @ app @ (subst @ r @ a @ zero_zero_nat) @ as)))). % Beta.hyps(1)
thf(fact_5_lift__IT, axiom,
    ((![T : dB, I : nat]: ((it @ T) => (it @ (lift @ T @ I)))))). % lift_IT
thf(fact_6_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_7_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_8__092_060open_062IT_A_IAbs_A_Ir_091lift_Au_A0_PSuc_Ai_093_J_A_092_060degree_062_Aa_091u_Pi_093_A_092_060degree_062_092_060degree_062_Amap_A_I_092_060lambda_062t_O_At_091u_Pi_093_J_Aas_J_092_060close_062, axiom,
    ((it @ (foldl_dB_dB @ app @ (app @ (abs @ (subst @ r @ (lift @ u @ zero_zero_nat) @ (suc @ i))) @ (subst @ a @ u @ i)) @ (map_dB_dB @ (^[T2 : dB]: (subst @ T2 @ u @ i)) @ as))))). % \<open>IT (Abs (r[lift u 0/Suc i]) \<degree> a[u/i] \<degree>\<degree> map (\<lambda>t. t[u/i]) as)\<close>
thf(fact_9_IT_OBeta, 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))))))). % IT.Beta
thf(fact_10_dB_Oinject_I3_J, axiom,
    ((![X3 : dB, Y3 : dB]: (((abs @ X3) = (abs @ Y3)) = (X3 = Y3))))). % dB.inject(3)
thf(fact_11_dB_Oinject_I2_J, axiom,
    ((![X21 : dB, X22 : dB, Y21 : dB, Y22 : dB]: (((app @ X21 @ X22) = (app @ Y21 @ Y22)) = (((X21 = Y21)) & ((X22 = Y22))))))). % dB.inject(2)
thf(fact_12_Lambda, axiom,
    ((![R : dB]: ((it @ R) => (it @ (abs @ R)))))). % Lambda
thf(fact_13_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_14_dB_Odistinct_I5_J, axiom,
    ((![X21 : dB, X22 : dB, X3 : dB]: (~ (((app @ X21 @ X22) = (abs @ X3))))))). % dB.distinct(5)
thf(fact_15_T, axiom,
    ((typing @ (shift_type @ e @ i @ t2) @ (foldl_dB_dB @ app @ (app @ (abs @ r) @ a) @ as) @ t))). % T
thf(fact_16_subst__lift, axiom,
    ((![T : dB, K : nat, S : dB]: ((subst @ (lift @ T @ K) @ S @ K) = T)))). % subst_lift
thf(fact_17_uT, axiom,
    ((typing @ e @ u @ t2))). % uT
thf(fact_18_Beta_Oprems_I3_J, axiom,
    ((typing @ e1 @ u1 @ t2))). % Beta.prems(3)
thf(fact_19_SI2, axiom,
    ((![E : nat > type, I : nat, T3 : type, U : dB]: ((typing @ (shift_type @ E @ I @ t2) @ a @ T3) => ((it @ U) => ((typing @ E @ U @ t2) => (it @ (subst @ a @ U @ I)))))))). % SI2
thf(fact_20_SI1, axiom,
    ((![E : nat > type, I : nat, T3 : type, U : dB]: ((typing @ (shift_type @ E @ I @ t2) @ (foldl_dB_dB @ app @ (subst @ r @ a @ zero_zero_nat) @ as) @ T3) => ((it @ U) => ((typing @ E @ U @ t2) => (it @ (subst @ (foldl_dB_dB @ app @ (subst @ r @ a @ zero_zero_nat) @ as) @ U @ I)))))))). % SI1
thf(fact_21_subst__map, axiom,
    ((![T : dB, Ts : list_dB, U : dB, I : nat]: ((subst @ (foldl_dB_dB @ app @ T @ Ts) @ U @ I) = (foldl_dB_dB @ app @ (subst @ T @ U @ I) @ (map_dB_dB @ (^[T2 : dB]: (subst @ T2 @ U @ I)) @ Ts)))))). % subst_map
thf(fact_22_lift__map, axiom,
    ((![T : dB, Ts : list_dB, I : nat]: ((lift @ (foldl_dB_dB @ app @ T @ Ts) @ I) = (foldl_dB_dB @ app @ (lift @ T @ I) @ (map_dB_dB @ (^[T2 : dB]: (lift @ T2 @ I)) @ Ts)))))). % lift_map
thf(fact_23_Beta_Oprems_I1_J, axiom,
    ((typing @ (shift_type @ e1 @ i1 @ t2) @ (foldl_dB_dB @ app @ (app @ (abs @ r) @ a) @ as) @ t_1))). % Beta.prems(1)
thf(fact_24_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_25_MI2, axiom,
    ((![T1 : type, T22 : type, T : dB, E : nat > type, I : nat, T4 : type, U : dB]: ((t2 = (fun @ T1 @ T22)) => ((it @ T) => ((typing @ (shift_type @ E @ I @ T22) @ T @ T4) => ((it @ U) => ((typing @ E @ U @ T22) => (it @ (subst @ T @ U @ I)))))))))). % MI2
thf(fact_26_MI1, axiom,
    ((![T1 : type, T22 : type, T : dB, E : nat > type, I : nat, T4 : type, U : dB]: ((t2 = (fun @ T1 @ T22)) => ((it @ T) => ((typing @ (shift_type @ E @ I @ T1) @ T @ T4) => ((it @ U) => ((typing @ E @ U @ T1) => (it @ (subst @ T @ U @ I)))))))))). % MI1
thf(fact_27_lift__type, axiom,
    ((![E : nat > type, T : dB, T4 : type, I : nat, U2 : type]: ((typing @ E @ T @ T4) => (typing @ (shift_type @ E @ I @ U2) @ (lift @ T @ I) @ T4))))). % lift_type
thf(fact_28_shift__commute, axiom,
    ((![E : nat > type, I : nat, U2 : type, T4 : type]: ((shift_type @ (shift_type @ E @ I @ U2) @ zero_zero_nat @ T4) = (shift_type @ (shift_type @ E @ zero_zero_nat @ T4) @ (suc @ I) @ U2))))). % shift_commute
thf(fact_29_abs__typeE, axiom,
    ((![E : nat > type, T : dB, T4 : type]: ((typing @ E @ (abs @ T) @ T4) => (~ ((![U3 : type, V : type]: (~ ((typing @ (shift_type @ E @ zero_zero_nat @ U3) @ T @ V)))))))))). % abs_typeE
thf(fact_30_subst__lemma, axiom,
    ((![E : nat > type, T : dB, T4 : type, E2 : nat > type, U : dB, U2 : type, I : nat]: ((typing @ E @ T @ T4) => ((typing @ E2 @ U @ U2) => ((E = (shift_type @ E2 @ I @ U2)) => (typing @ E2 @ (subst @ T @ U @ I) @ T4))))))). % subst_lemma
thf(fact_31_map__ident, axiom,
    (((map_dB_dB @ (^[X : dB]: X)) = (^[Xs : list_dB]: Xs)))). % map_ident
thf(fact_32_shift__eq, axiom,
    ((![I : nat, J : nat, E : nat > type, T4 : type]: ((I = J) => ((shift_type @ E @ I @ T4 @ J) = T4))))). % shift_eq
thf(fact_33_foldl__map, axiom,
    ((![G : nat > nat > nat, A : nat, F : nat > nat, Xs2 : list_nat]: ((foldl_nat_nat @ G @ A @ (map_nat_nat @ F @ Xs2)) = (foldl_nat_nat @ (^[A2 : nat]: (^[X : nat]: (G @ A2 @ (F @ X)))) @ A @ Xs2))))). % foldl_map
thf(fact_34_foldl__map, axiom,
    ((![G : dB > dB > dB, A : dB, F : dB > dB, Xs2 : list_dB]: ((foldl_dB_dB @ G @ A @ (map_dB_dB @ F @ Xs2)) = (foldl_dB_dB @ (^[A2 : dB]: (^[X : dB]: (G @ A2 @ (F @ X)))) @ A @ Xs2))))). % foldl_map
thf(fact_35_foldl__map, axiom,
    ((![G : dB > nat > dB, A : dB, F : dB > nat, Xs2 : list_dB]: ((foldl_dB_nat @ G @ A @ (map_dB_nat @ F @ Xs2)) = (foldl_dB_dB @ (^[A2 : dB]: (^[X : dB]: (G @ A2 @ (F @ X)))) @ A @ Xs2))))). % foldl_map
thf(fact_36_foldl__map, axiom,
    ((![G : nat > nat > nat, A : nat, F : dB > nat, Xs2 : list_dB]: ((foldl_nat_nat @ G @ A @ (map_dB_nat @ F @ Xs2)) = (foldl_nat_dB @ (^[A2 : nat]: (^[X : dB]: (G @ A2 @ (F @ X)))) @ A @ Xs2))))). % foldl_map
thf(fact_37_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_38_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_39_App, axiom,
    ((![Env : nat > type, S : dB, T4 : type, U2 : type, T : dB]: ((typing @ Env @ S @ (fun @ T4 @ U2)) => ((typing @ Env @ T @ T4) => (typing @ Env @ (app @ S @ T) @ U2)))))). % App
thf(fact_40_typing__elims_I2_J, axiom,
    ((![E : nat > type, T : dB, U : dB, T4 : type]: ((typing @ E @ (app @ T @ U) @ T4) => (~ ((![T5 : type]: ((typing @ E @ T @ (fun @ T5 @ T4)) => (~ ((typing @ E @ U @ T5))))))))))). % typing_elims(2)
thf(fact_41_Abs, axiom,
    ((![Env : nat > type, T4 : type, T : dB, U2 : type]: ((typing @ (shift_type @ Env @ zero_zero_nat @ T4) @ T @ U2) => (typing @ Env @ (abs @ T) @ (fun @ T4 @ U2)))))). % Abs
thf(fact_42_typing__elims_I3_J, axiom,
    ((![E : nat > type, T : dB, T4 : type]: ((typing @ E @ (abs @ T) @ T4) => (~ ((![T5 : type, U3 : type]: ((T4 = (fun @ T5 @ U3)) => (~ ((typing @ (shift_type @ E @ zero_zero_nat @ T5) @ T @ U3))))))))))). % typing_elims(3)
thf(fact_43_list_Omap__ident, axiom,
    ((![T : list_dB]: ((map_dB_dB @ (^[X : dB]: X) @ T) = T)))). % list.map_ident
thf(fact_44_old_Onat_Oinject, axiom,
    ((![Nat : nat, Nat2 : nat]: (((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))). % old.nat.inject
thf(fact_45_nat_Oinject, axiom,
    ((![X2 : nat, Y2 : nat]: (((suc @ X2) = (suc @ Y2)) = (X2 = Y2))))). % nat.inject
thf(fact_46_substs__lemma, axiom,
    ((![E : nat > type, U : dB, T4 : type, I : nat, Ts : list_dB, Ts2 : list_type]: ((typing @ E @ U @ T4) => ((typings @ (shift_type @ E @ I @ T4) @ Ts @ Ts2) => (typings @ E @ (map_dB_dB @ (^[T2 : dB]: (subst @ T2 @ U @ I)) @ Ts) @ Ts2)))))). % substs_lemma
thf(fact_47_typing_Oinducts, axiom,
    ((![X1 : nat > type, X2 : dB, X3 : type, P : (nat > type) > dB > type > $o]: ((typing @ X1 @ X2 @ X3) => ((![Env2 : nat > type, X4 : nat, T5 : type]: (((Env2 @ X4) = T5) => (P @ Env2 @ (var @ X4) @ T5))) => ((![Env2 : nat > type, T5 : type, T6 : dB, U3 : type]: ((typing @ (shift_type @ Env2 @ zero_zero_nat @ T5) @ T6 @ U3) => ((P @ (shift_type @ Env2 @ zero_zero_nat @ T5) @ T6 @ U3) => (P @ Env2 @ (abs @ T6) @ (fun @ T5 @ U3))))) => ((![Env2 : nat > type, S2 : dB, T5 : type, U3 : type, T6 : dB]: ((typing @ Env2 @ S2 @ (fun @ T5 @ U3)) => ((P @ Env2 @ S2 @ (fun @ T5 @ U3)) => ((typing @ Env2 @ T6 @ T5) => ((P @ Env2 @ T6 @ T5) => (P @ Env2 @ (app @ S2 @ T6) @ U3)))))) => (P @ X1 @ X2 @ X3)))))))). % typing.inducts
thf(fact_48_typing_Osimps, axiom,
    ((typing = (^[A1 : nat > type]: (^[A22 : dB]: (^[A3 : type]: (((?[Env3 : nat > type]: (?[X : nat]: (?[T7 : type]: (((A1 = Env3)) & ((((A22 = (var @ X))) & ((((A3 = T7)) & (((Env3 @ X) = T7))))))))))) | ((((?[Env3 : nat > type]: (?[T7 : type]: (?[T2 : dB]: (?[U4 : type]: (((A1 = Env3)) & ((((A22 = (abs @ T2))) & ((((A3 = (fun @ T7 @ U4))) & ((typing @ (shift_type @ Env3 @ zero_zero_nat @ T7) @ T2 @ U4)))))))))))) | ((?[Env3 : nat > type]: (?[S3 : dB]: (?[T7 : type]: (?[U4 : type]: (?[T2 : dB]: (((A1 = Env3)) & ((((A22 = (app @ S3 @ T2))) & ((((A3 = U4)) & ((((typing @ Env3 @ S3 @ (fun @ T7 @ U4))) & ((typing @ Env3 @ T2 @ T7)))))))))))))))))))))))). % typing.simps
thf(fact_49_dB_Oinject_I1_J, axiom,
    ((![X1 : nat, Y1 : nat]: (((var @ X1) = (var @ Y1)) = (X1 = Y1))))). % dB.inject(1)
thf(fact_50_subst__eq, axiom,
    ((![K : nat, U : dB]: ((subst @ (var @ K) @ U @ K) = U)))). % subst_eq
thf(fact_51_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_52_dB_Odistinct_I1_J, axiom,
    ((![X1 : nat, X21 : dB, X22 : dB]: (~ (((var @ X1) = (app @ X21 @ X22))))))). % dB.distinct(1)
thf(fact_53_dB_Odistinct_I3_J, axiom,
    ((![X1 : nat, X3 : dB]: (~ (((var @ X1) = (abs @ X3))))))). % dB.distinct(3)
thf(fact_54_typing__elims_I1_J, axiom,
    ((![E : nat > type, I : nat, T4 : type]: ((typing @ E @ (var @ I) @ T4) => ((E @ I) = T4))))). % typing_elims(1)
thf(fact_55_typing_OVar, axiom,
    ((![Env : nat > type, X5 : nat, T4 : type]: (((Env @ X5) = T4) => (typing @ Env @ (var @ X5) @ T4))))). % typing.Var
thf(fact_56_Var__IT, axiom,
    ((![N : nat]: (it @ (var @ N))))). % Var_IT
thf(fact_57_dB_Oexhaust, axiom,
    ((![Y : dB]: ((![X12 : nat]: (~ ((Y = (var @ X12))))) => ((![X212 : dB, X222 : dB]: (~ ((Y = (app @ X212 @ X222))))) => (~ ((![X32 : dB]: (~ ((Y = (abs @ X32)))))))))))). % dB.exhaust
thf(fact_58_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_59_app__Var__IT, axiom,
    ((![T : dB, I : nat]: ((it @ T) => (it @ (app @ T @ (var @ I))))))). % app_Var_IT
thf(fact_60_subst__Var__IT, axiom,
    ((![R : dB, I : nat, J : nat]: ((it @ R) => (it @ (subst @ R @ (var @ I) @ J)))))). % subst_Var_IT
thf(fact_61_zero__reorient, axiom,
    ((![X5 : nat]: ((zero_zero_nat = X5) = (X5 = zero_zero_nat))))). % zero_reorient
thf(fact_62_Suc__inject, axiom,
    ((![X5 : nat, Y : nat]: (((suc @ X5) = (suc @ Y)) => (X5 = Y))))). % Suc_inject
thf(fact_63_n__not__Suc__n, axiom,
    ((![N : nat]: (~ ((N = (suc @ N))))))). % n_not_Suc_n
thf(fact_64_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_65_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_66_ex__head__tail, axiom,
    ((![T : dB]: (?[Ts3 : list_dB, H : dB]: ((T = (foldl_dB_dB @ app @ H @ Ts3)) & ((?[N2 : nat]: (H = (var @ N2))) | (?[U5 : dB]: (H = (abs @ U5))))))))). % ex_head_tail
thf(fact_67_var__app__type__eq, axiom,
    ((![E : nat > type, I : nat, Ts : list_dB, T4 : type, U2 : type]: ((typing @ E @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ T4) => ((typing @ E @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ U2) => (T4 = U2)))))). % var_app_type_eq
thf(fact_68_lift__types, axiom,
    ((![E : nat > type, Ts : list_dB, Ts2 : list_type, I : nat, U2 : type]: ((typings @ E @ Ts @ Ts2) => (typings @ (shift_type @ E @ I @ U2) @ (map_dB_dB @ (^[T2 : dB]: (lift @ T2 @ I)) @ Ts) @ Ts2))))). % lift_types
thf(fact_69_nat_Odistinct_I1_J, axiom,
    ((![X2 : nat]: (~ ((zero_zero_nat = (suc @ X2))))))). % nat.distinct(1)
thf(fact_70_old_Onat_Odistinct_I2_J, axiom,
    ((![Nat2 : nat]: (~ (((suc @ Nat2) = zero_zero_nat)))))). % old.nat.distinct(2)
thf(fact_71_old_Onat_Odistinct_I1_J, axiom,
    ((![Nat2 : nat]: (~ ((zero_zero_nat = (suc @ Nat2))))))). % old.nat.distinct(1)
thf(fact_72_nat_OdiscI, axiom,
    ((![Nat : nat, X2 : nat]: ((Nat = (suc @ X2)) => (~ ((Nat = zero_zero_nat))))))). % nat.discI
thf(fact_73_nat__induct, axiom,
    ((![P : nat > $o, N : nat]: ((P @ zero_zero_nat) => ((![N2 : nat]: ((P @ N2) => (P @ (suc @ N2)))) => (P @ N)))))). % nat_induct
thf(fact_74_diff__induct, axiom,
    ((![P : nat > nat > $o, M : nat, N : nat]: ((![X4 : nat]: (P @ X4 @ zero_zero_nat)) => ((![Y4 : nat]: (P @ zero_zero_nat @ (suc @ Y4))) => ((![X4 : nat, Y4 : nat]: ((P @ X4 @ Y4) => (P @ (suc @ X4) @ (suc @ Y4)))) => (P @ M @ N))))))). % diff_induct
thf(fact_75_zero__induct, axiom,
    ((![P : nat > $o, K : nat]: ((P @ K) => ((![N2 : nat]: ((P @ (suc @ N2)) => (P @ N2))) => (P @ zero_zero_nat)))))). % zero_induct
thf(fact_76_Suc__neq__Zero, axiom,
    ((![M : nat]: (~ (((suc @ M) = zero_zero_nat)))))). % Suc_neq_Zero
thf(fact_77_Zero__neq__Suc, axiom,
    ((![M : nat]: (~ ((zero_zero_nat = (suc @ M))))))). % Zero_neq_Suc
thf(fact_78_Zero__not__Suc, axiom,
    ((![M : nat]: (~ ((zero_zero_nat = (suc @ M))))))). % Zero_not_Suc
thf(fact_79_old_Onat_Oexhaust, axiom,
    ((![Y : nat]: ((~ ((Y = zero_zero_nat))) => (~ ((![Nat3 : nat]: (~ ((Y = (suc @ Nat3))))))))))). % old.nat.exhaust
thf(fact_80_old_Onat_Oinducts, axiom,
    ((![P : nat > $o, Nat : nat]: ((P @ zero_zero_nat) => ((![Nat3 : nat]: ((P @ Nat3) => (P @ (suc @ Nat3)))) => (P @ Nat)))))). % old.nat.inducts
thf(fact_81_not0__implies__Suc, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (?[M2 : nat]: (N = (suc @ M2))))))). % not0_implies_Suc
thf(fact_82_typing_Ocases, axiom,
    ((![A12 : nat > type, A23 : dB, A32 : type]: ((typing @ A12 @ A23 @ A32) => ((![X4 : nat]: ((A23 = (var @ X4)) => (~ (((A12 @ X4) = A32))))) => ((![T5 : type, T6 : dB]: ((A23 = (abs @ T6)) => (![U3 : type]: ((A32 = (fun @ T5 @ U3)) => (~ ((typing @ (shift_type @ A12 @ zero_zero_nat @ T5) @ T6 @ U3))))))) => (~ ((![S2 : dB, T5 : type, U3 : type, T6 : dB]: ((A23 = (app @ S2 @ T6)) => ((A32 = U3) => ((typing @ A12 @ S2 @ (fun @ T5 @ U3)) => (~ ((typing @ A12 @ T6 @ T5))))))))))))))). % typing.cases
thf(fact_83_IT_Oinducts, axiom,
    ((![X5 : dB, P : dB > $o]: ((it @ X5) => ((![Rs2 : list_dB, N2 : nat]: ((listsp_dB @ (^[X : dB]: (((it @ X)) & ((P @ X)))) @ Rs2) => (P @ (foldl_dB_dB @ app @ (var @ N2) @ Rs2)))) => ((![R2 : dB]: ((it @ R2) => ((P @ R2) => (P @ (abs @ R2))))) => ((![R2 : dB, S2 : dB, Ss2 : list_dB]: ((it @ (foldl_dB_dB @ app @ (subst @ R2 @ S2 @ zero_zero_nat) @ Ss2)) => ((P @ (foldl_dB_dB @ app @ (subst @ R2 @ S2 @ zero_zero_nat) @ Ss2)) => ((it @ S2) => ((P @ S2) => (P @ (foldl_dB_dB @ app @ (app @ (abs @ R2) @ S2) @ Ss2))))))) => (P @ X5)))))))). % IT.inducts
thf(fact_84_IT_Osimps, axiom,
    ((it = (^[A2 : dB]: (((?[Rs3 : list_dB]: (?[N3 : nat]: (((A2 = (foldl_dB_dB @ app @ (var @ N3) @ Rs3))) & ((listsp_dB @ it @ Rs3)))))) | ((((?[R3 : dB]: (((A2 = (abs @ R3))) & ((it @ R3))))) | ((?[R3 : dB]: (?[S3 : dB]: (?[Ss3 : list_dB]: (((A2 = (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_85_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_86_var__app__types, axiom,
    ((![E : nat > type, I : nat, Ts : list_dB, Us : list_dB, T4 : type, Ts2 : list_type, U2 : type]: ((typing @ E @ (foldl_dB_dB @ app @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ Us) @ T4) => ((typings @ E @ Ts @ Ts2) => ((typing @ E @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ U2) => (?[Us2 : list_type]: ((U2 = (foldr_type_type @ fun @ Us2 @ T4)) & (typings @ E @ Us @ Us2))))))))). % var_app_types
thf(fact_87_var__app__typesE, axiom,
    ((![E : nat > type, I : nat, Ts : list_dB, T4 : type]: ((typing @ E @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ T4) => (~ ((![Ts4 : list_type]: ((typing @ E @ (var @ I) @ (foldr_type_type @ fun @ Ts4 @ T4)) => (~ ((typings @ E @ Ts @ Ts4))))))))))). % var_app_typesE
thf(fact_88_list__app__typeI, axiom,
    ((![E : nat > type, T : dB, Ts2 : list_type, T4 : type, Ts : list_dB]: ((typing @ E @ T @ (foldr_type_type @ fun @ Ts2 @ T4)) => ((typings @ E @ Ts @ Ts2) => (typing @ E @ (foldl_dB_dB @ app @ T @ Ts) @ T4)))))). % list_app_typeI
thf(fact_89_listsp__conj__eq, axiom,
    ((![A4 : dB > $o, B : dB > $o]: ((listsp_dB @ (^[X : dB]: (((A4 @ X)) & ((B @ X))))) = (^[X : list_dB]: (((listsp_dB @ A4 @ X)) & ((listsp_dB @ B @ X)))))))). % listsp_conj_eq
thf(fact_90_IT_OVar, axiom,
    ((![Rs : list_dB, N : nat]: ((listsp_dB @ it @ Rs) => (it @ (foldl_dB_dB @ app @ (var @ N) @ Rs)))))). % IT.Var
thf(fact_91_lifts__IT, axiom,
    ((![Ts : list_dB]: ((listsp_dB @ it @ Ts) => (listsp_dB @ it @ (map_dB_dB @ (^[T2 : dB]: (lift @ T2 @ zero_zero_nat)) @ Ts)))))). % lifts_IT
thf(fact_92_list__app__typeD, axiom,
    ((![E : nat > type, T : dB, Ts : list_dB, T4 : type]: ((typing @ E @ (foldl_dB_dB @ app @ T @ Ts) @ T4) => (?[Ts4 : list_type]: ((typing @ E @ T @ (foldr_type_type @ fun @ Ts4 @ T4)) & (typings @ E @ Ts @ Ts4))))))). % list_app_typeD
thf(fact_93_list__app__typeE, axiom,
    ((![E : nat > type, T : dB, Ts : list_dB, T4 : type]: ((typing @ E @ (foldl_dB_dB @ app @ T @ Ts) @ T4) => (~ ((![Ts4 : list_type]: ((typing @ E @ T @ (foldr_type_type @ fun @ Ts4 @ T4)) => (~ ((typings @ E @ Ts @ Ts4))))))))))). % list_app_typeE
thf(fact_94_liftn__lift, axiom,
    ((![N : nat, T : dB, K : nat]: ((liftn @ (suc @ N) @ T @ K) = (lift @ (liftn @ N @ T @ K) @ K))))). % liftn_lift
thf(fact_95_dB_Osize__gen_I1_J, axiom,
    ((![X1 : nat]: ((size_dB @ (var @ X1)) = zero_zero_nat)))). % dB.size_gen(1)
thf(fact_96_liftn__0, axiom,
    ((![T : dB, K : nat]: ((liftn @ zero_zero_nat @ T @ K) = T)))). % liftn_0
thf(fact_97_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_98_substn__subst__n, axiom,
    ((substn = (^[T2 : dB]: (^[S3 : dB]: (^[N3 : nat]: (subst @ T2 @ (liftn @ N3 @ S3 @ zero_zero_nat) @ N3))))))). % substn_subst_n
thf(fact_99_dB_Osize__gen_I3_J, axiom,
    ((![X3 : dB]: ((size_dB @ (abs @ X3)) = (plus_plus_nat @ (size_dB @ X3) @ (suc @ zero_zero_nat)))))). % dB.size_gen(3)
thf(fact_100_map__eq__map__tailrec, axiom,
    ((map_dB_dB = map_tailrec_dB_dB))). % map_eq_map_tailrec
thf(fact_101_map__eq__map__tailrec, axiom,
    ((map_dB_nat = map_tailrec_dB_nat))). % map_eq_map_tailrec
thf(fact_102_add__left__cancel, axiom,
    ((![A : nat, B2 : nat, C : nat]: (((plus_plus_nat @ A @ B2) = (plus_plus_nat @ A @ C)) = (B2 = C))))). % add_left_cancel
thf(fact_103_add__right__cancel, axiom,
    ((![B2 : nat, A : nat, C : nat]: (((plus_plus_nat @ B2 @ A) = (plus_plus_nat @ C @ A)) = (B2 = C))))). % add_right_cancel
thf(fact_104_zero__eq__add__iff__both__eq__0, axiom,
    ((![X5 : nat, Y : nat]: ((zero_zero_nat = (plus_plus_nat @ X5 @ Y)) = (((X5 = zero_zero_nat)) & ((Y = zero_zero_nat))))))). % zero_eq_add_iff_both_eq_0
thf(fact_105_add__eq__0__iff__both__eq__0, axiom,
    ((![X5 : nat, Y : nat]: (((plus_plus_nat @ X5 @ Y) = zero_zero_nat) = (((X5 = zero_zero_nat)) & ((Y = zero_zero_nat))))))). % add_eq_0_iff_both_eq_0
thf(fact_106_add__cancel__right__right, axiom,
    ((![A : nat, B2 : nat]: ((A = (plus_plus_nat @ A @ B2)) = (B2 = zero_zero_nat))))). % add_cancel_right_right
thf(fact_107_add__cancel__right__left, axiom,
    ((![A : nat, B2 : nat]: ((A = (plus_plus_nat @ B2 @ A)) = (B2 = zero_zero_nat))))). % add_cancel_right_left
thf(fact_108_add__cancel__left__right, axiom,
    ((![A : nat, B2 : nat]: (((plus_plus_nat @ A @ B2) = A) = (B2 = zero_zero_nat))))). % add_cancel_left_right
thf(fact_109_add__cancel__left__left, axiom,
    ((![B2 : nat, A : nat]: (((plus_plus_nat @ B2 @ A) = A) = (B2 = zero_zero_nat))))). % add_cancel_left_left
thf(fact_110_add_Oright__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ A @ zero_zero_nat) = A)))). % add.right_neutral
thf(fact_111_add_Oleft__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ zero_zero_nat @ A) = A)))). % add.left_neutral
thf(fact_112_Nat_Oadd__0__right, axiom,
    ((![M : nat]: ((plus_plus_nat @ M @ zero_zero_nat) = M)))). % Nat.add_0_right
thf(fact_113_add__is__0, axiom,
    ((![M : nat, N : nat]: (((plus_plus_nat @ M @ N) = zero_zero_nat) = (((M = zero_zero_nat)) & ((N = zero_zero_nat))))))). % add_is_0
thf(fact_114_add__Suc__right, axiom,
    ((![M : nat, N : nat]: ((plus_plus_nat @ M @ (suc @ N)) = (suc @ (plus_plus_nat @ M @ N)))))). % add_Suc_right
thf(fact_115_ab__semigroup__add__class_Oadd__ac_I1_J, axiom,
    ((![A : nat, B2 : nat, C : nat]: ((plus_plus_nat @ (plus_plus_nat @ A @ B2) @ C) = (plus_plus_nat @ A @ (plus_plus_nat @ B2 @ C)))))). % ab_semigroup_add_class.add_ac(1)
thf(fact_116_add__mono__thms__linordered__semiring_I4_J, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: (((I = J) & (K = L)) => ((plus_plus_nat @ I @ K) = (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_semiring(4)
thf(fact_117_group__cancel_Oadd1, axiom,
    ((![A4 : nat, K : nat, A : nat, B2 : nat]: ((A4 = (plus_plus_nat @ K @ A)) => ((plus_plus_nat @ A4 @ B2) = (plus_plus_nat @ K @ (plus_plus_nat @ A @ B2))))))). % group_cancel.add1
thf(fact_118_group__cancel_Oadd2, axiom,
    ((![B : nat, K : nat, B2 : nat, A : nat]: ((B = (plus_plus_nat @ K @ B2)) => ((plus_plus_nat @ A @ B) = (plus_plus_nat @ K @ (plus_plus_nat @ A @ B2))))))). % group_cancel.add2
thf(fact_119_add_Oassoc, axiom,
    ((![A : nat, B2 : nat, C : nat]: ((plus_plus_nat @ (plus_plus_nat @ A @ B2) @ C) = (plus_plus_nat @ A @ (plus_plus_nat @ B2 @ C)))))). % add.assoc
thf(fact_120_add_Ocommute, axiom,
    ((plus_plus_nat = (^[A2 : nat]: (^[B3 : nat]: (plus_plus_nat @ B3 @ A2)))))). % add.commute
thf(fact_121_add_Oleft__commute, axiom,
    ((![B2 : nat, A : nat, C : nat]: ((plus_plus_nat @ B2 @ (plus_plus_nat @ A @ C)) = (plus_plus_nat @ A @ (plus_plus_nat @ B2 @ C)))))). % add.left_commute
thf(fact_122_add__left__imp__eq, axiom,
    ((![A : nat, B2 : nat, C : nat]: (((plus_plus_nat @ A @ B2) = (plus_plus_nat @ A @ C)) => (B2 = C))))). % add_left_imp_eq
thf(fact_123_add__right__imp__eq, axiom,
    ((![B2 : nat, A : nat, C : nat]: (((plus_plus_nat @ B2 @ A) = (plus_plus_nat @ C @ A)) => (B2 = C))))). % add_right_imp_eq
thf(fact_124_add__Suc, axiom,
    ((![M : nat, N : nat]: ((plus_plus_nat @ (suc @ M) @ N) = (suc @ (plus_plus_nat @ M @ N)))))). % add_Suc
thf(fact_125_nat__arith_Osuc1, axiom,
    ((![A4 : nat, K : nat, A : nat]: ((A4 = (plus_plus_nat @ K @ A)) => ((suc @ A4) = (plus_plus_nat @ K @ (suc @ A))))))). % nat_arith.suc1
thf(fact_126_add__Suc__shift, axiom,
    ((![M : nat, N : nat]: ((plus_plus_nat @ (suc @ M) @ N) = (plus_plus_nat @ M @ (suc @ N)))))). % add_Suc_shift
thf(fact_127_add__eq__self__zero, axiom,
    ((![M : nat, N : nat]: (((plus_plus_nat @ M @ N) = M) => (N = zero_zero_nat))))). % add_eq_self_zero
thf(fact_128_plus__nat_Oadd__0, axiom,
    ((![N : nat]: ((plus_plus_nat @ zero_zero_nat @ N) = N)))). % plus_nat.add_0
thf(fact_129_add_Ocomm__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ A @ zero_zero_nat) = A)))). % add.comm_neutral
thf(fact_130_comm__monoid__add__class_Oadd__0, axiom,
    ((![A : nat]: ((plus_plus_nat @ zero_zero_nat @ A) = A)))). % comm_monoid_add_class.add_0
thf(fact_131_one__is__add, axiom,
    ((![M : nat, N : nat]: (((suc @ zero_zero_nat) = (plus_plus_nat @ M @ N)) = (((((M = (suc @ zero_zero_nat))) & ((N = zero_zero_nat)))) | ((((M = zero_zero_nat)) & ((N = (suc @ zero_zero_nat)))))))))). % one_is_add
thf(fact_132_add__is__1, axiom,
    ((![M : nat, N : nat]: (((plus_plus_nat @ M @ N) = (suc @ zero_zero_nat)) = (((((M = (suc @ zero_zero_nat))) & ((N = zero_zero_nat)))) | ((((M = zero_zero_nat)) & ((N = (suc @ zero_zero_nat)))))))))). % add_is_1
thf(fact_133_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_134_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_135_dB_Osize__gen_I2_J, axiom,
    ((![X21 : dB, X22 : dB]: ((size_dB @ (app @ X21 @ X22)) = (plus_plus_nat @ (plus_plus_nat @ (size_dB @ X21) @ (size_dB @ X22)) @ (suc @ zero_zero_nat)))))). % dB.size_gen(2)
thf(fact_136_type_Osize__gen_I2_J, axiom,
    ((![X21 : type, X22 : type]: ((size_type @ (fun @ X21 @ X22)) = (plus_plus_nat @ (plus_plus_nat @ (size_type @ X21) @ (size_type @ X22)) @ (suc @ zero_zero_nat)))))). % type.size_gen(2)
thf(fact_137_type_Osize_I4_J, axiom,
    ((![X21 : type, X22 : type]: ((size_size_type @ (fun @ X21 @ X22)) = (plus_plus_nat @ (plus_plus_nat @ (size_size_type @ X21) @ (size_size_type @ X22)) @ (suc @ zero_zero_nat)))))). % type.size(4)
thf(fact_138_Euclid__induct, axiom,
    ((![P : nat > nat > $o, A : nat, B2 : nat]: ((![A5 : nat, B4 : nat]: ((P @ A5 @ B4) = (P @ B4 @ A5))) => ((![A5 : nat]: (P @ A5 @ zero_zero_nat)) => ((![A5 : nat, B4 : nat]: ((P @ A5 @ B4) => (P @ A5 @ (plus_plus_nat @ A5 @ B4)))) => (P @ A @ B2))))))). % Euclid_induct
thf(fact_139_size__neq__size__imp__neq, axiom,
    ((![X5 : type, Y : type]: ((~ (((size_size_type @ X5) = (size_size_type @ Y)))) => (~ ((X5 = Y))))))). % size_neq_size_imp_neq
thf(fact_140_size__neq__size__imp__neq, axiom,
    ((![X5 : dB, Y : dB]: ((~ (((size_size_dB @ X5) = (size_size_dB @ Y)))) => (~ ((X5 = Y))))))). % size_neq_size_imp_neq
thf(fact_141_size__neq__size__imp__neq, axiom,
    ((![X5 : list_dB, Y : list_dB]: ((~ (((size_size_list_dB @ X5) = (size_size_list_dB @ Y)))) => (~ ((X5 = Y))))))). % size_neq_size_imp_neq
thf(fact_142_dB_Osize_I6_J, axiom,
    ((![X3 : dB]: ((size_size_dB @ (abs @ X3)) = (plus_plus_nat @ (size_size_dB @ X3) @ (suc @ zero_zero_nat)))))). % dB.size(6)
thf(fact_143_dB_Osize_I5_J, axiom,
    ((![X21 : dB, X22 : dB]: ((size_size_dB @ (app @ X21 @ X22)) = (plus_plus_nat @ (plus_plus_nat @ (size_size_dB @ X21) @ (size_size_dB @ X22)) @ (suc @ zero_zero_nat)))))). % dB.size(5)
thf(fact_144_length__map, axiom,
    ((![F : dB > nat, Xs2 : list_dB]: ((size_size_list_nat @ (map_dB_nat @ F @ Xs2)) = (size_size_list_dB @ Xs2))))). % length_map
thf(fact_145_length__map, axiom,
    ((![F : dB > dB, Xs2 : list_dB]: ((size_size_list_dB @ (map_dB_dB @ F @ Xs2)) = (size_size_list_dB @ Xs2))))). % length_map
thf(fact_146_size__apps, axiom,
    ((![R : dB, Rs : list_dB]: ((size_size_dB @ (foldl_dB_dB @ app @ R @ Rs)) = (plus_plus_nat @ (plus_plus_nat @ (size_size_dB @ R) @ (foldl_nat_nat @ plus_plus_nat @ zero_zero_nat @ (map_dB_nat @ size_size_dB @ Rs))) @ (size_size_list_dB @ Rs)))))). % size_apps

% Conjectures (1)
thf(conj_0, conjecture,
    ((it @ (subst @ (foldl_dB_dB @ app @ (app @ (abs @ r) @ a) @ as) @ u @ i)))).
