% 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_272__5212326_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:38:50.099

% Could-be-implicit typings (9)
thf(ty_n_t__List__Olist_It__List__Olist_It__LambdaType__Otype_J_J, type,
    list_list_type : $tType).
thf(ty_n_t__List__Olist_It__List__Olist_It__Lambda__OdB_J_J, type,
    list_list_dB : $tType).
thf(ty_n_t__List__Olist_It__LambdaType__Otype_J, type,
    list_type : $tType).
thf(ty_n_t__List__Olist_It__Lambda__OdB_J, type,
    list_dB : $tType).
thf(ty_n_t__Set__Oset_It__Lambda__OdB_J, type,
    set_dB : $tType).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J, type,
    set_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 (53)
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_InductTermi_OIT, type,
    it : dB > $o).
thf(sy_c_LambdaType_Oshift_001t__LambdaType__Otype, type,
    shift_type : (nat > type) > nat > type > nat > type).
thf(sy_c_LambdaType_Otype_OAtom, type,
    atom : 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_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_Oappend_001t__LambdaType__Otype, type,
    append_type : list_type > list_type > list_type).
thf(sy_c_List_Oappend_001t__Lambda__OdB, type,
    append_dB : list_dB > list_dB > list_dB).
thf(sy_c_List_Ocount__list_001t__LambdaType__Otype, type,
    count_list_type : list_type > type > nat).
thf(sy_c_List_Ocount__list_001t__Lambda__OdB, type,
    count_list_dB : list_dB > dB > nat).
thf(sy_c_List_Ofoldl_001t__Lambda__OdB_001t__Lambda__OdB, type,
    foldl_dB_dB : (dB > dB > dB) > dB > list_dB > dB).
thf(sy_c_List_Ofoldr_001t__LambdaType__Otype_001t__LambdaType__Otype, type,
    foldr_type_type : (type > type > type) > list_type > type > type).
thf(sy_c_List_Olist_OCons_001t__LambdaType__Otype, type,
    cons_type : type > list_type > list_type).
thf(sy_c_List_Olist_OCons_001t__Lambda__OdB, type,
    cons_dB : dB > list_dB > list_dB).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__LambdaType__Otype_J, type,
    cons_list_type : list_type > list_list_type > list_list_type).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Lambda__OdB_J, type,
    cons_list_dB : list_dB > list_list_dB > list_list_dB).
thf(sy_c_List_Olist_ONil_001t__LambdaType__Otype, type,
    nil_type : list_type).
thf(sy_c_List_Olist_ONil_001t__Lambda__OdB, type,
    nil_dB : list_dB).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__LambdaType__Otype_J, type,
    nil_list_type : list_list_type).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Lambda__OdB_J, type,
    nil_list_dB : list_list_dB).
thf(sy_c_List_Olist_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_On__lists_001t__LambdaType__Otype, type,
    n_lists_type : nat > list_type > list_list_type).
thf(sy_c_List_On__lists_001t__Lambda__OdB, type,
    n_lists_dB : nat > list_dB > list_list_dB).
thf(sy_c_List_Onths_001t__LambdaType__Otype, type,
    nths_type : list_type > set_nat > list_type).
thf(sy_c_List_Onths_001t__Lambda__OdB, type,
    nths_dB : list_dB > set_nat > list_dB).
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__LambdaType__Otype_J, type,
    size_size_list_type : list_type > 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_Orderings_Oord__class_Oless_001t__Nat__Onat, type,
    ord_less_nat : nat > nat > $o).
thf(sy_c_member_001t__Lambda__OdB, type,
    member_dB : dB > set_dB > $o).
thf(sy_c_member_001t__Nat__Onat, type,
    member_nat : nat > set_nat > $o).
thf(sy_v_T, type,
    t : type).
thf(sy_v_Ta____, type,
    ta : type).
thf(sy_v_U____, type,
    u : type).
thf(sy_v_e, type,
    e : nat > type).
thf(sy_v_ea____, type,
    ea : nat > type).
thf(sy_v_s____, type,
    s : dB).
thf(sy_v_t, type,
    t2 : dB).
thf(sy_v_ta____, type,
    ta2 : dB).

% Relevant facts (146)
thf(fact_0_assms, axiom,
    ((typing @ e @ t2 @ t))). % assms
thf(fact_1_App_Ohyps_I1_J, axiom,
    ((typing @ ea @ s @ (fun @ ta @ u)))). % App.hyps(1)
thf(fact_2_App_Ohyps_I2_J, axiom,
    ((it @ s))). % App.hyps(2)
thf(fact_3_App_Ohyps_I3_J, axiom,
    ((typing @ ea @ ta2 @ ta))). % App.hyps(3)
thf(fact_4_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_5__092_060open_062e_092_060langle_0620_058T_A_092_060Rightarrow_062_AU_092_060rangle_062_A_092_060turnstile_062_AVar_A0_A_058_AT_A_092_060Rightarrow_062_AU_092_060close_062, axiom,
    ((typing @ (shift_type @ ea @ zero_zero_nat @ (fun @ ta @ u)) @ (var @ zero_zero_nat) @ (fun @ ta @ u)))). % \<open>e\<langle>0:T \<Rightarrow> U\<rangle> \<turnstile> Var 0 : T \<Rightarrow> U\<close>
thf(fact_6_type__induct, axiom,
    ((![P : type > $o, T : type]: ((![T2 : type]: ((![T1 : type, T22 : type]: ((T2 = (fun @ T1 @ T22)) => (P @ T1))) => ((![T1 : type, T22 : type]: ((T2 = (fun @ T1 @ T22)) => (P @ T22))) => (P @ T2)))) => (P @ T))))). % type_induct
thf(fact_7__092_060open_062e_092_060langle_0620_058T_A_092_060Rightarrow_062_AU_092_060rangle_062_A_092_060turnstile_062_Alift_At_A0_A_058_AT_092_060close_062, axiom,
    ((typing @ (shift_type @ ea @ zero_zero_nat @ (fun @ ta @ u)) @ (lift @ ta2 @ zero_zero_nat) @ ta))). % \<open>e\<langle>0:T \<Rightarrow> U\<rangle> \<turnstile> lift t 0 : T\<close>
thf(fact_8__092_060open_062e_092_060langle_0620_058T_A_092_060Rightarrow_062_AU_092_060rangle_062_A_092_060turnstile_062_AVar_A0_A_092_060degree_062_Alift_At_A0_A_058_AU_092_060close_062, axiom,
    ((typing @ (shift_type @ ea @ zero_zero_nat @ (fun @ ta @ u)) @ (app @ (var @ zero_zero_nat) @ (lift @ ta2 @ zero_zero_nat)) @ u))). % \<open>e\<langle>0:T \<Rightarrow> U\<rangle> \<turnstile> Var 0 \<degree> lift t 0 : U\<close>
thf(fact_9_typing__elims_I2_J, axiom,
    ((![E : nat > type, T3 : dB, U : dB, T : type]: ((typing @ E @ (app @ T3 @ U) @ T) => (~ ((![T2 : type]: ((typing @ E @ T3 @ (fun @ T2 @ T)) => (~ ((typing @ E @ U @ T2))))))))))). % typing_elims(2)
thf(fact_10_typing_OApp, axiom,
    ((![Env : nat > type, S : dB, T : type, U2 : type, T3 : dB]: ((typing @ Env @ S @ (fun @ T @ U2)) => ((typing @ Env @ T3 @ T) => (typing @ Env @ (app @ S @ T3) @ U2)))))). % typing.App
thf(fact_11_type_Odistinct_I1_J, axiom,
    ((![X1 : nat, X21 : type, X22 : type]: (~ (((atom @ X1) = (fun @ X21 @ X22))))))). % type.distinct(1)
thf(fact_12_type_Oinduct, axiom,
    ((![P : type > $o, Type : type]: ((![X : nat]: (P @ (atom @ X))) => ((![X1a : type, X2 : type]: ((P @ X1a) => ((P @ X2) => (P @ (fun @ X1a @ X2))))) => (P @ Type)))))). % type.induct
thf(fact_13_type_Oexhaust, axiom,
    ((![Y : type]: ((![X12 : nat]: (~ ((Y = (atom @ X12))))) => (~ ((![X212 : type, X222 : type]: (~ ((Y = (fun @ X212 @ X222))))))))))). % type.exhaust
thf(fact_14_App_Ohyps_I4_J, axiom,
    ((it @ ta2))). % App.hyps(4)
thf(fact_15__092_060open_062IT_A_Ilift_At_A0_J_092_060close_062, axiom,
    ((it @ (lift @ ta2 @ zero_zero_nat)))). % \<open>IT (lift t 0)\<close>
thf(fact_16_shift__eq, axiom,
    ((![I : nat, J : nat, E : nat > type, T : type]: ((I = J) => ((shift_type @ E @ I @ T @ J) = T))))). % shift_eq
thf(fact_17_type_Oinject_I1_J, axiom,
    ((![X1 : nat, Y1 : nat]: (((atom @ X1) = (atom @ Y1)) = (X1 = Y1))))). % type.inject(1)
thf(fact_18__092_060open_062IT_A_IVar_A0_A_092_060degree_062_Alift_At_A0_J_092_060close_062, axiom,
    ((it @ (app @ (var @ zero_zero_nat) @ (lift @ ta2 @ zero_zero_nat))))). % \<open>IT (Var 0 \<degree> lift t 0)\<close>
thf(fact_19_lift__IT, axiom,
    ((![T3 : dB, I : nat]: ((it @ T3) => (it @ (lift @ T3 @ I)))))). % lift_IT
thf(fact_20_lift__type, axiom,
    ((![E : nat > type, T3 : dB, T : type, I : nat, U2 : type]: ((typing @ E @ T3 @ T) => (typing @ (shift_type @ E @ I @ U2) @ (lift @ T3 @ I) @ T))))). % lift_type
thf(fact_21_Var__IT, axiom,
    ((![N : nat]: (it @ (var @ N))))). % Var_IT
thf(fact_22_app__Var__IT, axiom,
    ((![T3 : dB, I : nat]: ((it @ T3) => (it @ (app @ T3 @ (var @ I))))))). % app_Var_IT
thf(fact_23_typing_OVar, axiom,
    ((![Env : nat > type, X3 : nat, T : type]: (((Env @ X3) = T) => (typing @ Env @ (var @ X3) @ T))))). % typing.Var
thf(fact_24_typing__elims_I1_J, axiom,
    ((![E : nat > type, I : nat, T : type]: ((typing @ E @ (var @ I) @ T) => ((E @ I) = T))))). % typing_elims(1)
thf(fact_25_dB_Oinject_I1_J, axiom,
    ((![X1 : nat, Y1 : nat]: (((var @ X1) = (var @ Y1)) = (X1 = Y1))))). % dB.inject(1)
thf(fact_26_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_27_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_28_dB_Odistinct_I1_J, axiom,
    ((![X1 : nat, X21 : dB, X22 : dB]: (~ (((var @ X1) = (app @ X21 @ X22))))))). % dB.distinct(1)
thf(fact_29_typing_Ocases, axiom,
    ((![A1 : nat > type, A2 : dB, A3 : type]: ((typing @ A1 @ A2 @ A3) => ((![X : nat]: ((A2 = (var @ X)) => (~ (((A1 @ X) = A3))))) => ((![T2 : type, T4 : dB]: ((A2 = (abs @ T4)) => (![U3 : type]: ((A3 = (fun @ T2 @ U3)) => (~ ((typing @ (shift_type @ A1 @ zero_zero_nat @ T2) @ T4 @ U3))))))) => (~ ((![S2 : dB, T2 : type, U3 : type, T4 : dB]: ((A2 = (app @ S2 @ T4)) => ((A3 = U3) => ((typing @ A1 @ S2 @ (fun @ T2 @ U3)) => (~ ((typing @ A1 @ T4 @ T2))))))))))))))). % typing.cases
thf(fact_30_typing_Osimps, axiom,
    ((typing = (^[A12 : nat > type]: (^[A22 : dB]: (^[A32 : type]: (((?[Env2 : nat > type]: (?[X4 : nat]: (?[T5 : type]: (((A12 = Env2)) & ((((A22 = (var @ X4))) & ((((A32 = T5)) & (((Env2 @ X4) = T5))))))))))) | ((((?[Env2 : nat > type]: (?[T5 : type]: (?[T6 : dB]: (?[U4 : type]: (((A12 = Env2)) & ((((A22 = (abs @ T6))) & ((((A32 = (fun @ T5 @ U4))) & ((typing @ (shift_type @ Env2 @ zero_zero_nat @ T5) @ T6 @ U4)))))))))))) | ((?[Env2 : nat > type]: (?[S3 : dB]: (?[T5 : type]: (?[U4 : type]: (?[T6 : dB]: (((A12 = Env2)) & ((((A22 = (app @ S3 @ T6))) & ((((A32 = U4)) & ((((typing @ Env2 @ S3 @ (fun @ T5 @ U4))) & ((typing @ Env2 @ T6 @ T5)))))))))))))))))))))))). % typing.simps
thf(fact_31_typing_Oinducts, axiom,
    ((![X1 : nat > type, X23 : dB, X32 : type, P : (nat > type) > dB > type > $o]: ((typing @ X1 @ X23 @ X32) => ((![Env3 : nat > type, X : nat, T2 : type]: (((Env3 @ X) = T2) => (P @ Env3 @ (var @ X) @ T2))) => ((![Env3 : nat > type, T2 : type, T4 : dB, U3 : type]: ((typing @ (shift_type @ Env3 @ zero_zero_nat @ T2) @ T4 @ U3) => ((P @ (shift_type @ Env3 @ zero_zero_nat @ T2) @ T4 @ U3) => (P @ Env3 @ (abs @ T4) @ (fun @ T2 @ U3))))) => ((![Env3 : nat > type, S2 : dB, T2 : type, U3 : type, T4 : dB]: ((typing @ Env3 @ S2 @ (fun @ T2 @ U3)) => ((P @ Env3 @ S2 @ (fun @ T2 @ U3)) => ((typing @ Env3 @ T4 @ T2) => ((P @ Env3 @ T4 @ T2) => (P @ Env3 @ (app @ S2 @ T4) @ U3)))))) => (P @ X1 @ X23 @ X32)))))))). % typing.inducts
thf(fact_32_type_Osize__gen_I1_J, axiom,
    ((![X1 : nat]: ((size_type @ (atom @ X1)) = zero_zero_nat)))). % type.size_gen(1)
thf(fact_33_dB_Oinject_I3_J, axiom,
    ((![X32 : dB, Y3 : dB]: (((abs @ X32) = (abs @ Y3)) = (X32 = Y3))))). % dB.inject(3)
thf(fact_34_dB_Odistinct_I5_J, axiom,
    ((![X21 : dB, X22 : dB, X32 : dB]: (~ (((app @ X21 @ X22) = (abs @ X32))))))). % dB.distinct(5)
thf(fact_35_dB_Odistinct_I3_J, axiom,
    ((![X1 : nat, X32 : dB]: (~ (((var @ X1) = (abs @ X32))))))). % dB.distinct(3)
thf(fact_36_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_37_dB_Oinduct, axiom,
    ((![P : dB > $o, DB : dB]: ((![X : nat]: (P @ (var @ X))) => ((![X1a : dB, X2 : dB]: ((P @ X1a) => ((P @ X2) => (P @ (app @ X1a @ X2))))) => ((![X : dB]: ((P @ X) => (P @ (abs @ X)))) => (P @ DB))))))). % dB.induct
thf(fact_38_zero__reorient, axiom,
    ((![X3 : nat]: ((zero_zero_nat = X3) = (X3 = zero_zero_nat))))). % zero_reorient
thf(fact_39_abs__typeE, axiom,
    ((![E : nat > type, T3 : dB, T : type]: ((typing @ E @ (abs @ T3) @ T) => (~ ((![U3 : type, V : type]: (~ ((typing @ (shift_type @ E @ zero_zero_nat @ U3) @ T3 @ V)))))))))). % abs_typeE
thf(fact_40_typing__elims_I3_J, axiom,
    ((![E : nat > type, T3 : dB, T : type]: ((typing @ E @ (abs @ T3) @ T) => (~ ((![T2 : type, U3 : type]: ((T = (fun @ T2 @ U3)) => (~ ((typing @ (shift_type @ E @ zero_zero_nat @ T2) @ T3 @ U3))))))))))). % typing_elims(3)
thf(fact_41_Abs, axiom,
    ((![Env : nat > type, T : type, T3 : dB, U2 : type]: ((typing @ (shift_type @ Env @ zero_zero_nat @ T) @ T3 @ U2) => (typing @ Env @ (abs @ T3) @ (fun @ T @ U2)))))). % Abs
thf(fact_42_Lambda, axiom,
    ((![R : dB]: ((it @ R) => (it @ (abs @ R)))))). % Lambda
thf(fact_43_type_Osize_I3_J, axiom,
    ((![X1 : nat]: ((size_size_type @ (atom @ X1)) = zero_zero_nat)))). % type.size(3)
thf(fact_44_dB_Osize__gen_I1_J, axiom,
    ((![X1 : nat]: ((size_dB @ (var @ X1)) = zero_zero_nat)))). % dB.size_gen(1)
thf(fact_45_subst__type__IT, axiom,
    ((![T3 : dB, E : nat > type, I : nat, U2 : type, T : type, U : dB]: ((it @ T3) => ((typing @ (shift_type @ E @ I @ U2) @ T3 @ T) => ((it @ U) => ((typing @ E @ U @ U2) => (it @ (subst @ T3 @ U @ I))))))))). % subst_type_IT
thf(fact_46_dB_Osize_I4_J, axiom,
    ((![X1 : nat]: ((size_size_dB @ (var @ X1)) = zero_zero_nat)))). % dB.size(4)
thf(fact_47_zero__natural_Orsp, axiom,
    ((zero_zero_nat = zero_zero_nat))). % zero_natural.rsp
thf(fact_48_subst__eq, axiom,
    ((![K : nat, U : dB]: ((subst @ (var @ K) @ U @ K) = U)))). % subst_eq
thf(fact_49_subst__lift, axiom,
    ((![T3 : dB, K : nat, S : dB]: ((subst @ (lift @ T3 @ K) @ S @ K) = T3)))). % subst_lift
thf(fact_50_size__neq__size__imp__neq, axiom,
    ((![X3 : type, Y : type]: ((~ (((size_size_type @ X3) = (size_size_type @ Y)))) => (~ ((X3 = Y))))))). % size_neq_size_imp_neq
thf(fact_51_size__neq__size__imp__neq, axiom,
    ((![X3 : dB, Y : dB]: ((~ (((size_size_dB @ X3) = (size_size_dB @ Y)))) => (~ ((X3 = Y))))))). % size_neq_size_imp_neq
thf(fact_52_subst__App, axiom,
    ((![T3 : dB, U : dB, S : dB, K : nat]: ((subst @ (app @ T3 @ U) @ S @ K) = (app @ (subst @ T3 @ S @ K) @ (subst @ U @ S @ K)))))). % subst_App
thf(fact_53_subst__Var__IT, axiom,
    ((![R : dB, I : nat, J : nat]: ((it @ R) => (it @ (subst @ R @ (var @ I) @ J)))))). % subst_Var_IT
thf(fact_54_subst__lemma, axiom,
    ((![E : nat > type, T3 : dB, T : type, E2 : nat > type, U : dB, U2 : type, I : nat]: ((typing @ E @ T3 @ T) => ((typing @ E2 @ U @ U2) => ((E = (shift_type @ E2 @ I @ U2)) => (typing @ E2 @ (subst @ T3 @ U @ I) @ T))))))). % subst_lemma
thf(fact_55_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_56_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_57_beta_Oinducts, axiom,
    ((![X1 : dB, X23 : dB, P : dB > dB > $o]: ((beta @ X1 @ X23) => ((![S2 : dB, T4 : dB]: (P @ (app @ (abs @ S2) @ T4) @ (subst @ S2 @ T4 @ zero_zero_nat))) => ((![S2 : dB, T4 : dB, U5 : dB]: ((beta @ S2 @ T4) => ((P @ S2 @ T4) => (P @ (app @ S2 @ U5) @ (app @ T4 @ U5))))) => ((![S2 : dB, T4 : dB, U5 : dB]: ((beta @ S2 @ T4) => ((P @ S2 @ T4) => (P @ (app @ U5 @ S2) @ (app @ U5 @ T4))))) => ((![S2 : dB, T4 : dB]: ((beta @ S2 @ T4) => ((P @ S2 @ T4) => (P @ (abs @ S2) @ (abs @ T4))))) => (P @ X1 @ X23))))))))). % beta.inducts
thf(fact_58_beta_Osimps, axiom,
    ((beta = (^[A12 : dB]: (^[A22 : dB]: (((?[S3 : dB]: (?[T6 : dB]: (((A12 = (app @ (abs @ S3) @ T6))) & ((A22 = (subst @ S3 @ T6 @ zero_zero_nat))))))) | ((((?[S3 : dB]: (?[T6 : dB]: (?[U6 : dB]: (((A12 = (app @ S3 @ U6))) & ((((A22 = (app @ T6 @ U6))) & ((beta @ S3 @ T6))))))))) | ((((?[S3 : dB]: (?[T6 : dB]: (?[U6 : dB]: (((A12 = (app @ U6 @ S3))) & ((((A22 = (app @ U6 @ T6))) & ((beta @ S3 @ T6))))))))) | ((?[S3 : dB]: (?[T6 : dB]: (((A12 = (abs @ S3))) & ((((A22 = (abs @ T6))) & ((beta @ S3 @ T6)))))))))))))))))). % beta.simps
thf(fact_59_beta_Ocases, axiom,
    ((![A1 : dB, A2 : dB]: ((beta @ A1 @ A2) => ((![S2 : dB, T4 : dB]: ((A1 = (app @ (abs @ S2) @ T4)) => (~ ((A2 = (subst @ S2 @ T4 @ zero_zero_nat)))))) => ((![S2 : dB, T4 : dB, U5 : dB]: ((A1 = (app @ S2 @ U5)) => ((A2 = (app @ T4 @ U5)) => (~ ((beta @ S2 @ T4)))))) => ((![S2 : dB, T4 : dB, U5 : dB]: ((A1 = (app @ U5 @ S2)) => ((A2 = (app @ U5 @ T4)) => (~ ((beta @ S2 @ T4)))))) => (~ ((![S2 : dB]: ((A1 = (abs @ S2)) => (![T4 : dB]: ((A2 = (abs @ T4)) => (~ ((beta @ S2 @ T4)))))))))))))))). % beta.cases
thf(fact_60_appR, axiom,
    ((![S : dB, T3 : dB, U : dB]: ((beta @ S @ T3) => (beta @ (app @ U @ S) @ (app @ U @ T3)))))). % appR
thf(fact_61_appL, axiom,
    ((![S : dB, T3 : dB, U : dB]: ((beta @ S @ T3) => (beta @ (app @ S @ U) @ (app @ T3 @ U)))))). % appL
thf(fact_62_beta__cases_I1_J, axiom,
    ((![I : nat, T3 : dB]: (~ ((beta @ (var @ I) @ T3)))))). % beta_cases(1)
thf(fact_63_abs, axiom,
    ((![S : dB, T3 : dB]: ((beta @ S @ T3) => (beta @ (abs @ S) @ (abs @ T3)))))). % abs
thf(fact_64_beta__cases_I2_J, axiom,
    ((![R : dB, S : dB]: ((beta @ (abs @ R) @ S) => (~ ((![T4 : dB]: ((S = (abs @ T4)) => (~ ((beta @ R @ T4))))))))))). % beta_cases(2)
thf(fact_65_subject__reduction, axiom,
    ((![E : nat > type, T3 : dB, T : type, T7 : dB]: ((typing @ E @ T3 @ T) => ((beta @ T3 @ T7) => (typing @ E @ T7 @ T)))))). % subject_reduction
thf(fact_66_subst__preserves__beta, axiom,
    ((![R : dB, S : dB, T3 : dB, I : nat]: ((beta @ R @ S) => (beta @ (subst @ R @ T3 @ I) @ (subst @ S @ T3 @ I)))))). % subst_preserves_beta
thf(fact_67_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_68_substn_Osimps_I2_J, axiom,
    ((![T3 : dB, U : dB, S : dB, K : nat]: ((substn @ (app @ T3 @ U) @ S @ K) = (app @ (substn @ T3 @ S @ K) @ (substn @ U @ S @ K)))))). % substn.simps(2)
thf(fact_69_var__app__type__eq, axiom,
    ((![E : nat > type, I : nat, Ts : list_dB, T : type, U2 : type]: ((typing @ E @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ T) => ((typing @ E @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ U2) => (T = U2)))))). % var_app_type_eq
thf(fact_70_beta__cases_I3_J, axiom,
    ((![S : dB, T3 : dB, U : dB]: ((beta @ (app @ S @ T3) @ U) => ((![S2 : dB]: ((S = (abs @ S2)) => (~ ((U = (subst @ S2 @ T3 @ zero_zero_nat)))))) => ((![T4 : dB]: ((U = (app @ T4 @ T3)) => (~ ((beta @ S @ T4))))) => (~ ((![T4 : dB]: ((U = (app @ S @ T4)) => (~ ((beta @ T3 @ T4))))))))))))). % beta_cases(3)
thf(fact_71_beta, axiom,
    ((![S : dB, T3 : dB]: (beta @ (app @ (abs @ S) @ T3) @ (subst @ S @ T3 @ zero_zero_nat))))). % beta
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_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_74_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_75_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_76_ex__head__tail, axiom,
    ((![T3 : dB]: (?[Ts2 : list_dB, H : dB]: ((T3 = (foldl_dB_dB @ app @ H @ Ts2)) & ((?[N2 : nat]: (H = (var @ N2))) | (?[U5 : dB]: (H = (abs @ U5))))))))). % ex_head_tail
thf(fact_77_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_78_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_79_substn__subst__n, axiom,
    ((substn = (^[T6 : dB]: (^[S3 : dB]: (^[N3 : nat]: (subst @ T6 @ (liftn @ N3 @ S3 @ zero_zero_nat) @ N3))))))). % substn_subst_n
thf(fact_80_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_81_IT_Osimps, axiom,
    ((it = (^[A4 : dB]: (((?[Rs3 : list_dB]: (?[N3 : nat]: (((A4 = (foldl_dB_dB @ app @ (var @ N3) @ Rs3))) & ((listsp_dB @ it @ Rs3)))))) | ((((?[R3 : dB]: (((A4 = (abs @ R3))) & ((it @ R3))))) | ((?[R3 : dB]: (?[S3 : dB]: (?[Ss3 : list_dB]: (((A4 = (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_82_liftn__0, axiom,
    ((![T3 : dB, K : nat]: ((liftn @ zero_zero_nat @ T3 @ K) = T3)))). % liftn_0
thf(fact_83_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_84_IT_OVar, axiom,
    ((![Rs : list_dB, N : nat]: ((listsp_dB @ it @ Rs) => (it @ (foldl_dB_dB @ app @ (var @ N) @ Rs)))))). % IT.Var
thf(fact_85__092_060open_062IT_A_IVar_A0_A_092_060degree_062_092_060degree_062_A_091lift_At_A0_093_J_092_060close_062, axiom,
    ((it @ (foldl_dB_dB @ app @ (var @ zero_zero_nat) @ (cons_dB @ (lift @ ta2 @ zero_zero_nat) @ nil_dB))))). % \<open>IT (Var 0 \<degree>\<degree> [lift t 0])\<close>
thf(fact_86_lem, axiom,
    ((![P : dB > $o, T3 : dB, N : nat]: ((![N2 : nat, Ts2 : list_dB]: ((![X5 : dB]: ((member_dB @ X5 @ (set_dB2 @ Ts2)) => (P @ X5))) => (P @ (foldl_dB_dB @ app @ (var @ N2) @ Ts2)))) => ((![U5 : dB]: ((P @ U5) => (![Ts2 : list_dB]: ((![X5 : dB]: ((member_dB @ X5 @ (set_dB2 @ Ts2)) => (P @ X5))) => (P @ (foldl_dB_dB @ app @ (abs @ U5) @ Ts2)))))) => (((size_size_dB @ T3) = N) => (P @ T3))))))). % lem
thf(fact_87_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_88__092_060open_062listsp_AIT_A_091lift_At_A0_093_092_060close_062, axiom,
    ((listsp_dB @ it @ (cons_dB @ (lift @ ta2 @ zero_zero_nat) @ nil_dB)))). % \<open>listsp IT [lift t 0]\<close>
thf(fact_89__092_060open_062Var_A0_A_092_060degree_062_092_060degree_062_A_091lift_At_A0_093_A_061_AVar_A0_A_092_060degree_062_Alift_At_A0_092_060close_062, axiom,
    (((foldl_dB_dB @ app @ (var @ zero_zero_nat) @ (cons_dB @ (lift @ ta2 @ zero_zero_nat) @ nil_dB)) = (app @ (var @ zero_zero_nat) @ (lift @ ta2 @ zero_zero_nat))))). % \<open>Var 0 \<degree>\<degree> [lift t 0] = Var 0 \<degree> lift t 0\<close>
thf(fact_90_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_91_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_92_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_93_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)))))) => (~ ((![T4 : dB]: ((R = (abs @ T4)) => (![U5 : dB, Us : list_dB]: ((Rs = (cons_dB @ U5 @ Us)) => (~ ((S = (foldl_dB_dB @ app @ (subst @ T4 @ U5 @ zero_zero_nat) @ Us)))))))))))))))). % apps_betasE
thf(fact_94_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_95_Apps__dB__induct, axiom,
    ((![P : dB > $o, T3 : dB]: ((![N2 : nat, Ts2 : list_dB]: ((![X5 : dB]: ((member_dB @ X5 @ (set_dB2 @ Ts2)) => (P @ X5))) => (P @ (foldl_dB_dB @ app @ (var @ N2) @ Ts2)))) => ((![U5 : dB]: ((P @ U5) => (![Ts2 : list_dB]: ((![X5 : dB]: ((member_dB @ X5 @ (set_dB2 @ Ts2)) => (P @ X5))) => (P @ (foldl_dB_dB @ app @ (abs @ U5) @ Ts2)))))) => (P @ T3)))))). % Apps_dB_induct
thf(fact_96_length__0__conv, axiom,
    ((![Xs : list_dB]: (((size_size_list_dB @ Xs) = zero_zero_nat) = (Xs = nil_dB))))). % length_0_conv
thf(fact_97_length__0__conv, axiom,
    ((![Xs : list_type]: (((size_size_list_type @ Xs) = zero_zero_nat) = (Xs = nil_type))))). % length_0_conv
thf(fact_98_foldl__cong, axiom,
    ((![A : dB, B : dB, L : list_dB, K : list_dB, F : dB > dB > dB, G : dB > dB > dB]: ((A = B) => ((L = K) => ((![A5 : dB, X : dB]: ((member_dB @ X @ (set_dB2 @ L)) => ((F @ A5 @ X) = (G @ A5 @ X)))) => ((foldl_dB_dB @ F @ A @ L) = (foldl_dB_dB @ G @ B @ K)))))))). % foldl_cong
thf(fact_99_foldl__Nil, axiom,
    ((![F : dB > dB > dB, A : dB]: ((foldl_dB_dB @ F @ A @ nil_dB) = A)))). % foldl_Nil
thf(fact_100_list_Osize_I3_J, axiom,
    (((size_size_list_dB @ nil_dB) = zero_zero_nat))). % list.size(3)
thf(fact_101_list_Osize_I3_J, axiom,
    (((size_size_list_type @ nil_type) = zero_zero_nat))). % list.size(3)
thf(fact_102_foldl__Cons, axiom,
    ((![F : dB > dB > dB, A : dB, X3 : dB, Xs : list_dB]: ((foldl_dB_dB @ F @ A @ (cons_dB @ X3 @ Xs)) = (foldl_dB_dB @ F @ (F @ A @ X3) @ Xs))))). % foldl_Cons
thf(fact_103_count__notin, axiom,
    ((![X3 : dB, Xs : list_dB]: ((~ ((member_dB @ X3 @ (set_dB2 @ Xs)))) => ((count_list_dB @ Xs @ X3) = zero_zero_nat))))). % count_notin
thf(fact_104_app__last, axiom,
    ((![T3 : dB, Ts : list_dB, U : dB]: ((app @ (foldl_dB_dB @ app @ T3 @ Ts) @ U) = (foldl_dB_dB @ app @ T3 @ (append_dB @ Ts @ (cons_dB @ U @ nil_dB))))))). % app_last
thf(fact_105_foldl__append, axiom,
    ((![F : dB > dB > dB, A : dB, Xs : list_dB, Ys : list_dB]: ((foldl_dB_dB @ F @ A @ (append_dB @ Xs @ Ys)) = (foldl_dB_dB @ F @ (foldl_dB_dB @ F @ A @ Xs) @ Ys))))). % foldl_append
thf(fact_106_rev__exhaust2, axiom,
    ((![Xs : list_dB]: ((~ ((Xs = nil_dB))) => (~ ((![Ys2 : list_dB, Y2 : dB]: (~ ((Xs = (append_dB @ Ys2 @ (cons_dB @ Y2 @ nil_dB)))))))))))). % rev_exhaust2
thf(fact_107_rev__exhaust2, axiom,
    ((![Xs : list_type]: ((~ ((Xs = nil_type))) => (~ ((![Ys2 : list_type, Y2 : type]: (~ ((Xs = (append_type @ Ys2 @ (cons_type @ Y2 @ nil_type)))))))))))). % rev_exhaust2
thf(fact_108_count__list_Osimps_I1_J, axiom,
    ((![Y : dB]: ((count_list_dB @ nil_dB @ Y) = zero_zero_nat)))). % count_list.simps(1)
thf(fact_109_count__list_Osimps_I1_J, axiom,
    ((![Y : type]: ((count_list_type @ nil_type @ Y) = zero_zero_nat)))). % count_list.simps(1)
thf(fact_110_App__eq__foldl__conv, axiom,
    ((![R : dB, S : dB, T3 : dB, Ts : list_dB]: (((app @ R @ S) = (foldl_dB_dB @ app @ T3 @ Ts)) = (((((Ts = nil_dB)) => (((app @ R @ S) = T3)))) & ((((~ ((Ts = nil_dB)))) => ((?[Ss3 : list_dB]: (((Ts = (append_dB @ Ss3 @ (cons_dB @ S @ nil_dB)))) & ((R = (foldl_dB_dB @ app @ T3 @ Ss3))))))))))))). % App_eq_foldl_conv
thf(fact_111_n__lists__Nil, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((n_lists_dB @ N @ nil_dB) = (cons_list_dB @ nil_dB @ nil_list_dB))) & ((~ ((N = zero_zero_nat))) => ((n_lists_dB @ N @ nil_dB) = nil_list_dB)))))). % n_lists_Nil
thf(fact_112_n__lists__Nil, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((n_lists_type @ N @ nil_type) = (cons_list_type @ nil_type @ nil_list_type))) & ((~ ((N = zero_zero_nat))) => ((n_lists_type @ N @ nil_type) = nil_list_type)))))). % n_lists_Nil
thf(fact_113_nths__singleton, axiom,
    ((![A6 : set_nat, X3 : dB]: (((member_nat @ zero_zero_nat @ A6) => ((nths_dB @ (cons_dB @ X3 @ nil_dB) @ A6) = (cons_dB @ X3 @ nil_dB))) & ((~ ((member_nat @ zero_zero_nat @ A6))) => ((nths_dB @ (cons_dB @ X3 @ nil_dB) @ A6) = nil_dB)))))). % nths_singleton
thf(fact_114_nths__singleton, axiom,
    ((![A6 : set_nat, X3 : type]: (((member_nat @ zero_zero_nat @ A6) => ((nths_type @ (cons_type @ X3 @ nil_type) @ A6) = (cons_type @ X3 @ nil_type))) & ((~ ((member_nat @ zero_zero_nat @ A6))) => ((nths_type @ (cons_type @ X3 @ nil_type) @ A6) = nil_type)))))). % nths_singleton
thf(fact_115_n__lists_Osimps_I1_J, axiom,
    ((![Xs : list_dB]: ((n_lists_dB @ zero_zero_nat @ Xs) = (cons_list_dB @ nil_dB @ nil_list_dB))))). % n_lists.simps(1)
thf(fact_116_n__lists_Osimps_I1_J, axiom,
    ((![Xs : list_type]: ((n_lists_type @ zero_zero_nat @ Xs) = (cons_list_type @ nil_type @ nil_list_type))))). % n_lists.simps(1)
thf(fact_117_types__snoc, axiom,
    ((![E : nat > type, Ts : list_dB, Ts3 : list_type, T3 : dB, T : type]: ((typings @ E @ Ts @ Ts3) => ((typing @ E @ T3 @ T) => (typings @ E @ (append_dB @ Ts @ (cons_dB @ T3 @ nil_dB)) @ (append_type @ Ts3 @ (cons_type @ T @ nil_type)))))))). % types_snoc
thf(fact_118_types__snocE, axiom,
    ((![E : nat > type, Ts : list_dB, T3 : dB, Ts3 : list_type]: ((typings @ E @ (append_dB @ Ts @ (cons_dB @ T3 @ nil_dB)) @ Ts3) => (~ ((![Us2 : list_type, U3 : type]: ((Ts3 = (append_type @ Us2 @ (cons_type @ U3 @ nil_type))) => ((typings @ E @ Ts @ Us2) => (~ ((typing @ E @ T3 @ U3)))))))))))). % types_snocE
thf(fact_119_typings_Osimps_I1_J, axiom,
    ((![E : nat > type, Ts3 : list_type]: ((typings @ E @ nil_dB @ Ts3) = (Ts3 = nil_type))))). % typings.simps(1)
thf(fact_120_types__snoc__eq, axiom,
    ((![E : nat > type, Ts : list_dB, T3 : dB, Ts3 : list_type, T : type]: ((typings @ E @ (append_dB @ Ts @ (cons_dB @ T3 @ nil_dB)) @ (append_type @ Ts3 @ (cons_type @ T @ nil_type))) = (((typings @ E @ Ts @ Ts3)) & ((typing @ E @ T3 @ T))))))). % types_snoc_eq
thf(fact_121_var__app__types, axiom,
    ((![E : nat > type, I : nat, Ts : list_dB, Us3 : list_dB, T : type, Ts3 : list_type, U2 : type]: ((typing @ E @ (foldl_dB_dB @ app @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ Us3) @ T) => ((typings @ E @ Ts @ Ts3) => ((typing @ E @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ U2) => (?[Us2 : list_type]: ((U2 = (foldr_type_type @ fun @ Us2 @ T)) & (typings @ E @ Us3 @ Us2))))))))). % var_app_types
thf(fact_122_var__app__typesE, axiom,
    ((![E : nat > type, I : nat, Ts : list_dB, T : type]: ((typing @ E @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ T) => (~ ((![Ts4 : list_type]: ((typing @ E @ (var @ I) @ (foldr_type_type @ fun @ Ts4 @ T)) => (~ ((typings @ E @ Ts @ Ts4))))))))))). % var_app_typesE
thf(fact_123_list__app__typeD, axiom,
    ((![E : nat > type, T3 : dB, Ts : list_dB, T : type]: ((typing @ E @ (foldl_dB_dB @ app @ T3 @ Ts) @ T) => (?[Ts4 : list_type]: ((typing @ E @ T3 @ (foldr_type_type @ fun @ Ts4 @ T)) & (typings @ E @ Ts @ Ts4))))))). % list_app_typeD
thf(fact_124_list__app__typeE, axiom,
    ((![E : nat > type, T3 : dB, Ts : list_dB, T : type]: ((typing @ E @ (foldl_dB_dB @ app @ T3 @ Ts) @ T) => (~ ((![Ts4 : list_type]: ((typing @ E @ T3 @ (foldr_type_type @ fun @ Ts4 @ T)) => (~ ((typings @ E @ Ts @ Ts4))))))))))). % list_app_typeE
thf(fact_125_list__app__typeI, axiom,
    ((![E : nat > type, T3 : dB, Ts3 : list_type, T : type, Ts : list_dB]: ((typing @ E @ T3 @ (foldr_type_type @ fun @ Ts3 @ T)) => ((typings @ E @ Ts @ Ts3) => (typing @ E @ (foldl_dB_dB @ app @ T3 @ Ts) @ T)))))). % list_app_typeI
thf(fact_126_IT_Oinducts, axiom,
    ((![X3 : dB, P : dB > $o]: ((it @ X3) => ((![Rs2 : list_dB, N2 : nat]: ((listsp_dB @ (^[X4 : dB]: (((it @ X4)) & ((P @ X4)))) @ 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 @ X3)))))))). % IT.inducts
thf(fact_127_length__greater__0__conv, axiom,
    ((![Xs : list_dB]: ((ord_less_nat @ zero_zero_nat @ (size_size_list_dB @ Xs)) = (~ ((Xs = nil_dB))))))). % length_greater_0_conv
thf(fact_128_length__greater__0__conv, axiom,
    ((![Xs : list_type]: ((ord_less_nat @ zero_zero_nat @ (size_size_list_type @ Xs)) = (~ ((Xs = nil_type))))))). % length_greater_0_conv
thf(fact_129_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_130_neq0__conv, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N))))). % neq0_conv
thf(fact_131_less__nat__zero__code, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_132_bot__nat__0_Onot__eq__extremum, axiom,
    ((![A : nat]: ((~ ((A = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ A))))). % bot_nat_0.not_eq_extremum
thf(fact_133_shift__gt, axiom,
    ((![J : nat, I : nat, E : nat > type, T : type]: ((ord_less_nat @ J @ I) => ((shift_type @ E @ I @ T @ J) = (E @ J)))))). % shift_gt
thf(fact_134_subst__lt, axiom,
    ((![J : nat, I : nat, U : dB]: ((ord_less_nat @ J @ I) => ((subst @ (var @ J) @ U @ I) = (var @ J)))))). % subst_lt
thf(fact_135_zero__less__iff__neq__zero, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) = (~ ((N = zero_zero_nat))))))). % zero_less_iff_neq_zero
thf(fact_136_gr__implies__not__zero, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not_zero
thf(fact_137_not__less__zero, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less_zero
thf(fact_138_gr__zeroI, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr_zeroI
thf(fact_139_gr0I, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr0I
thf(fact_140_not__gr0, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr0
thf(fact_141_not__less0, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less0
thf(fact_142_less__zeroE, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_zeroE
thf(fact_143_gr__implies__not0, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not0
thf(fact_144_infinite__descent0, axiom,
    ((![P : nat > $o, N : nat]: ((P @ zero_zero_nat) => ((![N2 : nat]: ((ord_less_nat @ zero_zero_nat @ N2) => ((~ ((P @ N2))) => (?[M2 : nat]: ((ord_less_nat @ M2 @ N2) & (~ ((P @ M2)))))))) => (P @ N)))))). % infinite_descent0
thf(fact_145_bot__nat__0_Oextremum__strict, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ zero_zero_nat)))))). % bot_nat_0.extremum_strict

% Conjectures (1)
thf(conj_0, conjecture,
    ((typing @ ea @ s @ (fun @ ta @ u)))).
