% 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_100__5210176_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:37:31.509

% 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__LambdaType__Otype, type,
    type : $tType).
thf(ty_n_t__Lambda__OdB, type,
    dB : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).
thf(ty_n_t__Int__Oint, type,
    int : $tType).

% Explicit typings (33)
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint, type,
    uminus_uminus_int : int > int).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint, type,
    zero_zero_int : int).
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_Int_Onat, type,
    nat2 : int > nat).
thf(sy_c_LambdaType_Oshift_001t__LambdaType__Otype, type,
    shift_type : (nat > type) > nat > type > nat > type).
thf(sy_c_LambdaType_Otype_OFun, type,
    fun : type > type > type).
thf(sy_c_LambdaType_Otyping, type,
    typing : (nat > type) > dB > type > $o).
thf(sy_c_LambdaType_Otypings, type,
    typings : (nat > type) > list_dB > list_type > $o).
thf(sy_c_Lambda_OdB_OAbs, type,
    abs : dB > dB).
thf(sy_c_Lambda_OdB_OApp, type,
    app : dB > dB > dB).
thf(sy_c_Lambda_OdB_OVar, type,
    var : nat > dB).
thf(sy_c_Lambda_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_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_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_Osemiring__1__class_Oof__nat_001t__Int__Oint, type,
    semiri2019852685at_int : nat > int).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat, type,
    semiri1382578993at_nat : nat > nat).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint, type,
    ord_less_int : int > int > $o).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat, type,
    ord_less_nat : nat > nat > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint, type,
    ord_less_eq_int : int > int > $o).
thf(sy_v_T_H, type,
    t : type).
thf(sy_v_T____, type,
    t2 : type).
thf(sy_v_e, type,
    e : nat > type).
thf(sy_v_i, type,
    i : nat).
thf(sy_v_t____, type,
    t3 : dB).
thf(sy_v_u, type,
    u : dB).

% Relevant facts (150)
thf(fact_0__092_060open_062IT_At_092_060close_062, axiom,
    ((it @ t3))). % \<open>IT t\<close>
thf(fact_1_MI1, axiom,
    ((![T1 : type, T2 : type, T : dB, E : nat > type, I : nat, T3 : type, U : dB]: ((t2 = (fun @ T1 @ T2)) => ((it @ T) => ((typing @ (shift_type @ E @ I @ T1) @ T @ T3) => ((it @ U) => ((typing @ E @ U @ T1) => (it @ (subst @ T @ U @ I)))))))))). % MI1
thf(fact_2_MI2, axiom,
    ((![T1 : type, T2 : type, T : dB, E : nat > type, I : nat, T3 : type, U : dB]: ((t2 = (fun @ T1 @ T2)) => ((it @ T) => ((typing @ (shift_type @ E @ I @ T2) @ T @ T3) => ((it @ U) => ((typing @ E @ U @ T2) => (it @ (subst @ T @ U @ I)))))))))). % MI2
thf(fact_3_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_4_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_5_subst__Var__IT, axiom,
    ((![R : dB, I : nat, J : nat]: ((it @ R) => (it @ (subst @ R @ (var @ I) @ J)))))). % subst_Var_IT
thf(fact_6_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_7_lift__IT, axiom,
    ((![T : dB, I : nat]: ((it @ T) => (it @ (lift @ T @ I)))))). % lift_IT
thf(fact_8_substs__lemma, axiom,
    ((![E : nat > type, U : dB, T3 : type, I : nat, Ts : list_dB, Ts2 : list_type]: ((typing @ E @ U @ T3) => ((typings @ (shift_type @ E @ I @ T3) @ Ts @ Ts2) => (typings @ E @ (map_dB_dB @ (^[T4 : dB]: (subst @ T4 @ U @ I)) @ Ts) @ Ts2)))))). % substs_lemma
thf(fact_9_Var__IT, axiom,
    ((![N : nat]: (it @ (var @ N))))). % Var_IT
thf(fact_10_subst__lift, axiom,
    ((![T : dB, K : nat, S : dB]: ((subst @ (lift @ T @ K) @ S @ K) = T)))). % subst_lift
thf(fact_11_shift__gt, axiom,
    ((![J : nat, I : nat, E : nat > type, T3 : type]: ((ord_less_nat @ J @ I) => ((shift_type @ E @ I @ T3 @ J) = (E @ J)))))). % shift_gt
thf(fact_12_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_13_dB_Oinject_I3_J, axiom,
    ((![X3 : dB, Y3 : dB]: (((abs @ X3) = (abs @ Y3)) = (X3 = Y3))))). % dB.inject(3)
thf(fact_14_dB_Oinject_I1_J, axiom,
    ((![X1 : nat, Y1 : nat]: (((var @ X1) = (var @ Y1)) = (X1 = Y1))))). % dB.inject(1)
thf(fact_15_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_16_subst__eq, axiom,
    ((![K : nat, U : dB]: ((subst @ (var @ K) @ U @ K) = U)))). % subst_eq
thf(fact_17_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_18_dB_Odistinct_I3_J, axiom,
    ((![X1 : nat, X3 : dB]: (~ (((var @ X1) = (abs @ X3))))))). % dB.distinct(3)
thf(fact_19_type__induct, axiom,
    ((![P : type > $o, T3 : type]: ((![T5 : type]: ((![T12 : type, T22 : type]: ((T5 = (fun @ T12 @ T22)) => (P @ T12))) => ((![T12 : type, T22 : type]: ((T5 = (fun @ T12 @ T22)) => (P @ T22))) => (P @ T5)))) => (P @ T3))))). % type_induct
thf(fact_20_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 @ (^[T4 : dB]: (lift @ T4 @ I)) @ Ts) @ Ts2))))). % lift_types
thf(fact_21_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_22_typing__elims_I3_J, axiom,
    ((![E : nat > type, T : dB, T3 : type]: ((typing @ E @ (abs @ T) @ T3) => (~ ((![T5 : type, U3 : type]: ((T3 = (fun @ T5 @ U3)) => (~ ((typing @ (shift_type @ E @ zero_zero_nat @ T5) @ T @ U3))))))))))). % typing_elims(3)
thf(fact_23_typing_OVar, axiom,
    ((![Env : nat > type, X : nat, T3 : type]: (((Env @ X) = T3) => (typing @ Env @ (var @ X) @ T3))))). % typing.Var
thf(fact_24_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_25_neq0__conv, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N))))). % neq0_conv
thf(fact_26_less__nat__zero__code, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_27_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_28_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_29_map__ident, axiom,
    (((map_dB_dB @ (^[X2 : dB]: X2)) = (^[Xs : list_dB]: Xs)))). % map_ident
thf(fact_30_Lambda, axiom,
    ((![R : dB]: ((it @ R) => (it @ (abs @ R)))))). % Lambda
thf(fact_31_gr0I, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr0I
thf(fact_32_not__gr0, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr0
thf(fact_33_not__less0, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less0
thf(fact_34_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_35_zero__reorient, axiom,
    ((![X : int]: ((zero_zero_int = X) = (X = zero_zero_int))))). % zero_reorient
thf(fact_36_linorder__neqE__nat, axiom,
    ((![X : nat, Y : nat]: ((~ ((X = Y))) => ((~ ((ord_less_nat @ X @ Y))) => (ord_less_nat @ Y @ X)))))). % linorder_neqE_nat
thf(fact_37_infinite__descent, axiom,
    ((![P : nat > $o, N : nat]: ((![N2 : nat]: ((~ ((P @ N2))) => (?[M : nat]: ((ord_less_nat @ M @ N2) & (~ ((P @ M))))))) => (P @ N))))). % infinite_descent
thf(fact_38_nat__less__induct, axiom,
    ((![P : nat > $o, N : nat]: ((![N2 : nat]: ((![M : nat]: ((ord_less_nat @ M @ N2) => (P @ M))) => (P @ N2))) => (P @ N))))). % nat_less_induct
thf(fact_39_less__irrefl__nat, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_irrefl_nat
thf(fact_40_less__not__refl3, axiom,
    ((![S : nat, T : nat]: ((ord_less_nat @ S @ T) => (~ ((S = T))))))). % less_not_refl3
thf(fact_41_less__not__refl2, axiom,
    ((![N : nat, M2 : nat]: ((ord_less_nat @ N @ M2) => (~ ((M2 = N))))))). % less_not_refl2
thf(fact_42_less__not__refl, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_not_refl
thf(fact_43_nat__neq__iff, axiom,
    ((![M2 : nat, N : nat]: ((~ ((M2 = N))) = (((ord_less_nat @ M2 @ N)) | ((ord_less_nat @ N @ M2))))))). % nat_neq_iff
thf(fact_44_list_Omap__ident, axiom,
    ((![T : list_dB]: ((map_dB_dB @ (^[X2 : dB]: X2) @ T) = T)))). % list.map_ident
thf(fact_45_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_46_gr__implies__not__zero, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_nat @ M2 @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not_zero
thf(fact_47_not__less__zero, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less_zero
thf(fact_48_gr__zeroI, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr_zeroI
thf(fact_49_bot__nat__0_Oextremum__strict, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ zero_zero_nat)))))). % bot_nat_0.extremum_strict
thf(fact_50_infinite__descent0, axiom,
    ((![P : nat > $o, N : nat]: ((P @ zero_zero_nat) => ((![N2 : nat]: ((ord_less_nat @ zero_zero_nat @ N2) => ((~ ((P @ N2))) => (?[M : nat]: ((ord_less_nat @ M @ N2) & (~ ((P @ M)))))))) => (P @ N)))))). % infinite_descent0
thf(fact_51_gr__implies__not0, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_nat @ M2 @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not0
thf(fact_52_less__zeroE, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_zeroE
thf(fact_53_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_nat @ zero_zero_nat @ zero_zero_nat))))). % less_numeral_extra(3)
thf(fact_54_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_int @ zero_zero_int @ zero_zero_int))))). % less_numeral_extra(3)
thf(fact_55_typing_Oinducts, axiom,
    ((![X1 : nat > type, X23 : dB, X3 : type, P : (nat > type) > dB > type > $o]: ((typing @ X1 @ X23 @ 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 @ X23 @ X3)))))))). % typing.inducts
thf(fact_56_typing_Osimps, axiom,
    ((typing = (^[A1 : nat > type]: (^[A2 : dB]: (^[A3 : type]: (((?[Env3 : nat > type]: (?[X2 : nat]: (?[T7 : type]: (((A1 = Env3)) & ((((A2 = (var @ X2))) & ((((A3 = T7)) & (((Env3 @ X2) = T7))))))))))) | ((((?[Env3 : nat > type]: (?[T7 : type]: (?[T4 : dB]: (?[U4 : type]: (((A1 = Env3)) & ((((A2 = (abs @ T4))) & ((((A3 = (fun @ T7 @ U4))) & ((typing @ (shift_type @ Env3 @ zero_zero_nat @ T7) @ T4 @ U4)))))))))))) | ((?[Env3 : nat > type]: (?[S3 : dB]: (?[T7 : type]: (?[U4 : type]: (?[T4 : dB]: (((A1 = Env3)) & ((((A2 = (app @ S3 @ T4))) & ((((A3 = U4)) & ((((typing @ Env3 @ S3 @ (fun @ T7 @ U4))) & ((typing @ Env3 @ T4 @ T7)))))))))))))))))))))))). % typing.simps
thf(fact_57_typing_Ocases, axiom,
    ((![A12 : nat > type, A22 : dB, A32 : type]: ((typing @ A12 @ A22 @ A32) => ((![X4 : nat]: ((A22 = (var @ X4)) => (~ (((A12 @ X4) = A32))))) => ((![T5 : type, T6 : dB]: ((A22 = (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]: ((A22 = (app @ S2 @ T6)) => ((A32 = U3) => ((typing @ A12 @ S2 @ (fun @ T5 @ U3)) => (~ ((typing @ A12 @ T6 @ T5))))))))))))))). % typing.cases
thf(fact_58_lifts__IT, axiom,
    ((![Ts : list_dB]: ((listsp_dB @ it @ Ts) => (listsp_dB @ it @ (map_dB_dB @ (^[T4 : dB]: (lift @ T4 @ zero_zero_nat)) @ Ts)))))). % lifts_IT
thf(fact_59_dB_Osize__gen_I1_J, axiom,
    ((![X1 : nat]: ((size_dB @ (var @ X1)) = zero_zero_nat)))). % dB.size_gen(1)
thf(fact_60_map__eq__map__tailrec, axiom,
    ((map_dB_dB = map_tailrec_dB_dB))). % map_eq_map_tailrec
thf(fact_61_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_62_listsp__conj__eq, axiom,
    ((![A4 : dB > $o, B : dB > $o]: ((listsp_dB @ (^[X2 : dB]: (((A4 @ X2)) & ((B @ X2))))) = (^[X2 : list_dB]: (((listsp_dB @ A4 @ X2)) & ((listsp_dB @ B @ X2)))))))). % listsp_conj_eq
thf(fact_63_dB_Odistinct_I5_J, axiom,
    ((![X21 : dB, X22 : dB, X3 : dB]: (~ (((app @ X21 @ X22) = (abs @ X3))))))). % dB.distinct(5)
thf(fact_64_dB_Odistinct_I1_J, axiom,
    ((![X1 : nat, X21 : dB, X22 : dB]: (~ (((var @ X1) = (app @ X21 @ X22))))))). % dB.distinct(1)
thf(fact_65_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_66_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_67_dB_Oinduct, axiom,
    ((![P : dB > $o, DB : dB]: ((![X4 : nat]: (P @ (var @ X4))) => ((![X1a : dB, X24 : dB]: ((P @ X1a) => ((P @ X24) => (P @ (app @ X1a @ X24))))) => ((![X4 : dB]: ((P @ X4) => (P @ (abs @ X4)))) => (P @ DB))))))). % dB.induct
thf(fact_68_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_69_app__Var__IT, axiom,
    ((![T : dB, I : nat]: ((it @ T) => (it @ (app @ T @ (var @ I))))))). % app_Var_IT
thf(fact_70_typing__elims_I2_J, axiom,
    ((![E : nat > type, T : dB, U : dB, T3 : type]: ((typing @ E @ (app @ T @ U) @ T3) => (~ ((![T5 : type]: ((typing @ E @ T @ (fun @ T5 @ T3)) => (~ ((typing @ E @ U @ T5))))))))))). % typing_elims(2)
thf(fact_71_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_72_IT_Oinducts, axiom,
    ((![X : dB, P : dB > $o]: ((it @ X) => ((![Rs : list_dB, N2 : nat]: ((listsp_dB @ (^[X2 : dB]: (((it @ X2)) & ((P @ X2)))) @ Rs) => (P @ (foldl_dB_dB @ app @ (var @ N2) @ Rs)))) => ((![R2 : dB]: ((it @ R2) => ((P @ R2) => (P @ (abs @ R2))))) => ((![R2 : dB, S2 : dB, Ss : list_dB]: ((it @ (foldl_dB_dB @ app @ (subst @ R2 @ S2 @ zero_zero_nat) @ Ss)) => ((P @ (foldl_dB_dB @ app @ (subst @ R2 @ S2 @ zero_zero_nat) @ Ss)) => ((it @ S2) => ((P @ S2) => (P @ (foldl_dB_dB @ app @ (app @ (abs @ R2) @ S2) @ Ss))))))) => (P @ X)))))))). % IT.inducts
thf(fact_73_IT_Ocases, axiom,
    ((![A : dB]: ((it @ A) => ((![Rs : list_dB]: ((?[N2 : nat]: (A = (foldl_dB_dB @ app @ (var @ N2) @ Rs))) => (~ ((listsp_dB @ it @ Rs))))) => ((![R2 : dB]: ((A = (abs @ R2)) => (~ ((it @ R2))))) => (~ ((![R2 : dB, S2 : dB, Ss : list_dB]: ((A = (foldl_dB_dB @ app @ (app @ (abs @ R2) @ S2) @ Ss)) => ((it @ (foldl_dB_dB @ app @ (subst @ R2 @ S2 @ zero_zero_nat) @ Ss)) => (~ ((it @ S2)))))))))))))). % IT.cases
thf(fact_74_IT_Osimps, axiom,
    ((it = (^[A5 : dB]: (((?[Rs2 : list_dB]: (?[N3 : nat]: (((A5 = (foldl_dB_dB @ app @ (var @ N3) @ Rs2))) & ((listsp_dB @ it @ Rs2)))))) | ((((?[R3 : dB]: (((A5 = (abs @ R3))) & ((it @ R3))))) | ((?[R3 : dB]: (?[S3 : dB]: (?[Ss2 : list_dB]: (((A5 = (foldl_dB_dB @ app @ (app @ (abs @ R3) @ S3) @ Ss2))) & ((((it @ (foldl_dB_dB @ app @ (subst @ R3 @ S3 @ zero_zero_nat) @ Ss2))) & ((it @ S3)))))))))))))))). % IT.simps
thf(fact_75_Beta, axiom,
    ((![R : dB, S : dB, Ss3 : list_dB]: ((it @ (foldl_dB_dB @ app @ (subst @ R @ S @ zero_zero_nat) @ Ss3)) => ((it @ S) => (it @ (foldl_dB_dB @ app @ (app @ (abs @ R) @ S) @ Ss3))))))). % Beta
thf(fact_76_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 @ (^[A5 : dB]: (^[X2 : dB]: (G @ A5 @ (F @ X2)))) @ A @ Xs2))))). % foldl_map
thf(fact_77_var__app__type__eq, axiom,
    ((![E : nat > type, I : nat, Ts : list_dB, T3 : type, U2 : type]: ((typing @ E @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ T3) => ((typing @ E @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ U2) => (T3 = U2)))))). % var_app_type_eq
thf(fact_78_IT_OVar, axiom,
    ((![Rs3 : list_dB, N : nat]: ((listsp_dB @ it @ Rs3) => (it @ (foldl_dB_dB @ app @ (var @ N) @ Rs3)))))). % IT.Var
thf(fact_79_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 @ (^[T4 : dB]: (lift @ T4 @ I)) @ Ts)))))). % lift_map
thf(fact_80_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 @ (^[T4 : dB]: (subst @ T4 @ U @ I)) @ Ts)))))). % subst_map
thf(fact_81_Var__apps__eq__Var__apps__conv, axiom,
    ((![M2 : nat, Rs3 : list_dB, N : nat, Ss3 : list_dB]: (((foldl_dB_dB @ app @ (var @ M2) @ Rs3) = (foldl_dB_dB @ app @ (var @ N) @ Ss3)) = (((M2 = N)) & ((Rs3 = Ss3))))))). % Var_apps_eq_Var_apps_conv
thf(fact_82_Abs__apps__eq__Abs__apps__conv, axiom,
    ((![R : dB, Rs3 : list_dB, S : dB, Ss3 : list_dB]: (((foldl_dB_dB @ app @ (abs @ R) @ Rs3) = (foldl_dB_dB @ app @ (abs @ S) @ Ss3)) = (((R = S)) & ((Rs3 = Ss3))))))). % Abs_apps_eq_Abs_apps_conv
thf(fact_83_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_84_Var__apps__neq__Abs__apps, axiom,
    ((![N : nat, Ts : list_dB, R : dB, Ss3 : list_dB]: (~ (((foldl_dB_dB @ app @ (var @ N) @ Ts) = (foldl_dB_dB @ app @ (abs @ R) @ Ss3))))))). % Var_apps_neq_Abs_apps
thf(fact_85_Abs__App__neq__Var__apps, axiom,
    ((![S : dB, T : dB, N : nat, Ss3 : list_dB]: (~ (((app @ (abs @ S) @ T) = (foldl_dB_dB @ app @ (var @ N) @ Ss3))))))). % Abs_App_neq_Var_apps
thf(fact_86_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_87_var__app__types, axiom,
    ((![E : nat > type, I : nat, Ts : list_dB, Us : list_dB, T3 : type, Ts2 : list_type, U2 : type]: ((typing @ E @ (foldl_dB_dB @ app @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ Us) @ T3) => ((typings @ E @ Ts @ Ts2) => ((typing @ E @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ U2) => (?[Us2 : list_type]: ((U2 = (foldr_type_type @ fun @ Us2 @ T3)) & (typings @ E @ Us @ Us2))))))))). % var_app_types
thf(fact_88_var__app__typesE, axiom,
    ((![E : nat > type, I : nat, Ts : list_dB, T3 : type]: ((typing @ E @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ T3) => (~ ((![Ts4 : list_type]: ((typing @ E @ (var @ I) @ (foldr_type_type @ fun @ Ts4 @ T3)) => (~ ((typings @ E @ Ts @ Ts4))))))))))). % var_app_typesE
thf(fact_89_list__app__typeD, axiom,
    ((![E : nat > type, T : dB, Ts : list_dB, T3 : type]: ((typing @ E @ (foldl_dB_dB @ app @ T @ Ts) @ T3) => (?[Ts4 : list_type]: ((typing @ E @ T @ (foldr_type_type @ fun @ Ts4 @ T3)) & (typings @ E @ Ts @ Ts4))))))). % list_app_typeD
thf(fact_90_list__app__typeI, axiom,
    ((![E : nat > type, T : dB, Ts2 : list_type, T3 : type, Ts : list_dB]: ((typing @ E @ T @ (foldr_type_type @ fun @ Ts2 @ T3)) => ((typings @ E @ Ts @ Ts2) => (typing @ E @ (foldl_dB_dB @ app @ T @ Ts) @ T3)))))). % list_app_typeI
thf(fact_91_list__app__typeE, axiom,
    ((![E : nat > type, T : dB, Ts : list_dB, T3 : type]: ((typing @ E @ (foldl_dB_dB @ app @ T @ Ts) @ T3) => (~ ((![Ts4 : list_type]: ((typing @ E @ T @ (foldr_type_type @ fun @ Ts4 @ T3)) => (~ ((typings @ E @ Ts @ Ts4))))))))))). % list_app_typeE
thf(fact_92_of__nat__0__less__iff, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ (semiri1382578993at_nat @ N)) = (ord_less_nat @ zero_zero_nat @ N))))). % of_nat_0_less_iff
thf(fact_93_of__nat__0__less__iff, axiom,
    ((![N : nat]: ((ord_less_int @ zero_zero_int @ (semiri2019852685at_int @ N)) = (ord_less_nat @ zero_zero_nat @ N))))). % of_nat_0_less_iff
thf(fact_94_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_95_of__nat__eq__iff, axiom,
    ((![M2 : nat, N : nat]: (((semiri2019852685at_int @ M2) = (semiri2019852685at_int @ N)) = (M2 = N))))). % of_nat_eq_iff
thf(fact_96_of__nat__0, axiom,
    (((semiri1382578993at_nat @ zero_zero_nat) = zero_zero_nat))). % of_nat_0
thf(fact_97_of__nat__0, axiom,
    (((semiri2019852685at_int @ zero_zero_nat) = zero_zero_int))). % of_nat_0
thf(fact_98_of__nat__0__eq__iff, axiom,
    ((![N : nat]: ((zero_zero_nat = (semiri1382578993at_nat @ N)) = (zero_zero_nat = N))))). % of_nat_0_eq_iff
thf(fact_99_of__nat__0__eq__iff, axiom,
    ((![N : nat]: ((zero_zero_int = (semiri2019852685at_int @ N)) = (zero_zero_nat = N))))). % of_nat_0_eq_iff
thf(fact_100_of__nat__eq__0__iff, axiom,
    ((![M2 : nat]: (((semiri1382578993at_nat @ M2) = zero_zero_nat) = (M2 = zero_zero_nat))))). % of_nat_eq_0_iff
thf(fact_101_of__nat__eq__0__iff, axiom,
    ((![M2 : nat]: (((semiri2019852685at_int @ M2) = zero_zero_int) = (M2 = zero_zero_nat))))). % of_nat_eq_0_iff
thf(fact_102_of__nat__less__iff, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_nat @ (semiri1382578993at_nat @ M2) @ (semiri1382578993at_nat @ N)) = (ord_less_nat @ M2 @ N))))). % of_nat_less_iff
thf(fact_103_of__nat__less__iff, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_int @ (semiri2019852685at_int @ M2) @ (semiri2019852685at_int @ N)) = (ord_less_nat @ M2 @ N))))). % of_nat_less_iff
thf(fact_104_of__nat__less__0__iff, axiom,
    ((![M2 : nat]: (~ ((ord_less_nat @ (semiri1382578993at_nat @ M2) @ zero_zero_nat)))))). % of_nat_less_0_iff
thf(fact_105_of__nat__less__0__iff, axiom,
    ((![M2 : nat]: (~ ((ord_less_int @ (semiri2019852685at_int @ M2) @ zero_zero_int)))))). % of_nat_less_0_iff
thf(fact_106_less__imp__of__nat__less, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_nat @ M2 @ N) => (ord_less_nat @ (semiri1382578993at_nat @ M2) @ (semiri1382578993at_nat @ N)))))). % less_imp_of_nat_less
thf(fact_107_less__imp__of__nat__less, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_nat @ M2 @ N) => (ord_less_int @ (semiri2019852685at_int @ M2) @ (semiri2019852685at_int @ N)))))). % less_imp_of_nat_less
thf(fact_108_of__nat__less__imp__less, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_nat @ (semiri1382578993at_nat @ M2) @ (semiri1382578993at_nat @ N)) => (ord_less_nat @ M2 @ N))))). % of_nat_less_imp_less
thf(fact_109_of__nat__less__imp__less, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_int @ (semiri2019852685at_int @ M2) @ (semiri2019852685at_int @ N)) => (ord_less_nat @ M2 @ N))))). % of_nat_less_imp_less
thf(fact_110_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_111_substn__subst__n, axiom,
    ((substn = (^[T4 : dB]: (^[S3 : dB]: (^[N3 : nat]: (subst @ T4 @ (liftn @ N3 @ S3 @ zero_zero_nat) @ N3))))))). % substn_subst_n
thf(fact_112_pos__int__cases, axiom,
    ((![K : int]: ((ord_less_int @ zero_zero_int @ K) => (~ ((![N2 : nat]: ((K = (semiri2019852685at_int @ N2)) => (~ ((ord_less_nat @ zero_zero_nat @ N2))))))))))). % pos_int_cases
thf(fact_113_zero__less__imp__eq__int, axiom,
    ((![K : int]: ((ord_less_int @ zero_zero_int @ K) => (?[N2 : nat]: ((ord_less_nat @ zero_zero_nat @ N2) & (K = (semiri2019852685at_int @ N2)))))))). % zero_less_imp_eq_int
thf(fact_114_liftn__0, axiom,
    ((![T : dB, K : nat]: ((liftn @ zero_zero_nat @ T @ K) = T)))). % liftn_0
thf(fact_115_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_116_nat__less__as__int, axiom,
    ((ord_less_nat = (^[A5 : nat]: (^[B2 : nat]: (ord_less_int @ (semiri2019852685at_int @ A5) @ (semiri2019852685at_int @ B2))))))). % nat_less_as_int
thf(fact_117_nat__int__comparison_I2_J, axiom,
    ((ord_less_nat = (^[A5 : nat]: (^[B2 : nat]: (ord_less_int @ (semiri2019852685at_int @ A5) @ (semiri2019852685at_int @ B2))))))). % nat_int_comparison(2)
thf(fact_118_verit__comp__simplify1_I1_J, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ A)))))). % verit_comp_simplify1(1)
thf(fact_119_verit__comp__simplify1_I1_J, axiom,
    ((![A : int]: (~ ((ord_less_int @ A @ A)))))). % verit_comp_simplify1(1)
thf(fact_120_int__ops_I1_J, axiom,
    (((semiri2019852685at_int @ zero_zero_nat) = zero_zero_int))). % int_ops(1)
thf(fact_121_neg__int__cases, axiom,
    ((![K : int]: ((ord_less_int @ K @ zero_zero_int) => (~ ((![N2 : nat]: ((K = (uminus_uminus_int @ (semiri2019852685at_int @ N2))) => (~ ((ord_less_nat @ zero_zero_nat @ N2))))))))))). % neg_int_cases
thf(fact_122_zero__less__nat__eq, axiom,
    ((![Z : int]: ((ord_less_nat @ zero_zero_nat @ (nat2 @ Z)) = (ord_less_int @ zero_zero_int @ Z))))). % zero_less_nat_eq
thf(fact_123_add_Oinverse__inverse, axiom,
    ((![A : int]: ((uminus_uminus_int @ (uminus_uminus_int @ A)) = A)))). % add.inverse_inverse
thf(fact_124_neg__equal__iff__equal, axiom,
    ((![A : int, B3 : int]: (((uminus_uminus_int @ A) = (uminus_uminus_int @ B3)) = (A = B3))))). % neg_equal_iff_equal
thf(fact_125_neg__equal__zero, axiom,
    ((![A : int]: (((uminus_uminus_int @ A) = A) = (A = zero_zero_int))))). % neg_equal_zero
thf(fact_126_equal__neg__zero, axiom,
    ((![A : int]: ((A = (uminus_uminus_int @ A)) = (A = zero_zero_int))))). % equal_neg_zero
thf(fact_127_neg__equal__0__iff__equal, axiom,
    ((![A : int]: (((uminus_uminus_int @ A) = zero_zero_int) = (A = zero_zero_int))))). % neg_equal_0_iff_equal
thf(fact_128_neg__0__equal__iff__equal, axiom,
    ((![A : int]: ((zero_zero_int = (uminus_uminus_int @ A)) = (zero_zero_int = A))))). % neg_0_equal_iff_equal
thf(fact_129_add_Oinverse__neutral, axiom,
    (((uminus_uminus_int @ zero_zero_int) = zero_zero_int))). % add.inverse_neutral
thf(fact_130_neg__less__iff__less, axiom,
    ((![B3 : int, A : int]: ((ord_less_int @ (uminus_uminus_int @ B3) @ (uminus_uminus_int @ A)) = (ord_less_int @ A @ B3))))). % neg_less_iff_less
thf(fact_131_neg__less__0__iff__less, axiom,
    ((![A : int]: ((ord_less_int @ (uminus_uminus_int @ A) @ zero_zero_int) = (ord_less_int @ zero_zero_int @ A))))). % neg_less_0_iff_less
thf(fact_132_neg__0__less__iff__less, axiom,
    ((![A : int]: ((ord_less_int @ zero_zero_int @ (uminus_uminus_int @ A)) = (ord_less_int @ A @ zero_zero_int))))). % neg_0_less_iff_less
thf(fact_133_neg__less__pos, axiom,
    ((![A : int]: ((ord_less_int @ (uminus_uminus_int @ A) @ A) = (ord_less_int @ zero_zero_int @ A))))). % neg_less_pos
thf(fact_134_less__neg__neg, axiom,
    ((![A : int]: ((ord_less_int @ A @ (uminus_uminus_int @ A)) = (ord_less_int @ A @ zero_zero_int))))). % less_neg_neg
thf(fact_135_negative__eq__positive, axiom,
    ((![N : nat, M2 : nat]: (((uminus_uminus_int @ (semiri2019852685at_int @ N)) = (semiri2019852685at_int @ M2)) = (((N = zero_zero_nat)) & ((M2 = zero_zero_nat))))))). % negative_eq_positive
thf(fact_136_zless__nat__conj, axiom,
    ((![W : int, Z : int]: ((ord_less_nat @ (nat2 @ W) @ (nat2 @ Z)) = (((ord_less_int @ zero_zero_int @ Z)) & ((ord_less_int @ W @ Z))))))). % zless_nat_conj
thf(fact_137_nat__zminus__int, axiom,
    ((![N : nat]: ((nat2 @ (uminus_uminus_int @ (semiri2019852685at_int @ N))) = zero_zero_nat)))). % nat_zminus_int
thf(fact_138_verit__negate__coefficient_I2_J, axiom,
    ((![A : int, B3 : int]: ((ord_less_int @ A @ B3) => (ord_less_int @ (uminus_uminus_int @ B3) @ (uminus_uminus_int @ A)))))). % verit_negate_coefficient(2)
thf(fact_139_nat__zero__as__int, axiom,
    ((zero_zero_nat = (nat2 @ zero_zero_int)))). % nat_zero_as_int
thf(fact_140_equation__minus__iff, axiom,
    ((![A : int, B3 : int]: ((A = (uminus_uminus_int @ B3)) = (B3 = (uminus_uminus_int @ A)))))). % equation_minus_iff
thf(fact_141_minus__equation__iff, axiom,
    ((![A : int, B3 : int]: (((uminus_uminus_int @ A) = B3) = ((uminus_uminus_int @ B3) = A))))). % minus_equation_iff
thf(fact_142_minus__less__iff, axiom,
    ((![A : int, B3 : int]: ((ord_less_int @ (uminus_uminus_int @ A) @ B3) = (ord_less_int @ (uminus_uminus_int @ B3) @ A))))). % minus_less_iff
thf(fact_143_less__minus__iff, axiom,
    ((![A : int, B3 : int]: ((ord_less_int @ A @ (uminus_uminus_int @ B3)) = (ord_less_int @ B3 @ (uminus_uminus_int @ A)))))). % less_minus_iff
thf(fact_144_nat__mono__iff, axiom,
    ((![Z : int, W : int]: ((ord_less_int @ zero_zero_int @ Z) => ((ord_less_nat @ (nat2 @ W) @ (nat2 @ Z)) = (ord_less_int @ W @ Z)))))). % nat_mono_iff
thf(fact_145_zless__nat__eq__int__zless, axiom,
    ((![M2 : nat, Z : int]: ((ord_less_nat @ M2 @ (nat2 @ Z)) = (ord_less_int @ (semiri2019852685at_int @ M2) @ Z))))). % zless_nat_eq_int_zless
thf(fact_146_int__cases4, axiom,
    ((![M2 : int]: ((![N2 : nat]: (~ ((M2 = (semiri2019852685at_int @ N2))))) => (~ ((![N2 : nat]: ((ord_less_nat @ zero_zero_nat @ N2) => (~ ((M2 = (uminus_uminus_int @ (semiri2019852685at_int @ N2))))))))))))). % int_cases4
thf(fact_147_split__nat, axiom,
    ((![P : nat > $o, I : int]: ((P @ (nat2 @ I)) = (((![N3 : nat]: (((I = (semiri2019852685at_int @ N3))) => ((P @ N3))))) & ((((ord_less_int @ I @ zero_zero_int)) => ((P @ zero_zero_nat))))))))). % split_nat
thf(fact_148_int__cases3, axiom,
    ((![K : int]: ((~ ((K = zero_zero_int))) => ((![N2 : nat]: ((K = (semiri2019852685at_int @ N2)) => (~ ((ord_less_nat @ zero_zero_nat @ N2))))) => (~ ((![N2 : nat]: ((K = (uminus_uminus_int @ (semiri2019852685at_int @ N2))) => (~ ((ord_less_nat @ zero_zero_nat @ N2)))))))))))). % int_cases3
thf(fact_149_nat__less__iff, axiom,
    ((![W : int, M2 : nat]: ((ord_less_eq_int @ zero_zero_int @ W) => ((ord_less_nat @ (nat2 @ W) @ M2) = (ord_less_int @ W @ (semiri2019852685at_int @ M2))))))). % nat_less_iff

% Conjectures (4)
thf(conj_0, hypothesis,
    ((typing @ (shift_type @ e @ i @ t2) @ t3 @ t))).
thf(conj_1, hypothesis,
    ((it @ u))).
thf(conj_2, hypothesis,
    ((typing @ e @ u @ t2))).
thf(conj_3, conjecture,
    ((it @ (subst @ t3 @ u @ i)))).
