% 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_193__5211318_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:38:14.982

% Could-be-implicit typings (5)
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).

% Explicit typings (32)
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_InductTermi_OIT, type,
    it : dB > $o).
thf(sy_c_LambdaType_Oshift_001t__LambdaType__Otype, type,
    shift_type : (nat > type) > nat > type > nat > type).
thf(sy_c_LambdaType_Otype_OFun, type,
    fun : type > type > type).
thf(sy_c_LambdaType_Otyping, type,
    typing : (nat > type) > dB > type > $o).
thf(sy_c_LambdaType_Otypings, type,
    typings : (nat > type) > list_dB > list_type > $o).
thf(sy_c_Lambda_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_Olistsp_001t__Lambda__OdB, type,
    listsp_dB : (dB > $o) > list_dB > $o).
thf(sy_c_Nat_OSuc, type,
    suc : nat > nat).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat, type,
    ord_less_nat : nat > nat > $o).
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_e1____, type,
    e1 : nat > type).
thf(sy_v_e____, type,
    e : nat > type).
thf(sy_v_i1____, type,
    i1 : nat).
thf(sy_v_i____, type,
    i : nat).
thf(sy_v_n____, type,
    n : nat).
thf(sy_v_rs____, type,
    rs : list_dB).
thf(sy_v_t____, type,
    t3 : dB).
thf(sy_v_thesis____, type,
    thesis : $o).
thf(sy_v_u1____, type,
    u1 : dB).
thf(sy_v_u____, type,
    u : dB).

% Relevant facts (141)
thf(fact_0_False, axiom,
    ((~ ((n = i))))). % False
thf(fact_1_uT, axiom,
    ((typing @ e @ u @ t2))). % uT
thf(fact_2_nT, axiom,
    ((typing @ (shift_type @ e @ i @ t2) @ (foldl_dB_dB @ app @ (var @ n) @ rs) @ t))). % nT
thf(fact_3__092_060open_062IT_At_092_060close_062, axiom,
    ((it @ t3))). % \<open>IT t\<close>
thf(fact_4_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_5_Var_Oprems_I2_J, axiom,
    ((it @ u1))). % Var.prems(2)
thf(fact_6_uIT, axiom,
    ((it @ u))). % uIT
thf(fact_7_Var_Oprems_I1_J, axiom,
    ((typing @ (shift_type @ e1 @ i1 @ t2) @ (foldl_dB_dB @ app @ (var @ n) @ rs) @ t_1))). % Var.prems(1)
thf(fact_8_MI1, axiom,
    ((![T1 : type, T2 : type, T3 : dB, E : nat > type, I : nat, T : type, U : dB]: ((t2 = (fun @ T1 @ T2)) => ((it @ T3) => ((typing @ (shift_type @ E @ I @ T1) @ T3 @ T) => ((it @ U) => ((typing @ E @ U @ T1) => (it @ (subst @ T3 @ U @ I)))))))))). % MI1
thf(fact_9_MI2, axiom,
    ((![T1 : type, T2 : type, T3 : dB, E : nat > type, I : nat, T : type, U : dB]: ((t2 = (fun @ T1 @ T2)) => ((it @ T3) => ((typing @ (shift_type @ E @ I @ T2) @ T3 @ T) => ((it @ U) => ((typing @ E @ U @ T2) => (it @ (subst @ T3 @ U @ I)))))))))). % MI2
thf(fact_10_Var_Oprems_I3_J, axiom,
    ((typing @ e1 @ u1 @ t2))). % Var.prems(3)
thf(fact_11_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_12_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_13_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_14_shift__commute, axiom,
    ((![E : nat > type, I : nat, U2 : type, T : type]: ((shift_type @ (shift_type @ E @ I @ U2) @ zero_zero_nat @ T) = (shift_type @ (shift_type @ E @ zero_zero_nat @ T) @ (suc @ I) @ U2))))). % shift_commute
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_lift__IT, axiom,
    ((![T3 : dB, I : nat]: ((it @ T3) => (it @ (lift @ T3 @ I)))))). % lift_IT
thf(fact_17_typing__elims_I2_J, axiom,
    ((![E : nat > type, T3 : dB, U : dB, T : type]: ((typing @ E @ (app @ T3 @ U) @ T) => (~ ((![T4 : type]: ((typing @ E @ T3 @ (fun @ T4 @ T)) => (~ ((typing @ E @ U @ T4))))))))))). % typing_elims(2)
thf(fact_18_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_19_App, 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)))))). % App
thf(fact_20_typing_OVar, axiom,
    ((![Env : nat > type, X : nat, T : type]: (((Env @ X) = T) => (typing @ Env @ (var @ X) @ T))))). % typing.Var
thf(fact_21_type__induct, axiom,
    ((![P : type > $o, T : type]: ((![T4 : type]: ((![T12 : type, T22 : type]: ((T4 = (fun @ T12 @ T22)) => (P @ T12))) => ((![T12 : type, T22 : type]: ((T4 = (fun @ T12 @ T22)) => (P @ T22))) => (P @ T4)))) => (P @ T))))). % type_induct
thf(fact_22_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_23_Var__IT, axiom,
    ((![N : nat]: (it @ (var @ N))))). % Var_IT
thf(fact_24_app__Var__IT, axiom,
    ((![T3 : dB, I : nat]: ((it @ T3) => (it @ (app @ T3 @ (var @ I))))))). % app_Var_IT
thf(fact_25_subst__Var__IT, axiom,
    ((![R : dB, I : nat, J : nat]: ((it @ R) => (it @ (subst @ R @ (var @ I) @ J)))))). % subst_Var_IT
thf(fact_26_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_27_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_28_less__Suc0, axiom,
    ((![N : nat]: ((ord_less_nat @ N @ (suc @ zero_zero_nat)) = (N = zero_zero_nat))))). % less_Suc0
thf(fact_29_zero__less__Suc, axiom,
    ((![N : nat]: (ord_less_nat @ zero_zero_nat @ (suc @ N))))). % zero_less_Suc
thf(fact_30_subst__lift, axiom,
    ((![T3 : dB, K : nat, S : dB]: ((subst @ (lift @ T3 @ K) @ S @ K) = T3)))). % subst_lift
thf(fact_31_subst__eq, axiom,
    ((![K : nat, U : dB]: ((subst @ (var @ K) @ U @ K) = U)))). % subst_eq
thf(fact_32_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_33_lessI, axiom,
    ((![N : nat]: (ord_less_nat @ N @ (suc @ N))))). % lessI
thf(fact_34_Suc__mono, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (ord_less_nat @ (suc @ M) @ (suc @ N)))))). % Suc_mono
thf(fact_35_Suc__less__eq, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ (suc @ M) @ (suc @ N)) = (ord_less_nat @ M @ N))))). % Suc_less_eq
thf(fact_36_old_Onat_Oinject, axiom,
    ((![Nat : nat, Nat2 : nat]: (((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))). % old.nat.inject
thf(fact_37_nat_Oinject, axiom,
    ((![X2 : nat, Y2 : nat]: (((suc @ X2) = (suc @ Y2)) = (X2 = Y2))))). % nat.inject
thf(fact_38_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_39_dB_Oinject_I1_J, axiom,
    ((![X1 : nat, Y1 : nat]: (((var @ X1) = (var @ Y1)) = (X1 = Y1))))). % dB.inject(1)
thf(fact_40_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_41_less__nat__zero__code, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_42_neq0__conv, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N))))). % neq0_conv
thf(fact_43_not__less__eq, axiom,
    ((![M : nat, N : nat]: ((~ ((ord_less_nat @ M @ N))) = (ord_less_nat @ N @ (suc @ M)))))). % not_less_eq
thf(fact_44_n__not__Suc__n, axiom,
    ((![N : nat]: (~ ((N = (suc @ N))))))). % n_not_Suc_n
thf(fact_45_Suc__inject, axiom,
    ((![X : nat, Y : nat]: (((suc @ X) = (suc @ Y)) => (X = Y))))). % Suc_inject
thf(fact_46_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_47_infinite__descent, axiom,
    ((![P : nat > $o, N : nat]: ((![N2 : nat]: ((~ ((P @ N2))) => (?[M2 : nat]: ((ord_less_nat @ M2 @ N2) & (~ ((P @ M2))))))) => (P @ N))))). % infinite_descent
thf(fact_48_nat__less__induct, axiom,
    ((![P : nat > $o, N : nat]: ((![N2 : nat]: ((![M2 : nat]: ((ord_less_nat @ M2 @ N2) => (P @ M2))) => (P @ N2))) => (P @ N))))). % nat_less_induct
thf(fact_49_less__irrefl__nat, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_irrefl_nat
thf(fact_50_less__not__refl3, axiom,
    ((![S : nat, T3 : nat]: ((ord_less_nat @ S @ T3) => (~ ((S = T3))))))). % less_not_refl3
thf(fact_51_less__not__refl2, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ N @ M) => (~ ((M = N))))))). % less_not_refl2
thf(fact_52_less__not__refl, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_not_refl
thf(fact_53_nat__neq__iff, axiom,
    ((![M : nat, N : nat]: ((~ ((M = N))) = (((ord_less_nat @ M @ N)) | ((ord_less_nat @ N @ M))))))). % nat_neq_iff
thf(fact_54_not0__implies__Suc, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (?[M3 : nat]: (N = (suc @ M3))))))). % not0_implies_Suc
thf(fact_55_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_56_old_Onat_Oexhaust, axiom,
    ((![Y : nat]: ((~ ((Y = zero_zero_nat))) => (~ ((![Nat3 : nat]: (~ ((Y = (suc @ Nat3))))))))))). % old.nat.exhaust
thf(fact_57_Zero__not__Suc, axiom,
    ((![M : nat]: (~ ((zero_zero_nat = (suc @ M))))))). % Zero_not_Suc
thf(fact_58_Zero__neq__Suc, axiom,
    ((![M : nat]: (~ ((zero_zero_nat = (suc @ M))))))). % Zero_neq_Suc
thf(fact_59_Suc__neq__Zero, axiom,
    ((![M : nat]: (~ (((suc @ M) = zero_zero_nat)))))). % Suc_neq_Zero
thf(fact_60_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_61_diff__induct, axiom,
    ((![P : nat > nat > $o, M : nat, N : nat]: ((![X3 : nat]: (P @ X3 @ zero_zero_nat)) => ((![Y3 : nat]: (P @ zero_zero_nat @ (suc @ Y3))) => ((![X3 : nat, Y3 : nat]: ((P @ X3 @ Y3) => (P @ (suc @ X3) @ (suc @ Y3)))) => (P @ M @ N))))))). % diff_induct
thf(fact_62_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_63_nat_OdiscI, axiom,
    ((![Nat : nat, X2 : nat]: ((Nat = (suc @ X2)) => (~ ((Nat = zero_zero_nat))))))). % nat.discI
thf(fact_64_old_Onat_Odistinct_I1_J, axiom,
    ((![Nat2 : nat]: (~ ((zero_zero_nat = (suc @ Nat2))))))). % old.nat.distinct(1)
thf(fact_65_old_Onat_Odistinct_I2_J, axiom,
    ((![Nat2 : nat]: (~ (((suc @ Nat2) = zero_zero_nat)))))). % old.nat.distinct(2)
thf(fact_66_nat_Odistinct_I1_J, axiom,
    ((![X2 : nat]: (~ ((zero_zero_nat = (suc @ X2))))))). % nat.distinct(1)
thf(fact_67_bot__nat__0_Oextremum__strict, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ zero_zero_nat)))))). % bot_nat_0.extremum_strict
thf(fact_68_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_69_gr__implies__not0, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not0
thf(fact_70_less__zeroE, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_zeroE
thf(fact_71_not__less0, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less0
thf(fact_72_not__gr0, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr0
thf(fact_73_gr0I, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr0I
thf(fact_74_not__less__less__Suc__eq, axiom,
    ((![N : nat, M : nat]: ((~ ((ord_less_nat @ N @ M))) => ((ord_less_nat @ N @ (suc @ M)) = (N = M)))))). % not_less_less_Suc_eq
thf(fact_75_strict__inc__induct, axiom,
    ((![I : nat, J : nat, P : nat > $o]: ((ord_less_nat @ I @ J) => ((![I2 : nat]: ((J = (suc @ I2)) => (P @ I2))) => ((![I2 : nat]: ((ord_less_nat @ I2 @ J) => ((P @ (suc @ I2)) => (P @ I2)))) => (P @ I))))))). % strict_inc_induct
thf(fact_76_less__Suc__induct, axiom,
    ((![I : nat, J : nat, P : nat > nat > $o]: ((ord_less_nat @ I @ J) => ((![I2 : nat]: (P @ I2 @ (suc @ I2))) => ((![I2 : nat, J2 : nat, K2 : nat]: ((ord_less_nat @ I2 @ J2) => ((ord_less_nat @ J2 @ K2) => ((P @ I2 @ J2) => ((P @ J2 @ K2) => (P @ I2 @ K2)))))) => (P @ I @ J))))))). % less_Suc_induct
thf(fact_77_less__trans__Suc, axiom,
    ((![I : nat, J : nat, K : nat]: ((ord_less_nat @ I @ J) => ((ord_less_nat @ J @ K) => (ord_less_nat @ (suc @ I) @ K)))))). % less_trans_Suc
thf(fact_78_Suc__less__SucD, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ (suc @ M) @ (suc @ N)) => (ord_less_nat @ M @ N))))). % Suc_less_SucD
thf(fact_79_less__antisym, axiom,
    ((![N : nat, M : nat]: ((~ ((ord_less_nat @ N @ M))) => ((ord_less_nat @ N @ (suc @ M)) => (M = N)))))). % less_antisym
thf(fact_80_Suc__less__eq2, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ (suc @ N) @ M) = (?[M4 : nat]: (((M = (suc @ M4))) & ((ord_less_nat @ N @ M4)))))))). % Suc_less_eq2
thf(fact_81_All__less__Suc, axiom,
    ((![N : nat, P : nat > $o]: ((![I3 : nat]: (((ord_less_nat @ I3 @ (suc @ N))) => ((P @ I3)))) = (((P @ N)) & ((![I3 : nat]: (((ord_less_nat @ I3 @ N)) => ((P @ I3)))))))))). % All_less_Suc
thf(fact_82_less__Suc__eq, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ (suc @ N)) = (((ord_less_nat @ M @ N)) | ((M = N))))))). % less_Suc_eq
thf(fact_83_Ex__less__Suc, axiom,
    ((![N : nat, P : nat > $o]: ((?[I3 : nat]: (((ord_less_nat @ I3 @ (suc @ N))) & ((P @ I3)))) = (((P @ N)) | ((?[I3 : nat]: (((ord_less_nat @ I3 @ N)) & ((P @ I3)))))))))). % Ex_less_Suc
thf(fact_84_less__SucI, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (ord_less_nat @ M @ (suc @ N)))))). % less_SucI
thf(fact_85_less__SucE, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ (suc @ N)) => ((~ ((ord_less_nat @ M @ N))) => (M = N)))))). % less_SucE
thf(fact_86_Suc__lessI, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => ((~ (((suc @ M) = N))) => (ord_less_nat @ (suc @ M) @ N)))))). % Suc_lessI
thf(fact_87_Suc__lessE, axiom,
    ((![I : nat, K : nat]: ((ord_less_nat @ (suc @ I) @ K) => (~ ((![J2 : nat]: ((ord_less_nat @ I @ J2) => (~ ((K = (suc @ J2)))))))))))). % Suc_lessE
thf(fact_88_Suc__lessD, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ (suc @ M) @ N) => (ord_less_nat @ M @ N))))). % Suc_lessD
thf(fact_89_Nat_OlessE, axiom,
    ((![I : nat, K : nat]: ((ord_less_nat @ I @ K) => ((~ ((K = (suc @ I)))) => (~ ((![J2 : nat]: ((ord_less_nat @ I @ J2) => (~ ((K = (suc @ J2))))))))))))). % Nat.lessE
thf(fact_90_dB_Odistinct_I1_J, axiom,
    ((![X1 : nat, X21 : dB, X22 : dB]: (~ (((var @ X1) = (app @ X21 @ X22))))))). % dB.distinct(1)
thf(fact_91_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_92_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_93_lift__Suc__mono__less__iff, axiom,
    ((![F : nat > nat, N : nat, M : nat]: ((![N2 : nat]: (ord_less_nat @ (F @ N2) @ (F @ (suc @ N2)))) => ((ord_less_nat @ (F @ N) @ (F @ M)) = (ord_less_nat @ N @ M)))))). % lift_Suc_mono_less_iff
thf(fact_94_lift__Suc__mono__less, axiom,
    ((![F : nat > nat, N : nat, N3 : nat]: ((![N2 : nat]: (ord_less_nat @ (F @ N2) @ (F @ (suc @ N2)))) => ((ord_less_nat @ N @ N3) => (ord_less_nat @ (F @ N) @ (F @ N3))))))). % lift_Suc_mono_less
thf(fact_95_less__Suc__eq__0__disj, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ (suc @ N)) = (((M = zero_zero_nat)) | ((?[J3 : nat]: (((M = (suc @ J3))) & ((ord_less_nat @ J3 @ N)))))))))). % less_Suc_eq_0_disj
thf(fact_96_gr0__implies__Suc, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) => (?[M3 : nat]: (N = (suc @ M3))))))). % gr0_implies_Suc
thf(fact_97_All__less__Suc2, axiom,
    ((![N : nat, P : nat > $o]: ((![I3 : nat]: (((ord_less_nat @ I3 @ (suc @ N))) => ((P @ I3)))) = (((P @ zero_zero_nat)) & ((![I3 : nat]: (((ord_less_nat @ I3 @ N)) => ((P @ (suc @ I3))))))))))). % All_less_Suc2
thf(fact_98_gr0__conv__Suc, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) = (?[M5 : nat]: (N = (suc @ M5))))))). % gr0_conv_Suc
thf(fact_99_Ex__less__Suc2, axiom,
    ((![N : nat, P : nat > $o]: ((?[I3 : nat]: (((ord_less_nat @ I3 @ (suc @ N))) & ((P @ I3)))) = (((P @ zero_zero_nat)) | ((?[I3 : nat]: (((ord_less_nat @ I3 @ N)) & ((P @ (suc @ I3))))))))))). % Ex_less_Suc2
thf(fact_100_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_101_var__app__typesE, axiom,
    ((![E : nat > type, I : nat, Ts : list_dB, T : type]: ((typing @ E @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ T) => (~ ((![Ts2 : list_type]: ((typing @ E @ (var @ I) @ (foldr_type_type @ fun @ Ts2 @ T)) => (~ ((typings @ E @ Ts @ Ts2))))))))))). % var_app_typesE
thf(fact_102_var__app__types, axiom,
    ((![E : nat > type, I : nat, Ts : list_dB, Us : list_dB, T : type, Ts3 : list_type, U2 : type]: ((typing @ E @ (foldl_dB_dB @ app @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ Us) @ 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 @ Us @ Us2))))))))). % var_app_types
thf(fact_103_typing_Oinducts, axiom,
    ((![X1 : nat > type, X2 : dB, X32 : type, P : (nat > type) > dB > type > $o]: ((typing @ X1 @ X2 @ X32) => ((![Env2 : nat > type, X3 : nat, T4 : type]: (((Env2 @ X3) = T4) => (P @ Env2 @ (var @ X3) @ T4))) => ((![Env2 : nat > type, T4 : type, T5 : dB, U3 : type]: ((typing @ (shift_type @ Env2 @ zero_zero_nat @ T4) @ T5 @ U3) => ((P @ (shift_type @ Env2 @ zero_zero_nat @ T4) @ T5 @ U3) => (P @ Env2 @ (abs @ T5) @ (fun @ T4 @ U3))))) => ((![Env2 : nat > type, S2 : dB, T4 : type, U3 : type, T5 : dB]: ((typing @ Env2 @ S2 @ (fun @ T4 @ U3)) => ((P @ Env2 @ S2 @ (fun @ T4 @ U3)) => ((typing @ Env2 @ T5 @ T4) => ((P @ Env2 @ T5 @ T4) => (P @ Env2 @ (app @ S2 @ T5) @ U3)))))) => (P @ X1 @ X2 @ X32)))))))). % typing.inducts
thf(fact_104_typing_Osimps, axiom,
    ((typing = (^[A1 : nat > type]: (^[A2 : dB]: (^[A3 : type]: (((?[Env3 : nat > type]: (?[X4 : nat]: (?[T6 : type]: (((A1 = Env3)) & ((((A2 = (var @ X4))) & ((((A3 = T6)) & (((Env3 @ X4) = T6))))))))))) | ((((?[Env3 : nat > type]: (?[T6 : type]: (?[T7 : dB]: (?[U4 : type]: (((A1 = Env3)) & ((((A2 = (abs @ T7))) & ((((A3 = (fun @ T6 @ U4))) & ((typing @ (shift_type @ Env3 @ zero_zero_nat @ T6) @ T7 @ U4)))))))))))) | ((?[Env3 : nat > type]: (?[S3 : dB]: (?[T6 : type]: (?[U4 : type]: (?[T7 : dB]: (((A1 = Env3)) & ((((A2 = (app @ S3 @ T7))) & ((((A3 = U4)) & ((((typing @ Env3 @ S3 @ (fun @ T6 @ U4))) & ((typing @ Env3 @ T7 @ T6)))))))))))))))))))))))). % typing.simps
thf(fact_105_typing_Ocases, axiom,
    ((![A12 : nat > type, A22 : dB, A32 : type]: ((typing @ A12 @ A22 @ A32) => ((![X3 : nat]: ((A22 = (var @ X3)) => (~ (((A12 @ X3) = A32))))) => ((![T4 : type, T5 : dB]: ((A22 = (abs @ T5)) => (![U3 : type]: ((A32 = (fun @ T4 @ U3)) => (~ ((typing @ (shift_type @ A12 @ zero_zero_nat @ T4) @ T5 @ U3))))))) => (~ ((![S2 : dB, T4 : type, U3 : type, T5 : dB]: ((A22 = (app @ S2 @ T5)) => ((A32 = U3) => ((typing @ A12 @ S2 @ (fun @ T4 @ U3)) => (~ ((typing @ A12 @ T5 @ T4))))))))))))))). % typing.cases
thf(fact_106_dB_Oinject_I3_J, axiom,
    ((![X32 : dB, Y32 : dB]: (((abs @ X32) = (abs @ Y32)) = (X32 = Y32))))). % dB.inject(3)
thf(fact_107_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_108_dB_Odistinct_I5_J, axiom,
    ((![X21 : dB, X22 : dB, X32 : dB]: (~ (((app @ X21 @ X22) = (abs @ X32))))))). % dB.distinct(5)
thf(fact_109_dB_Odistinct_I3_J, axiom,
    ((![X1 : nat, X32 : dB]: (~ (((var @ X1) = (abs @ X32))))))). % dB.distinct(3)
thf(fact_110_dB_Oinduct, axiom,
    ((![P : dB > $o, DB : dB]: ((![X3 : nat]: (P @ (var @ X3))) => ((![X1a : dB, X23 : dB]: ((P @ X1a) => ((P @ X23) => (P @ (app @ X1a @ X23))))) => ((![X3 : dB]: ((P @ X3) => (P @ (abs @ X3)))) => (P @ DB))))))). % dB.induct
thf(fact_111_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_112_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_113_ex__head__tail, axiom,
    ((![T3 : dB]: (?[Ts4 : list_dB, H : dB]: ((T3 = (foldl_dB_dB @ app @ H @ Ts4)) & ((?[N2 : nat]: (H = (var @ N2))) | (?[U5 : dB]: (H = (abs @ U5))))))))). % ex_head_tail
thf(fact_114_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_115_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_116_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_117_typing__elims_I3_J, axiom,
    ((![E : nat > type, T3 : dB, T : type]: ((typing @ E @ (abs @ T3) @ T) => (~ ((![T4 : type, U3 : type]: ((T = (fun @ T4 @ U3)) => (~ ((typing @ (shift_type @ E @ zero_zero_nat @ T4) @ T3 @ U3))))))))))). % typing_elims(3)
thf(fact_118_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_119_gr__zeroI, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr_zeroI
thf(fact_120_not__less__zero, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less_zero
thf(fact_121_gr__implies__not__zero, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not_zero
thf(fact_122_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_123_list__app__typeD, axiom,
    ((![E : nat > type, T3 : dB, Ts : list_dB, T : type]: ((typing @ E @ (foldl_dB_dB @ app @ T3 @ Ts) @ T) => (?[Ts2 : list_type]: ((typing @ E @ T3 @ (foldr_type_type @ fun @ Ts2 @ T)) & (typings @ E @ Ts @ Ts2))))))). % 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) => (~ ((![Ts2 : list_type]: ((typing @ E @ T3 @ (foldr_type_type @ fun @ Ts2 @ T)) => (~ ((typings @ E @ Ts @ Ts2))))))))))). % 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_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_127_Lambda, axiom,
    ((![R : dB]: ((it @ R) => (it @ (abs @ R)))))). % Lambda
thf(fact_128_IT_Oinducts, axiom,
    ((![X : dB, P : dB > $o]: ((it @ X) => ((![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 @ X)))))))). % IT.inducts
thf(fact_129_IT_Osimps, axiom,
    ((it = (^[A4 : dB]: (((?[Rs3 : list_dB]: (?[N4 : nat]: (((A4 = (foldl_dB_dB @ app @ (var @ N4) @ 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_130_IT_OVar, axiom,
    ((![Rs : list_dB, N : nat]: ((listsp_dB @ it @ Rs) => (it @ (foldl_dB_dB @ app @ (var @ N) @ Rs)))))). % IT.Var
thf(fact_131_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_132_listsp__conj__eq, axiom,
    ((![A5 : dB > $o, B : dB > $o]: ((listsp_dB @ (^[X4 : dB]: (((A5 @ X4)) & ((B @ X4))))) = (^[X4 : list_dB]: (((listsp_dB @ A5 @ X4)) & ((listsp_dB @ B @ X4)))))))). % listsp_conj_eq
thf(fact_133_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_nat @ zero_zero_nat @ zero_zero_nat))))). % less_numeral_extra(3)
thf(fact_134_liftn__lift, axiom,
    ((![N : nat, T3 : dB, K : nat]: ((liftn @ (suc @ N) @ T3 @ K) = (lift @ (liftn @ N @ T3 @ K) @ K))))). % liftn_lift
thf(fact_135_liftn__0, axiom,
    ((![T3 : dB, K : nat]: ((liftn @ zero_zero_nat @ T3 @ K) = T3)))). % liftn_0
thf(fact_136_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_137_substn__subst__n, axiom,
    ((substn = (^[T7 : dB]: (^[S3 : dB]: (^[N4 : nat]: (subst @ T7 @ (liftn @ N4 @ S3 @ zero_zero_nat) @ N4))))))). % substn_subst_n
thf(fact_138_dB_Osize__gen_I1_J, axiom,
    ((![X1 : nat]: ((size_dB @ (var @ X1)) = zero_zero_nat)))). % dB.size_gen(1)
thf(fact_139_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_140_substn__subst__0, axiom,
    ((![T3 : dB, S : dB]: ((substn @ T3 @ S @ zero_zero_nat) = (subst @ T3 @ S @ zero_zero_nat))))). % substn_subst_0

% Conjectures (2)
thf(conj_0, hypothesis,
    ((![Ts5 : list_type]: ((typings @ (shift_type @ e @ i @ t2) @ rs @ Ts5) => thesis)))).
thf(conj_1, conjecture,
    (thesis)).
