% 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_191__5211284_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:38:13.365

% 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 (37)
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_If_001t__Nat__Onat, type,
    if_nat : $o > nat > 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_Osubst, type,
    subst : 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_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_u1____, type,
    u1 : dB).
thf(sy_v_u____, type,
    u : dB).

% Relevant facts (152)
thf(fact_0_Var_Oprems_I2_J, axiom,
    ((it @ u1))). % Var.prems(2)
thf(fact_1__092_060open_062IT_At_092_060close_062, axiom,
    ((it @ t3))). % \<open>IT t\<close>
thf(fact_2_uIT, axiom,
    ((it @ u))). % uIT
thf(fact_3_Var_Oprems_I3_J, axiom,
    ((typing @ e1 @ u1 @ t2))). % Var.prems(3)
thf(fact_4_uT, axiom,
    ((typing @ e @ u @ t2))). % uT
thf(fact_5_Var_Ohyps, axiom,
    ((listsp_dB @ (^[X : dB]: (((it @ X)) & ((![Y : nat > type]: (![Z : type]: (![Aa : dB]: (![Ab : nat]: (((typing @ (shift_type @ Y @ Ab @ t2) @ X @ Z)) => ((((it @ Aa)) => ((((typing @ Y @ Aa @ t2)) => ((it @ (subst @ X @ Aa @ Ab))))))))))))))) @ rs))). % Var.hyps
thf(fact_6_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_7_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_8_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_9_listsp__conj__eq, axiom,
    ((![A : dB > $o, B : dB > $o]: ((listsp_dB @ (^[X : dB]: (((A @ X)) & ((B @ X))))) = (^[X : list_dB]: (((listsp_dB @ A @ X)) & ((listsp_dB @ B @ X)))))))). % listsp_conj_eq
thf(fact_10_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_11_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_12_subst__Var__IT, axiom,
    ((![R : dB, I : nat, J : nat]: ((it @ R) => (it @ (subst @ R @ (var @ I) @ J)))))). % subst_Var_IT
thf(fact_13_nT, axiom,
    ((typing @ (shift_type @ e @ i @ t2) @ (foldl_dB_dB @ app @ (var @ n) @ rs) @ t))). % nT
thf(fact_14_Var__IT, axiom,
    ((![N : nat]: (it @ (var @ N))))). % Var_IT
thf(fact_15_subst__eq, axiom,
    ((![K : nat, U : dB]: ((subst @ (var @ K) @ U @ K) = U)))). % subst_eq
thf(fact_16_app__Var__IT, axiom,
    ((![T : dB, I : nat]: ((it @ T) => (it @ (app @ T @ (var @ I))))))). % app_Var_IT
thf(fact_17_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_18_False, axiom,
    ((~ ((n = i))))). % False
thf(fact_19_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_20_dB_Oinject_I1_J, axiom,
    ((![X1 : nat, Y1 : nat]: (((var @ X1) = (var @ Y1)) = (X1 = Y1))))). % dB.inject(1)
thf(fact_21_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_22_subst__lift, axiom,
    ((![T : dB, K : nat, S : dB]: ((subst @ (lift @ T @ K) @ S @ K) = T)))). % subst_lift
thf(fact_23_lift__IT, axiom,
    ((![T : dB, I : nat]: ((it @ T) => (it @ (lift @ T @ I)))))). % lift_IT
thf(fact_24_lift_Osimps_I2_J, axiom,
    ((![S : dB, T : dB, K : nat]: ((lift @ (app @ S @ T) @ K) = (app @ (lift @ S @ K) @ (lift @ T @ K)))))). % lift.simps(2)
thf(fact_25_dB_Odistinct_I1_J, axiom,
    ((![X1 : nat, X21 : dB, X22 : dB]: (~ (((var @ X1) = (app @ X21 @ X22))))))). % dB.distinct(1)
thf(fact_26_type__induct, axiom,
    ((![P : type > $o, T3 : 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 @ T3))))). % type_induct
thf(fact_27_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_28_typing__elims_I2_J, axiom,
    ((![E : nat > type, T : dB, U : dB, T3 : type]: ((typing @ E @ (app @ T @ U) @ T3) => (~ ((![T4 : type]: ((typing @ E @ T @ (fun @ T4 @ T3)) => (~ ((typing @ E @ U @ T4))))))))))). % typing_elims(2)
thf(fact_29_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_30_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_31_typing_OVar, axiom,
    ((![Env : nat > type, X2 : nat, T3 : type]: (((Env @ X2) = T3) => (typing @ Env @ (var @ X2) @ T3))))). % typing.Var
thf(fact_32_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_33_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_34_IT_OVar, axiom,
    ((![Rs : list_dB, N : nat]: ((listsp_dB @ it @ Rs) => (it @ (foldl_dB_dB @ app @ (var @ N) @ Rs)))))). % IT.Var
thf(fact_35_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_36_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_37_var__app__typesE, axiom,
    ((![E : nat > type, I : nat, Ts : list_dB, T3 : type]: ((typing @ E @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ T3) => (~ ((![Ts3 : list_type]: ((typing @ E @ (var @ I) @ (foldr_type_type @ fun @ Ts3 @ T3)) => (~ ((typings @ E @ Ts @ Ts3))))))))))). % var_app_typesE
thf(fact_38_IT_Oinducts, axiom,
    ((![X2 : dB, P : dB > $o]: ((it @ X2) => ((![Rs2 : list_dB, N2 : nat]: ((listsp_dB @ (^[X : dB]: (((it @ X)) & ((P @ X)))) @ Rs2) => (P @ (foldl_dB_dB @ app @ (var @ N2) @ Rs2)))) => ((![R2 : dB]: ((it @ R2) => ((P @ R2) => (P @ (abs @ R2))))) => ((![R2 : dB, S2 : dB, Ss2 : list_dB]: ((it @ (foldl_dB_dB @ app @ (subst @ R2 @ S2 @ zero_zero_nat) @ Ss2)) => ((P @ (foldl_dB_dB @ app @ (subst @ R2 @ S2 @ zero_zero_nat) @ Ss2)) => ((it @ S2) => ((P @ S2) => (P @ (foldl_dB_dB @ app @ (app @ (abs @ R2) @ S2) @ Ss2))))))) => (P @ X2)))))))). % IT.inducts
thf(fact_39_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_40_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, T4 : type]: (((Env2 @ X4) = T4) => (P @ Env2 @ (var @ X4) @ 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 @ X23 @ X3)))))))). % typing.inducts
thf(fact_41_typing_Osimps, axiom,
    ((typing = (^[A1 : nat > type]: (^[A2 : dB]: (^[A3 : type]: (((?[Env3 : nat > type]: (?[X : nat]: (?[T6 : type]: (((A1 = Env3)) & ((((A2 = (var @ X))) & ((((A3 = T6)) & (((Env3 @ X) = 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_42_dB_Oinject_I3_J, axiom,
    ((![X3 : dB, Y3 : dB]: (((abs @ X3) = (abs @ Y3)) = (X3 = Y3))))). % dB.inject(3)
thf(fact_43_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_44_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_45_Lambda, axiom,
    ((![R : dB]: ((it @ R) => (it @ (abs @ R)))))). % Lambda
thf(fact_46_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_47_dB_Odistinct_I5_J, axiom,
    ((![X21 : dB, X22 : dB, X3 : dB]: (~ (((app @ X21 @ X22) = (abs @ X3))))))). % dB.distinct(5)
thf(fact_48_dB_Odistinct_I3_J, axiom,
    ((![X1 : nat, X3 : dB]: (~ (((var @ X1) = (abs @ X3))))))). % dB.distinct(3)
thf(fact_49_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_50_typing__elims_I3_J, axiom,
    ((![E : nat > type, T : dB, T3 : type]: ((typing @ E @ (abs @ T) @ T3) => (~ ((![T4 : type, U3 : type]: ((T3 = (fun @ T4 @ U3)) => (~ ((typing @ (shift_type @ E @ zero_zero_nat @ T4) @ T @ U3))))))))))). % typing_elims(3)
thf(fact_51_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_52_dB_Oexhaust, axiom,
    ((![Y2 : dB]: ((![X12 : nat]: (~ ((Y2 = (var @ X12))))) => ((![X212 : dB, X222 : dB]: (~ ((Y2 = (app @ X212 @ X222))))) => (~ ((![X32 : dB]: (~ ((Y2 = (abs @ X32)))))))))))). % dB.exhaust
thf(fact_53_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_54_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_55_Abs__App__neq__Var__apps, axiom,
    ((![S : dB, T : dB, N : nat, Ss : list_dB]: (~ (((app @ (abs @ S) @ T) = (foldl_dB_dB @ app @ (var @ N) @ Ss))))))). % Abs_App_neq_Var_apps
thf(fact_56_ex__head__tail, axiom,
    ((![T : dB]: (?[Ts4 : list_dB, H : dB]: ((T = (foldl_dB_dB @ app @ H @ Ts4)) & ((?[N2 : nat]: (H = (var @ N2))) | (?[U5 : dB]: (H = (abs @ U5))))))))). % ex_head_tail
thf(fact_57_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_58_list__app__typeE, axiom,
    ((![E : nat > type, T : dB, Ts : list_dB, T3 : type]: ((typing @ E @ (foldl_dB_dB @ app @ T @ Ts) @ T3) => (~ ((![Ts3 : list_type]: ((typing @ E @ T @ (foldr_type_type @ fun @ Ts3 @ T3)) => (~ ((typings @ E @ Ts @ Ts3))))))))))). % list_app_typeE
thf(fact_59_list__app__typeD, axiom,
    ((![E : nat > type, T : dB, Ts : list_dB, T3 : type]: ((typing @ E @ (foldl_dB_dB @ app @ T @ Ts) @ T3) => (?[Ts3 : list_type]: ((typing @ E @ T @ (foldr_type_type @ fun @ Ts3 @ T3)) & (typings @ E @ Ts @ Ts3))))))). % list_app_typeD
thf(fact_60_IT_Ocases, axiom,
    ((![A4 : dB]: ((it @ A4) => ((![Rs2 : list_dB]: ((?[N2 : nat]: (A4 = (foldl_dB_dB @ app @ (var @ N2) @ Rs2))) => (~ ((listsp_dB @ it @ Rs2))))) => ((![R2 : dB]: ((A4 = (abs @ R2)) => (~ ((it @ R2))))) => (~ ((![R2 : dB, S2 : dB, Ss2 : list_dB]: ((A4 = (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_61_IT_Osimps, axiom,
    ((it = (^[A5 : dB]: (((?[Rs3 : list_dB]: (?[N3 : nat]: (((A5 = (foldl_dB_dB @ app @ (var @ N3) @ Rs3))) & ((listsp_dB @ it @ Rs3)))))) | ((((?[R3 : dB]: (((A5 = (abs @ R3))) & ((it @ R3))))) | ((?[R3 : dB]: (?[S3 : dB]: (?[Ss3 : list_dB]: (((A5 = (foldl_dB_dB @ app @ (app @ (abs @ R3) @ S3) @ Ss3))) & ((((it @ (foldl_dB_dB @ app @ (subst @ R3 @ S3 @ zero_zero_nat) @ Ss3))) & ((it @ S3)))))))))))))))). % IT.simps
thf(fact_62_typing_Ocases, axiom,
    ((![A12 : nat > type, A22 : dB, A32 : type]: ((typing @ A12 @ A22 @ A32) => ((![X4 : nat]: ((A22 = (var @ X4)) => (~ (((A12 @ X4) = 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_63_bot__nat__0_Onot__eq__extremum, axiom,
    ((![A4 : nat]: ((~ ((A4 = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ A4))))). % bot_nat_0.not_eq_extremum
thf(fact_64_less__nat__zero__code, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_65_neq0__conv, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N))))). % neq0_conv
thf(fact_66_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_67_zero__reorient, axiom,
    ((![X2 : nat]: ((zero_zero_nat = X2) = (X2 = zero_zero_nat))))). % zero_reorient
thf(fact_68_zero__reorient, axiom,
    ((![X2 : int]: ((zero_zero_int = X2) = (X2 = zero_zero_int))))). % zero_reorient
thf(fact_69_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_70_less__not__refl, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_not_refl
thf(fact_71_less__not__refl2, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ N @ M) => (~ ((M = N))))))). % less_not_refl2
thf(fact_72_less__not__refl3, axiom,
    ((![S : nat, T : nat]: ((ord_less_nat @ S @ T) => (~ ((S = T))))))). % less_not_refl3
thf(fact_73_less__irrefl__nat, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_irrefl_nat
thf(fact_74_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_75_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_76_linorder__neqE__nat, axiom,
    ((![X2 : nat, Y2 : nat]: ((~ ((X2 = Y2))) => ((~ ((ord_less_nat @ X2 @ Y2))) => (ord_less_nat @ Y2 @ X2)))))). % linorder_neqE_nat
thf(fact_77_gr__zeroI, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr_zeroI
thf(fact_78_not__less__zero, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less_zero
thf(fact_79_gr__implies__not__zero, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not_zero
thf(fact_80_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_81_gr0I, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr0I
thf(fact_82_not__gr0, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr0
thf(fact_83_not__less0, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less0
thf(fact_84_less__zeroE, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_zeroE
thf(fact_85_gr__implies__not0, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not0
thf(fact_86_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_87_bot__nat__0_Oextremum__strict, axiom,
    ((![A4 : nat]: (~ ((ord_less_nat @ A4 @ zero_zero_nat)))))). % bot_nat_0.extremum_strict
thf(fact_88_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_nat @ zero_zero_nat @ zero_zero_nat))))). % less_numeral_extra(3)
thf(fact_89_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_int @ zero_zero_int @ zero_zero_int))))). % less_numeral_extra(3)
thf(fact_90_dB_Osize__gen_I1_J, axiom,
    ((![X1 : nat]: ((size_dB @ (var @ X1)) = zero_zero_nat)))). % dB.size_gen(1)
thf(fact_91_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 @ (^[T7 : dB]: (subst @ T7 @ U @ I)) @ Ts) @ Ts2)))))). % substs_lemma
thf(fact_92_lifts__IT, axiom,
    ((![Ts : list_dB]: ((listsp_dB @ it @ Ts) => (listsp_dB @ it @ (map_dB_dB @ (^[T7 : dB]: (lift @ T7 @ zero_zero_nat)) @ Ts)))))). % lifts_IT
thf(fact_93_map__ident, axiom,
    (((map_dB_dB @ (^[X : dB]: X)) = (^[Xs : list_dB]: Xs)))). % map_ident
thf(fact_94_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 @ (^[T7 : dB]: (subst @ T7 @ U @ I)) @ Ts)))))). % subst_map
thf(fact_95_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 @ (^[T7 : dB]: (lift @ T7 @ I)) @ Ts)))))). % lift_map
thf(fact_96_list_Omap__ident, axiom,
    ((![T : list_dB]: ((map_dB_dB @ (^[X : dB]: X) @ T) = T)))). % list.map_ident
thf(fact_97_foldl__map, axiom,
    ((![G : dB > dB > dB, A4 : dB, F : dB > dB, Xs2 : list_dB]: ((foldl_dB_dB @ G @ A4 @ (map_dB_dB @ F @ Xs2)) = (foldl_dB_dB @ (^[A5 : dB]: (^[X : dB]: (G @ A5 @ (F @ X)))) @ A4 @ Xs2))))). % foldl_map
thf(fact_98_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 @ (^[T7 : dB]: (lift @ T7 @ I)) @ Ts) @ Ts2))))). % lift_types
thf(fact_99_map__eq__map__tailrec, axiom,
    ((map_dB_dB = map_tailrec_dB_dB))). % map_eq_map_tailrec
thf(fact_100_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_101_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_102_of__nat__eq__iff, axiom,
    ((![M : nat, N : nat]: (((semiri2019852685at_int @ M) = (semiri2019852685at_int @ N)) = (M = N))))). % of_nat_eq_iff
thf(fact_103_of__nat__0, axiom,
    (((semiri1382578993at_nat @ zero_zero_nat) = zero_zero_nat))). % of_nat_0
thf(fact_104_of__nat__0, axiom,
    (((semiri2019852685at_int @ zero_zero_nat) = zero_zero_int))). % of_nat_0
thf(fact_105_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_106_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_107_of__nat__eq__0__iff, axiom,
    ((![M : nat]: (((semiri1382578993at_nat @ M) = zero_zero_nat) = (M = zero_zero_nat))))). % of_nat_eq_0_iff
thf(fact_108_of__nat__eq__0__iff, axiom,
    ((![M : nat]: (((semiri2019852685at_int @ M) = zero_zero_int) = (M = zero_zero_nat))))). % of_nat_eq_0_iff
thf(fact_109_of__nat__less__iff, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ (semiri1382578993at_nat @ M) @ (semiri1382578993at_nat @ N)) = (ord_less_nat @ M @ N))))). % of_nat_less_iff
thf(fact_110_of__nat__less__iff, axiom,
    ((![M : nat, N : nat]: ((ord_less_int @ (semiri2019852685at_int @ M) @ (semiri2019852685at_int @ N)) = (ord_less_nat @ M @ N))))). % of_nat_less_iff
thf(fact_111_of__nat__less__0__iff, axiom,
    ((![M : nat]: (~ ((ord_less_nat @ (semiri1382578993at_nat @ M) @ zero_zero_nat)))))). % of_nat_less_0_iff
thf(fact_112_of__nat__less__0__iff, axiom,
    ((![M : nat]: (~ ((ord_less_int @ (semiri2019852685at_int @ M) @ zero_zero_int)))))). % of_nat_less_0_iff
thf(fact_113_less__imp__of__nat__less, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (ord_less_nat @ (semiri1382578993at_nat @ M) @ (semiri1382578993at_nat @ N)))))). % less_imp_of_nat_less
thf(fact_114_less__imp__of__nat__less, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (ord_less_int @ (semiri2019852685at_int @ M) @ (semiri2019852685at_int @ N)))))). % less_imp_of_nat_less
thf(fact_115_of__nat__less__imp__less, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ (semiri1382578993at_nat @ M) @ (semiri1382578993at_nat @ N)) => (ord_less_nat @ M @ N))))). % of_nat_less_imp_less
thf(fact_116_of__nat__less__imp__less, axiom,
    ((![M : nat, N : nat]: ((ord_less_int @ (semiri2019852685at_int @ M) @ (semiri2019852685at_int @ N)) => (ord_less_nat @ M @ N))))). % of_nat_less_imp_less
thf(fact_117_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_118_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_119_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_120_nat__int__comparison_I1_J, axiom,
    (((^[Y4 : nat]: (^[Z2 : nat]: (Y4 = Z2))) = (^[A5 : nat]: (^[B2 : nat]: ((semiri2019852685at_int @ A5) = (semiri2019852685at_int @ B2))))))). % nat_int_comparison(1)
thf(fact_121_int__if, axiom,
    ((![P : $o, A4 : nat, B3 : nat]: ((P => ((semiri2019852685at_int @ (if_nat @ P @ A4 @ B3)) = (semiri2019852685at_int @ A4))) & ((~ (P)) => ((semiri2019852685at_int @ (if_nat @ P @ A4 @ B3)) = (semiri2019852685at_int @ B3))))))). % int_if
thf(fact_122_int__int__eq, axiom,
    ((![M : nat, N : nat]: (((semiri2019852685at_int @ M) = (semiri2019852685at_int @ N)) = (M = N))))). % int_int_eq
thf(fact_123_verit__comp__simplify1_I1_J, axiom,
    ((![A4 : nat]: (~ ((ord_less_nat @ A4 @ A4)))))). % verit_comp_simplify1(1)
thf(fact_124_verit__comp__simplify1_I1_J, axiom,
    ((![A4 : int]: (~ ((ord_less_int @ A4 @ A4)))))). % verit_comp_simplify1(1)
thf(fact_125_int__ops_I1_J, axiom,
    (((semiri2019852685at_int @ zero_zero_nat) = zero_zero_int))). % int_ops(1)
thf(fact_126_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_127_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_128_zero__less__nat__eq, axiom,
    ((![Z3 : int]: ((ord_less_nat @ zero_zero_nat @ (nat2 @ Z3)) = (ord_less_int @ zero_zero_int @ Z3))))). % zero_less_nat_eq
thf(fact_129_neg__equal__iff__equal, axiom,
    ((![A4 : int, B3 : int]: (((uminus_uminus_int @ A4) = (uminus_uminus_int @ B3)) = (A4 = B3))))). % neg_equal_iff_equal
thf(fact_130_add_Oinverse__inverse, axiom,
    ((![A4 : int]: ((uminus_uminus_int @ (uminus_uminus_int @ A4)) = A4)))). % add.inverse_inverse
thf(fact_131_neg__equal__zero, axiom,
    ((![A4 : int]: (((uminus_uminus_int @ A4) = A4) = (A4 = zero_zero_int))))). % neg_equal_zero
thf(fact_132_equal__neg__zero, axiom,
    ((![A4 : int]: ((A4 = (uminus_uminus_int @ A4)) = (A4 = zero_zero_int))))). % equal_neg_zero
thf(fact_133_neg__equal__0__iff__equal, axiom,
    ((![A4 : int]: (((uminus_uminus_int @ A4) = zero_zero_int) = (A4 = zero_zero_int))))). % neg_equal_0_iff_equal
thf(fact_134_neg__0__equal__iff__equal, axiom,
    ((![A4 : int]: ((zero_zero_int = (uminus_uminus_int @ A4)) = (zero_zero_int = A4))))). % neg_0_equal_iff_equal
thf(fact_135_add_Oinverse__neutral, axiom,
    (((uminus_uminus_int @ zero_zero_int) = zero_zero_int))). % add.inverse_neutral
thf(fact_136_neg__less__iff__less, axiom,
    ((![B3 : int, A4 : int]: ((ord_less_int @ (uminus_uminus_int @ B3) @ (uminus_uminus_int @ A4)) = (ord_less_int @ A4 @ B3))))). % neg_less_iff_less
thf(fact_137_nat__int, axiom,
    ((![N : nat]: ((nat2 @ (semiri2019852685at_int @ N)) = N)))). % nat_int
thf(fact_138_neg__less__0__iff__less, axiom,
    ((![A4 : int]: ((ord_less_int @ (uminus_uminus_int @ A4) @ zero_zero_int) = (ord_less_int @ zero_zero_int @ A4))))). % neg_less_0_iff_less
thf(fact_139_neg__0__less__iff__less, axiom,
    ((![A4 : int]: ((ord_less_int @ zero_zero_int @ (uminus_uminus_int @ A4)) = (ord_less_int @ A4 @ zero_zero_int))))). % neg_0_less_iff_less
thf(fact_140_neg__less__pos, axiom,
    ((![A4 : int]: ((ord_less_int @ (uminus_uminus_int @ A4) @ A4) = (ord_less_int @ zero_zero_int @ A4))))). % neg_less_pos
thf(fact_141_less__neg__neg, axiom,
    ((![A4 : int]: ((ord_less_int @ A4 @ (uminus_uminus_int @ A4)) = (ord_less_int @ A4 @ zero_zero_int))))). % less_neg_neg
thf(fact_142_negative__eq__positive, axiom,
    ((![N : nat, M : nat]: (((uminus_uminus_int @ (semiri2019852685at_int @ N)) = (semiri2019852685at_int @ M)) = (((N = zero_zero_nat)) & ((M = zero_zero_nat))))))). % negative_eq_positive
thf(fact_143_zless__nat__conj, axiom,
    ((![W : int, Z3 : int]: ((ord_less_nat @ (nat2 @ W) @ (nat2 @ Z3)) = (((ord_less_int @ zero_zero_int @ Z3)) & ((ord_less_int @ W @ Z3))))))). % zless_nat_conj
thf(fact_144_nat__zminus__int, axiom,
    ((![N : nat]: ((nat2 @ (uminus_uminus_int @ (semiri2019852685at_int @ N))) = zero_zero_nat)))). % nat_zminus_int
thf(fact_145_int__cases2, axiom,
    ((![Z3 : int]: ((![N2 : nat]: (~ ((Z3 = (semiri2019852685at_int @ N2))))) => (~ ((![N2 : nat]: (~ ((Z3 = (uminus_uminus_int @ (semiri2019852685at_int @ N2)))))))))))). % int_cases2
thf(fact_146_not__int__zless__negative, axiom,
    ((![N : nat, M : nat]: (~ ((ord_less_int @ (semiri2019852685at_int @ N) @ (uminus_uminus_int @ (semiri2019852685at_int @ M)))))))). % not_int_zless_negative
thf(fact_147_minus__less__iff, axiom,
    ((![A4 : int, B3 : int]: ((ord_less_int @ (uminus_uminus_int @ A4) @ B3) = (ord_less_int @ (uminus_uminus_int @ B3) @ A4))))). % minus_less_iff
thf(fact_148_less__minus__iff, axiom,
    ((![A4 : int, B3 : int]: ((ord_less_int @ A4 @ (uminus_uminus_int @ B3)) = (ord_less_int @ B3 @ (uminus_uminus_int @ A4)))))). % less_minus_iff
thf(fact_149_nat__zero__as__int, axiom,
    ((zero_zero_nat = (nat2 @ zero_zero_int)))). % nat_zero_as_int
thf(fact_150_verit__negate__coefficient_I2_J, axiom,
    ((![A4 : int, B3 : int]: ((ord_less_int @ A4 @ B3) => (ord_less_int @ (uminus_uminus_int @ B3) @ (uminus_uminus_int @ A4)))))). % verit_negate_coefficient(2)
thf(fact_151_minus__equation__iff, axiom,
    ((![A4 : int, B3 : int]: (((uminus_uminus_int @ A4) = B3) = ((uminus_uminus_int @ B3) = A4))))). % minus_equation_iff

% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T, axiom,
    ((![P : $o]: ((P = $true) | (P = $false))))).
thf(help_If_2_1_If_001t__Nat__Onat_T, axiom,
    ((![X2 : nat, Y2 : nat]: ((if_nat @ $false @ X2 @ Y2) = Y2)))).
thf(help_If_1_1_If_001t__Nat__Onat_T, axiom,
    ((![X2 : nat, Y2 : nat]: ((if_nat @ $true @ X2 @ Y2) = X2)))).

% Conjectures (1)
thf(conj_0, conjecture,
    ((listsp_dB @ (^[T7 : dB]: (![E3 : nat > type]: (![T8 : type]: (![U6 : dB]: (![I2 : nat]: (((typing @ (shift_type @ E3 @ I2 @ t2) @ T7 @ T8)) => ((((it @ U6)) => ((((typing @ E3 @ U6 @ t2)) => ((it @ (subst @ T7 @ U6 @ I2))))))))))))) @ rs))).
