% 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_21__5209590_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:36:54.746

% Could-be-implicit typings (3)
thf(ty_n_t__List__Olist_It__Lambda__OdB_J, type,
    list_dB : $tType).
thf(ty_n_t__Lambda__OdB, type,
    dB : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).

% Explicit typings (21)
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat, type,
    plus_plus_nat : nat > nat > nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_InductTermi_OIT, type,
    it : dB > $o).
thf(sy_c_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_Olist_ONil_001t__Lambda__OdB, type,
    nil_dB : list_dB).
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_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_ia, type,
    ia : nat).
thf(sy_v_n, type,
    n : nat).
thf(sy_v_rs, type,
    rs : list_dB).

% Relevant facts (138)
thf(fact_0_lift__map, axiom,
    ((![T : dB, Ts : list_dB, I : nat]: ((lift @ (foldl_dB_dB @ app @ T @ Ts) @ I) = (foldl_dB_dB @ app @ (lift @ T @ I) @ (map_dB_dB @ (^[T2 : dB]: (lift @ T2 @ I)) @ Ts)))))). % lift_map
thf(fact_1_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_2_IT_OVar, axiom,
    ((![Rs : list_dB, N : nat]: ((listsp_dB @ it @ Rs) => (it @ (foldl_dB_dB @ app @ (var @ N) @ Rs)))))). % IT.Var
thf(fact_3_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_4_lessI, axiom,
    ((![N : nat]: (ord_less_nat @ N @ (suc @ N))))). % lessI
thf(fact_5_Suc__mono, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (ord_less_nat @ (suc @ M) @ (suc @ N)))))). % Suc_mono
thf(fact_6_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_7_not__less__eq, axiom,
    ((![M : nat, N : nat]: ((~ ((ord_less_nat @ M @ N))) = (ord_less_nat @ N @ (suc @ M)))))). % not_less_eq
thf(fact_8_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_9_map__ident, axiom,
    (((map_dB_dB @ (^[X : dB]: X)) = (^[Xs : list_dB]: Xs)))). % map_ident
thf(fact_10_lift__Suc__mono__less, axiom,
    ((![F : nat > nat, N : nat, N2 : nat]: ((![N3 : nat]: (ord_less_nat @ (F @ N3) @ (F @ (suc @ N3)))) => ((ord_less_nat @ N @ N2) => (ord_less_nat @ (F @ N) @ (F @ N2))))))). % lift_Suc_mono_less
thf(fact_11_lift__Suc__mono__less__iff, axiom,
    ((![F : nat > nat, N : nat, M : nat]: ((![N3 : nat]: (ord_less_nat @ (F @ N3) @ (F @ (suc @ N3)))) => ((ord_less_nat @ (F @ N) @ (F @ M)) = (ord_less_nat @ N @ M)))))). % lift_Suc_mono_less_iff
thf(fact_12_old_Onat_Oinject, axiom,
    ((![Nat : nat, Nat2 : nat]: (((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))). % old.nat.inject
thf(fact_13_nat_Oinject, axiom,
    ((![X2 : nat, Y2 : nat]: (((suc @ X2) = (suc @ Y2)) = (X2 = Y2))))). % nat.inject
thf(fact_14_n__not__Suc__n, axiom,
    ((![N : nat]: (~ ((N = (suc @ N))))))). % n_not_Suc_n
thf(fact_15_Suc__inject, axiom,
    ((![X3 : nat, Y : nat]: (((suc @ X3) = (suc @ Y)) => (X3 = Y))))). % Suc_inject
thf(fact_16_linorder__neqE__nat, axiom,
    ((![X3 : nat, Y : nat]: ((~ ((X3 = Y))) => ((~ ((ord_less_nat @ X3 @ Y))) => (ord_less_nat @ Y @ X3)))))). % linorder_neqE_nat
thf(fact_17_infinite__descent, axiom,
    ((![P : nat > $o, N : nat]: ((![N3 : nat]: ((~ ((P @ N3))) => (?[M2 : nat]: ((ord_less_nat @ M2 @ N3) & (~ ((P @ M2))))))) => (P @ N))))). % infinite_descent
thf(fact_18_nat__less__induct, axiom,
    ((![P : nat > $o, N : nat]: ((![N3 : nat]: ((![M2 : nat]: ((ord_less_nat @ M2 @ N3) => (P @ M2))) => (P @ N3))) => (P @ N))))). % nat_less_induct
thf(fact_19_less__irrefl__nat, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_irrefl_nat
thf(fact_20_less__not__refl3, axiom,
    ((![S : nat, T : nat]: ((ord_less_nat @ S @ T) => (~ ((S = T))))))). % less_not_refl3
thf(fact_21_less__not__refl2, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ N @ M) => (~ ((M = N))))))). % less_not_refl2
thf(fact_22_less__not__refl, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_not_refl
thf(fact_23_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_24_list_Omap__ident, axiom,
    ((![T : list_dB]: ((map_dB_dB @ (^[X : dB]: X) @ T) = T)))). % list.map_ident
thf(fact_25_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_26_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_27_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, K : nat]: ((ord_less_nat @ I2 @ J2) => ((ord_less_nat @ J2 @ K) => ((P @ I2 @ J2) => ((P @ J2 @ K) => (P @ I2 @ K)))))) => (P @ I @ J))))))). % less_Suc_induct
thf(fact_28_less__trans__Suc, axiom,
    ((![I : nat, J : nat, K2 : nat]: ((ord_less_nat @ I @ J) => ((ord_less_nat @ J @ K2) => (ord_less_nat @ (suc @ I) @ K2)))))). % less_trans_Suc
thf(fact_29_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_30_less__antisym, axiom,
    ((![N : nat, M : nat]: ((~ ((ord_less_nat @ N @ M))) => ((ord_less_nat @ N @ (suc @ M)) => (M = N)))))). % less_antisym
thf(fact_31_Suc__less__eq2, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ (suc @ N) @ M) = (?[M3 : nat]: (((M = (suc @ M3))) & ((ord_less_nat @ N @ M3)))))))). % Suc_less_eq2
thf(fact_32_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_33_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_34_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_35_less__SucI, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (ord_less_nat @ M @ (suc @ N)))))). % less_SucI
thf(fact_36_less__SucE, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ (suc @ N)) => ((~ ((ord_less_nat @ M @ N))) => (M = N)))))). % less_SucE
thf(fact_37_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_38_Suc__lessE, axiom,
    ((![I : nat, K2 : nat]: ((ord_less_nat @ (suc @ I) @ K2) => (~ ((![J2 : nat]: ((ord_less_nat @ I @ J2) => (~ ((K2 = (suc @ J2)))))))))))). % Suc_lessE
thf(fact_39_Suc__lessD, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ (suc @ M) @ N) => (ord_less_nat @ M @ N))))). % Suc_lessD
thf(fact_40_Nat_OlessE, axiom,
    ((![I : nat, K2 : nat]: ((ord_less_nat @ I @ K2) => ((~ ((K2 = (suc @ I)))) => (~ ((![J2 : nat]: ((ord_less_nat @ I @ J2) => (~ ((K2 = (suc @ J2))))))))))))). % Nat.lessE
thf(fact_41_foldl__map, axiom,
    ((![G : dB > dB > dB, A2 : dB, F : dB > dB, Xs2 : list_dB]: ((foldl_dB_dB @ G @ A2 @ (map_dB_dB @ F @ Xs2)) = (foldl_dB_dB @ (^[A3 : dB]: (^[X : dB]: (G @ A3 @ (F @ X)))) @ A2 @ Xs2))))). % foldl_map
thf(fact_42_dB_Oinject_I1_J, axiom,
    ((![X1 : nat, Y1 : nat]: (((var @ X1) = (var @ Y1)) = (X1 = Y1))))). % dB.inject(1)
thf(fact_43_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_44_lift_Osimps_I2_J, axiom,
    ((![S : dB, T : dB, K2 : nat]: ((lift @ (app @ S @ T) @ K2) = (app @ (lift @ S @ K2) @ (lift @ T @ K2)))))). % lift.simps(2)
thf(fact_45_dB_Odistinct_I1_J, axiom,
    ((![X1 : nat, X21 : dB, X22 : dB]: (~ (((var @ X1) = (app @ X21 @ X22))))))). % dB.distinct(1)
thf(fact_46_subst__map, axiom,
    ((![T : dB, Ts : list_dB, U : dB, I : nat]: ((subst @ (foldl_dB_dB @ app @ T @ Ts) @ U @ I) = (foldl_dB_dB @ app @ (subst @ T @ U @ I) @ (map_dB_dB @ (^[T2 : dB]: (subst @ T2 @ U @ I)) @ Ts)))))). % subst_map
thf(fact_47_map__eq__map__tailrec, axiom,
    ((map_dB_dB = map_tailrec_dB_dB))). % map_eq_map_tailrec
thf(fact_48_Var__eq__apps__conv, axiom,
    ((![M : nat, S : dB, Ss : list_dB]: (((var @ M) = (foldl_dB_dB @ app @ S @ Ss)) = ((((var @ M) = S)) & ((Ss = nil_dB))))))). % Var_eq_apps_conv
thf(fact_49_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_50_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_51_ex__head__tail, axiom,
    ((![T : dB]: (?[Ts2 : list_dB, H : dB]: ((T = (foldl_dB_dB @ app @ H @ Ts2)) & ((?[N3 : nat]: (H = (var @ N3))) | (?[U2 : dB]: (H = (abs @ U2))))))))). % ex_head_tail
thf(fact_52_dB_Oinject_I3_J, axiom,
    ((![X32 : dB, Y3 : dB]: (((abs @ X32) = (abs @ Y3)) = (X32 = Y3))))). % dB.inject(3)
thf(fact_53_Nil__is__map__conv, axiom,
    ((![F : dB > dB, Xs2 : list_dB]: ((nil_dB = (map_dB_dB @ F @ Xs2)) = (Xs2 = nil_dB))))). % Nil_is_map_conv
thf(fact_54_map__is__Nil__conv, axiom,
    ((![F : dB > dB, Xs2 : list_dB]: (((map_dB_dB @ F @ Xs2) = nil_dB) = (Xs2 = nil_dB))))). % map_is_Nil_conv
thf(fact_55_list_Omap__disc__iff, axiom,
    ((![F : dB > dB, A2 : list_dB]: (((map_dB_dB @ F @ A2) = nil_dB) = (A2 = nil_dB))))). % list.map_disc_iff
thf(fact_56_listsp__simps_I1_J, axiom,
    ((![A : dB > $o]: (listsp_dB @ A @ nil_dB)))). % listsp_simps(1)
thf(fact_57_subst__eq, axiom,
    ((![K2 : nat, U : dB]: ((subst @ (var @ K2) @ U @ K2) = U)))). % subst_eq
thf(fact_58_subst__lift, axiom,
    ((![T : dB, K2 : nat, S : dB]: ((subst @ (lift @ T @ K2) @ S @ K2) = T)))). % subst_lift
thf(fact_59_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_60_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_61_Abs__eq__apps__conv, axiom,
    ((![R : dB, S : dB, Ss : list_dB]: (((abs @ R) = (foldl_dB_dB @ app @ S @ Ss)) = ((((abs @ R) = S)) & ((Ss = nil_dB))))))). % Abs_eq_apps_conv
thf(fact_62_apps__eq__Abs__conv, axiom,
    ((![S : dB, Ss : list_dB, R : dB]: (((foldl_dB_dB @ app @ S @ Ss) = (abs @ R)) = (((S = (abs @ R))) & ((Ss = nil_dB))))))). % apps_eq_Abs_conv
thf(fact_63_dB_Odistinct_I5_J, axiom,
    ((![X21 : dB, X22 : dB, X32 : dB]: (~ (((app @ X21 @ X22) = (abs @ X32))))))). % dB.distinct(5)
thf(fact_64_subst__App, axiom,
    ((![T : dB, U : dB, S : dB, K2 : nat]: ((subst @ (app @ T @ U) @ S @ K2) = (app @ (subst @ T @ S @ K2) @ (subst @ U @ S @ K2)))))). % subst_App
thf(fact_65_dB_Odistinct_I3_J, axiom,
    ((![X1 : nat, X32 : dB]: (~ (((var @ X1) = (abs @ X32))))))). % dB.distinct(3)
thf(fact_66_list_Osimps_I8_J, axiom,
    ((![F : dB > dB]: ((map_dB_dB @ F @ nil_dB) = nil_dB)))). % list.simps(8)
thf(fact_67_foldl__Nil, axiom,
    ((![F : dB > dB > dB, A2 : dB]: ((foldl_dB_dB @ F @ A2 @ nil_dB) = A2)))). % foldl_Nil
thf(fact_68_listsp_ONil, axiom,
    ((![A : dB > $o]: (listsp_dB @ A @ nil_dB)))). % listsp.Nil
thf(fact_69_Lambda, axiom,
    ((![R : dB]: ((it @ R) => (it @ (abs @ R)))))). % Lambda
thf(fact_70_dB_Oinduct, axiom,
    ((![P : dB > $o, DB : dB]: ((![X4 : nat]: (P @ (var @ X4))) => ((![X1a : dB, X23 : dB]: ((P @ X1a) => ((P @ X23) => (P @ (app @ X1a @ X23))))) => ((![X4 : dB]: ((P @ X4) => (P @ (abs @ X4)))) => (P @ DB))))))). % dB.induct
thf(fact_71_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_72_IT_Oinducts, axiom,
    ((![X3 : dB, P : dB > $o]: ((it @ X3) => ((![Rs2 : list_dB, N3 : nat]: ((listsp_dB @ (^[X : dB]: (((it @ X)) & ((P @ X)))) @ Rs2) => (P @ (foldl_dB_dB @ app @ (var @ N3) @ Rs2)))) => ((![R2 : dB]: ((it @ R2) => ((P @ R2) => (P @ (abs @ R2))))) => ((![R2 : dB, S2 : dB, Ss2 : list_dB]: ((it @ (foldl_dB_dB @ app @ (subst @ R2 @ S2 @ zero_zero_nat) @ Ss2)) => ((P @ (foldl_dB_dB @ app @ (subst @ R2 @ S2 @ zero_zero_nat) @ Ss2)) => ((it @ S2) => ((P @ S2) => (P @ (foldl_dB_dB @ app @ (app @ (abs @ R2) @ S2) @ Ss2))))))) => (P @ X3)))))))). % IT.inducts
thf(fact_73_IT_Ocases, axiom,
    ((![A2 : dB]: ((it @ A2) => ((![Rs2 : list_dB]: ((?[N3 : nat]: (A2 = (foldl_dB_dB @ app @ (var @ N3) @ Rs2))) => (~ ((listsp_dB @ it @ Rs2))))) => ((![R2 : dB]: ((A2 = (abs @ R2)) => (~ ((it @ R2))))) => (~ ((![R2 : dB, S2 : dB, Ss2 : list_dB]: ((A2 = (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_74_IT_Osimps, axiom,
    ((it = (^[A3 : dB]: (((?[Rs3 : list_dB]: (?[N4 : nat]: (((A3 = (foldl_dB_dB @ app @ (var @ N4) @ Rs3))) & ((listsp_dB @ it @ Rs3)))))) | ((((?[R3 : dB]: (((A3 = (abs @ R3))) & ((it @ R3))))) | ((?[R3 : dB]: (?[S3 : dB]: (?[Ss3 : list_dB]: (((A3 = (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_75_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_76_liftn__lift, axiom,
    ((![N : nat, T : dB, K2 : nat]: ((liftn @ (suc @ N) @ T @ K2) = (lift @ (liftn @ N @ T @ K2) @ K2))))). % liftn_lift
thf(fact_77_bot__nat__0_Onot__eq__extremum, axiom,
    ((![A2 : nat]: ((~ ((A2 = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ A2))))). % bot_nat_0.not_eq_extremum
thf(fact_78_less__nat__zero__code, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_79_neq0__conv, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N))))). % neq0_conv
thf(fact_80_liftn__0, axiom,
    ((![T : dB, K2 : nat]: ((liftn @ zero_zero_nat @ T @ K2) = T)))). % liftn_0
thf(fact_81_zero__less__Suc, axiom,
    ((![N : nat]: (ord_less_nat @ zero_zero_nat @ (suc @ N))))). % zero_less_Suc
thf(fact_82_less__Suc0, axiom,
    ((![N : nat]: ((ord_less_nat @ N @ (suc @ zero_zero_nat)) = (N = zero_zero_nat))))). % less_Suc0
thf(fact_83_not0__implies__Suc, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (?[M4 : nat]: (N = (suc @ M4))))))). % not0_implies_Suc
thf(fact_84_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_85_old_Onat_Oexhaust, axiom,
    ((![Y : nat]: ((~ ((Y = zero_zero_nat))) => (~ ((![Nat3 : nat]: (~ ((Y = (suc @ Nat3))))))))))). % old.nat.exhaust
thf(fact_86_Zero__not__Suc, axiom,
    ((![M : nat]: (~ ((zero_zero_nat = (suc @ M))))))). % Zero_not_Suc
thf(fact_87_Zero__neq__Suc, axiom,
    ((![M : nat]: (~ ((zero_zero_nat = (suc @ M))))))). % Zero_neq_Suc
thf(fact_88_Suc__neq__Zero, axiom,
    ((![M : nat]: (~ (((suc @ M) = zero_zero_nat)))))). % Suc_neq_Zero
thf(fact_89_zero__induct, axiom,
    ((![P : nat > $o, K2 : nat]: ((P @ K2) => ((![N3 : nat]: ((P @ (suc @ N3)) => (P @ N3))) => (P @ zero_zero_nat)))))). % zero_induct
thf(fact_90_diff__induct, axiom,
    ((![P : nat > nat > $o, M : nat, N : nat]: ((![X4 : nat]: (P @ X4 @ zero_zero_nat)) => ((![Y4 : nat]: (P @ zero_zero_nat @ (suc @ Y4))) => ((![X4 : nat, Y4 : nat]: ((P @ X4 @ Y4) => (P @ (suc @ X4) @ (suc @ Y4)))) => (P @ M @ N))))))). % diff_induct
thf(fact_91_nat__induct, axiom,
    ((![P : nat > $o, N : nat]: ((P @ zero_zero_nat) => ((![N3 : nat]: ((P @ N3) => (P @ (suc @ N3)))) => (P @ N)))))). % nat_induct
thf(fact_92_nat_OdiscI, axiom,
    ((![Nat : nat, X2 : nat]: ((Nat = (suc @ X2)) => (~ ((Nat = zero_zero_nat))))))). % nat.discI
thf(fact_93_old_Onat_Odistinct_I1_J, axiom,
    ((![Nat2 : nat]: (~ ((zero_zero_nat = (suc @ Nat2))))))). % old.nat.distinct(1)
thf(fact_94_old_Onat_Odistinct_I2_J, axiom,
    ((![Nat2 : nat]: (~ (((suc @ Nat2) = zero_zero_nat)))))). % old.nat.distinct(2)
thf(fact_95_nat_Odistinct_I1_J, axiom,
    ((![X2 : nat]: (~ ((zero_zero_nat = (suc @ X2))))))). % nat.distinct(1)
thf(fact_96_bot__nat__0_Oextremum__strict, axiom,
    ((![A2 : nat]: (~ ((ord_less_nat @ A2 @ zero_zero_nat)))))). % bot_nat_0.extremum_strict
thf(fact_97_infinite__descent0, axiom,
    ((![P : nat > $o, N : nat]: ((P @ zero_zero_nat) => ((![N3 : nat]: ((ord_less_nat @ zero_zero_nat @ N3) => ((~ ((P @ N3))) => (?[M2 : nat]: ((ord_less_nat @ M2 @ N3) & (~ ((P @ M2)))))))) => (P @ N)))))). % infinite_descent0
thf(fact_98_gr__implies__not0, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not0
thf(fact_99_less__zeroE, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_zeroE
thf(fact_100_not__less0, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less0
thf(fact_101_not__gr0, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr0
thf(fact_102_gr0I, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr0I
thf(fact_103_liftn_Osimps_I2_J, axiom,
    ((![N : nat, S : dB, T : dB, K2 : nat]: ((liftn @ N @ (app @ S @ T) @ K2) = (app @ (liftn @ N @ S @ K2) @ (liftn @ N @ T @ K2)))))). % liftn.simps(2)
thf(fact_104_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_105_gr0__implies__Suc, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) => (?[M4 : nat]: (N = (suc @ M4))))))). % gr0_implies_Suc
thf(fact_106_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_107_gr0__conv__Suc, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) = (?[M5 : nat]: (N = (suc @ M5))))))). % gr0_conv_Suc
thf(fact_108_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_109_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_110_substn__subst__n, axiom,
    ((substn = (^[T2 : dB]: (^[S3 : dB]: (^[N4 : nat]: (subst @ T2 @ (liftn @ N4 @ S3 @ zero_zero_nat) @ N4))))))). % substn_subst_n
thf(fact_111_substn_Osimps_I2_J, axiom,
    ((![T : dB, U : dB, S : dB, K2 : nat]: ((substn @ (app @ T @ U) @ S @ K2) = (app @ (substn @ T @ S @ K2) @ (substn @ U @ S @ K2)))))). % substn.simps(2)
thf(fact_112_zero__reorient, axiom,
    ((![X3 : nat]: ((zero_zero_nat = X3) = (X3 = zero_zero_nat))))). % zero_reorient
thf(fact_113_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_114_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_115_gr__implies__not__zero, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not_zero
thf(fact_116_not__less__zero, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less_zero
thf(fact_117_gr__zeroI, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr_zeroI
thf(fact_118_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_nat @ zero_zero_nat @ zero_zero_nat))))). % less_numeral_extra(3)
thf(fact_119_dB_Osize__gen_I1_J, axiom,
    ((![X1 : nat]: ((size_dB @ (var @ X1)) = zero_zero_nat)))). % dB.size_gen(1)
thf(fact_120_dB_Osize__gen_I3_J, axiom,
    ((![X32 : dB]: ((size_dB @ (abs @ X32)) = (plus_plus_nat @ (size_dB @ X32) @ (suc @ zero_zero_nat)))))). % dB.size_gen(3)
thf(fact_121_dB_Osize__gen_I2_J, axiom,
    ((![X21 : dB, X22 : dB]: ((size_dB @ (app @ X21 @ X22)) = (plus_plus_nat @ (plus_plus_nat @ (size_dB @ X21) @ (size_dB @ X22)) @ (suc @ zero_zero_nat)))))). % dB.size_gen(2)
thf(fact_122_add__left__cancel, axiom,
    ((![A2 : nat, B2 : nat, C : nat]: (((plus_plus_nat @ A2 @ B2) = (plus_plus_nat @ A2 @ C)) = (B2 = C))))). % add_left_cancel
thf(fact_123_add__right__cancel, axiom,
    ((![B2 : nat, A2 : nat, C : nat]: (((plus_plus_nat @ B2 @ A2) = (plus_plus_nat @ C @ A2)) = (B2 = C))))). % add_right_cancel
thf(fact_124_zero__eq__add__iff__both__eq__0, axiom,
    ((![X3 : nat, Y : nat]: ((zero_zero_nat = (plus_plus_nat @ X3 @ Y)) = (((X3 = zero_zero_nat)) & ((Y = zero_zero_nat))))))). % zero_eq_add_iff_both_eq_0
thf(fact_125_add__eq__0__iff__both__eq__0, axiom,
    ((![X3 : nat, Y : nat]: (((plus_plus_nat @ X3 @ Y) = zero_zero_nat) = (((X3 = zero_zero_nat)) & ((Y = zero_zero_nat))))))). % add_eq_0_iff_both_eq_0
thf(fact_126_add__cancel__right__right, axiom,
    ((![A2 : nat, B2 : nat]: ((A2 = (plus_plus_nat @ A2 @ B2)) = (B2 = zero_zero_nat))))). % add_cancel_right_right
thf(fact_127_add__cancel__right__left, axiom,
    ((![A2 : nat, B2 : nat]: ((A2 = (plus_plus_nat @ B2 @ A2)) = (B2 = zero_zero_nat))))). % add_cancel_right_left
thf(fact_128_add__cancel__left__right, axiom,
    ((![A2 : nat, B2 : nat]: (((plus_plus_nat @ A2 @ B2) = A2) = (B2 = zero_zero_nat))))). % add_cancel_left_right
thf(fact_129_add__cancel__left__left, axiom,
    ((![B2 : nat, A2 : nat]: (((plus_plus_nat @ B2 @ A2) = A2) = (B2 = zero_zero_nat))))). % add_cancel_left_left
thf(fact_130_add_Oright__neutral, axiom,
    ((![A2 : nat]: ((plus_plus_nat @ A2 @ zero_zero_nat) = A2)))). % add.right_neutral
thf(fact_131_add_Oleft__neutral, axiom,
    ((![A2 : nat]: ((plus_plus_nat @ zero_zero_nat @ A2) = A2)))). % add.left_neutral
thf(fact_132_add__less__cancel__left, axiom,
    ((![C : nat, A2 : nat, B2 : nat]: ((ord_less_nat @ (plus_plus_nat @ C @ A2) @ (plus_plus_nat @ C @ B2)) = (ord_less_nat @ A2 @ B2))))). % add_less_cancel_left
thf(fact_133_add__less__cancel__right, axiom,
    ((![A2 : nat, C : nat, B2 : nat]: ((ord_less_nat @ (plus_plus_nat @ A2 @ C) @ (plus_plus_nat @ B2 @ C)) = (ord_less_nat @ A2 @ B2))))). % add_less_cancel_right
thf(fact_134_add__is__0, axiom,
    ((![M : nat, N : nat]: (((plus_plus_nat @ M @ N) = zero_zero_nat) = (((M = zero_zero_nat)) & ((N = zero_zero_nat))))))). % add_is_0
thf(fact_135_Nat_Oadd__0__right, axiom,
    ((![M : nat]: ((plus_plus_nat @ M @ zero_zero_nat) = M)))). % Nat.add_0_right
thf(fact_136_add__Suc__right, axiom,
    ((![M : nat, N : nat]: ((plus_plus_nat @ M @ (suc @ N)) = (suc @ (plus_plus_nat @ M @ N)))))). % add_Suc_right
thf(fact_137_nat__add__left__cancel__less, axiom,
    ((![K2 : nat, M : nat, N : nat]: ((ord_less_nat @ (plus_plus_nat @ K2 @ M) @ (plus_plus_nat @ K2 @ N)) = (ord_less_nat @ M @ N))))). % nat_add_left_cancel_less

% Conjectures (2)
thf(conj_0, hypothesis,
    ((listsp_dB @ (^[X : dB]: (((it @ X)) & ((![Y5 : nat]: (it @ (lift @ X @ Y5)))))) @ rs))).
thf(conj_1, conjecture,
    ((((~ ((ord_less_nat @ n @ ia))) | (it @ (foldl_dB_dB @ app @ (var @ n) @ (map_dB_dB @ (^[T2 : dB]: (lift @ T2 @ ia)) @ rs)))) & ((~ ((ord_less_nat @ ia @ (suc @ n)))) | (it @ (foldl_dB_dB @ app @ (var @ (suc @ n)) @ (map_dB_dB @ (^[T2 : dB]: (lift @ T2 @ ia)) @ rs))))))).
