% 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_161__5210954_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:37:57.967

% Could-be-implicit typings (6)
thf(ty_n_t__List__Olist_It__Lambda__OdB_J, type,
    list_dB : $tType).
thf(ty_n_t__Set__Oset_It__Lambda__OdB_J, type,
    set_dB : $tType).
thf(ty_n_t__List__Olist_It__Nat__Onat_J, type,
    list_nat : $tType).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J, type,
    set_nat : $tType).
thf(ty_n_t__Lambda__OdB, type,
    dB : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).

% Explicit typings (51)
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat, type,
    one_one_nat : nat).
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_Obeta, type,
    beta : dB > dB > $o).
thf(sy_c_Lambda_OdB_OAbs, type,
    abs : dB > dB).
thf(sy_c_Lambda_OdB_OApp, type,
    app : dB > dB > dB).
thf(sy_c_Lambda_OdB_OVar, type,
    var : nat > dB).
thf(sy_c_Lambda_OdB_Osize__dB, type,
    size_dB : dB > nat).
thf(sy_c_Lambda_Olift, type,
    lift : dB > nat > dB).
thf(sy_c_Lambda_Oliftn, type,
    liftn : nat > dB > nat > dB).
thf(sy_c_Lambda_Osubst, type,
    subst : dB > dB > nat > dB).
thf(sy_c_Lambda_Osubstn, type,
    substn : dB > dB > nat > dB).
thf(sy_c_List_Obind_001t__Lambda__OdB_001t__Lambda__OdB, type,
    bind_dB_dB : list_dB > (dB > list_dB) > list_dB).
thf(sy_c_List_Obind_001t__Lambda__OdB_001t__Nat__Onat, type,
    bind_dB_nat : list_dB > (dB > list_nat) > list_nat).
thf(sy_c_List_Obind_001t__Nat__Onat_001t__Lambda__OdB, type,
    bind_nat_dB : list_nat > (nat > list_dB) > list_dB).
thf(sy_c_List_Obind_001t__Nat__Onat_001t__Nat__Onat, type,
    bind_nat_nat : list_nat > (nat > list_nat) > list_nat).
thf(sy_c_List_Ocan__select_001t__Lambda__OdB, type,
    can_select_dB : (dB > $o) > set_dB > $o).
thf(sy_c_List_Ocount__list_001t__Lambda__OdB, type,
    count_list_dB : list_dB > dB > nat).
thf(sy_c_List_Ocount__list_001t__Nat__Onat, type,
    count_list_nat : list_nat > nat > nat).
thf(sy_c_List_Ofoldl_001t__Lambda__OdB_001t__Lambda__OdB, type,
    foldl_dB_dB : (dB > dB > dB) > dB > list_dB > dB).
thf(sy_c_List_Ofoldl_001t__Lambda__OdB_001t__Nat__Onat, type,
    foldl_dB_nat : (dB > nat > dB) > dB > list_nat > dB).
thf(sy_c_List_Ofoldl_001t__Nat__Onat_001t__Lambda__OdB, type,
    foldl_nat_dB : (nat > dB > nat) > nat > list_dB > nat).
thf(sy_c_List_Ofoldl_001t__Nat__Onat_001t__Nat__Onat, type,
    foldl_nat_nat : (nat > nat > nat) > nat > list_nat > nat).
thf(sy_c_List_Olist_ONil_001t__Lambda__OdB, type,
    nil_dB : list_dB).
thf(sy_c_List_Olist_ONil_001t__Nat__Onat, type,
    nil_nat : list_nat).
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_Olist_Omap_001t__Lambda__OdB_001t__Nat__Onat, type,
    map_dB_nat : (dB > nat) > list_dB > list_nat).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Lambda__OdB, type,
    map_nat_dB : (nat > dB) > list_nat > list_dB).
thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat, type,
    map_nat_nat : (nat > nat) > list_nat > list_nat).
thf(sy_c_List_Olist_Oset_001t__Lambda__OdB, type,
    set_dB2 : list_dB > set_dB).
thf(sy_c_List_Olist_Oset_001t__Nat__Onat, type,
    set_nat2 : list_nat > set_nat).
thf(sy_c_List_Olist__ex1_001t__Lambda__OdB, type,
    list_ex1_dB : (dB > $o) > list_dB > $o).
thf(sy_c_List_Olist__ex1_001t__Nat__Onat, type,
    list_ex1_nat : (nat > $o) > list_nat > $o).
thf(sy_c_List_Olistsp_001t__Lambda__OdB, type,
    listsp_dB : (dB > $o) > list_dB > $o).
thf(sy_c_List_Olistsp_001t__Nat__Onat, type,
    listsp_nat : (nat > $o) > list_nat > $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_List_Omap__tailrec_001t__Lambda__OdB_001t__Nat__Onat, type,
    map_tailrec_dB_nat : (dB > nat) > list_dB > list_nat).
thf(sy_c_List_Omember_001t__Lambda__OdB, type,
    member_dB : list_dB > dB > $o).
thf(sy_c_List_Omember_001t__Nat__Onat, type,
    member_nat : list_nat > nat > $o).
thf(sy_c_Nat_Osize__class_Osize_001t__Lambda__OdB, type,
    size_size_dB : dB > nat).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Lambda__OdB_J, type,
    size_size_list_dB : list_dB > nat).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Nat__Onat_J, type,
    size_size_list_nat : list_nat > nat).
thf(sy_c_member_001t__Lambda__OdB, type,
    member_dB2 : dB > set_dB > $o).
thf(sy_c_member_001t__Nat__Onat, type,
    member_nat2 : nat > set_nat > $o).
thf(sy_v_a____, type,
    a : dB).
thf(sy_v_i____, type,
    i : nat).
thf(sy_v_n____, type,
    n : nat).
thf(sy_v_t____, type,
    t : dB).
thf(sy_v_u1____, type,
    u1 : dB).
thf(sy_v_u____, type,
    u : dB).

% Relevant facts (172)
thf(fact_0__092_060open_062IT_At_092_060close_062, axiom,
    ((it @ t))). % \<open>IT t\<close>
thf(fact_1_Var_Oprems_I2_J, axiom,
    ((it @ u1))). % Var.prems(2)
thf(fact_2_uIT, axiom,
    ((it @ u))). % uIT
thf(fact_3_True, axiom,
    ((n = i))). % True
thf(fact_4_lift__IT, axiom,
    ((![T : dB, I : nat]: ((it @ T) => (it @ (lift @ T @ I)))))). % lift_IT
thf(fact_5_lifts__IT, axiom,
    ((![Ts : list_dB]: ((listsp_dB @ it @ Ts) => (listsp_dB @ it @ (map_dB_dB @ (^[T2 : dB]: (lift @ T2 @ zero_zero_nat)) @ Ts)))))). % lifts_IT
thf(fact_6_subst__lift, axiom,
    ((![T : dB, K : nat, S : dB]: ((subst @ (lift @ T @ K) @ S @ K) = T)))). % subst_lift
thf(fact_7_listsp__simps_I1_J, axiom,
    ((![A : nat > $o]: (listsp_nat @ A @ nil_nat)))). % listsp_simps(1)
thf(fact_8_listsp__simps_I1_J, axiom,
    ((![A : dB > $o]: (listsp_dB @ A @ nil_dB)))). % listsp_simps(1)
thf(fact_9_Nil__is__map__conv, axiom,
    ((![F : nat > dB, Xs : list_nat]: ((nil_dB = (map_nat_dB @ F @ Xs)) = (Xs = nil_nat))))). % Nil_is_map_conv
thf(fact_10_Nil__is__map__conv, axiom,
    ((![F : nat > nat, Xs : list_nat]: ((nil_nat = (map_nat_nat @ F @ Xs)) = (Xs = nil_nat))))). % Nil_is_map_conv
thf(fact_11_Nil__is__map__conv, axiom,
    ((![F : dB > dB, Xs : list_dB]: ((nil_dB = (map_dB_dB @ F @ Xs)) = (Xs = nil_dB))))). % Nil_is_map_conv
thf(fact_12_Nil__is__map__conv, axiom,
    ((![F : dB > nat, Xs : list_dB]: ((nil_nat = (map_dB_nat @ F @ Xs)) = (Xs = nil_dB))))). % Nil_is_map_conv
thf(fact_13_map__is__Nil__conv, axiom,
    ((![F : nat > dB, Xs : list_nat]: (((map_nat_dB @ F @ Xs) = nil_dB) = (Xs = nil_nat))))). % map_is_Nil_conv
thf(fact_14_map__is__Nil__conv, axiom,
    ((![F : nat > nat, Xs : list_nat]: (((map_nat_nat @ F @ Xs) = nil_nat) = (Xs = nil_nat))))). % map_is_Nil_conv
thf(fact_15_map__is__Nil__conv, axiom,
    ((![F : dB > dB, Xs : list_dB]: (((map_dB_dB @ F @ Xs) = nil_dB) = (Xs = nil_dB))))). % map_is_Nil_conv
thf(fact_16_map__is__Nil__conv, axiom,
    ((![F : dB > nat, Xs : list_dB]: (((map_dB_nat @ F @ Xs) = nil_nat) = (Xs = nil_dB))))). % map_is_Nil_conv
thf(fact_17_list_Omap__disc__iff, axiom,
    ((![F : nat > dB, A2 : list_nat]: (((map_nat_dB @ F @ A2) = nil_dB) = (A2 = nil_nat))))). % list.map_disc_iff
thf(fact_18_list_Omap__disc__iff, axiom,
    ((![F : nat > nat, A2 : list_nat]: (((map_nat_nat @ F @ A2) = nil_nat) = (A2 = nil_nat))))). % list.map_disc_iff
thf(fact_19_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_20_list_Omap__disc__iff, axiom,
    ((![F : dB > nat, A2 : list_dB]: (((map_dB_nat @ F @ A2) = nil_nat) = (A2 = nil_dB))))). % list.map_disc_iff
thf(fact_21_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_22_map__ident, axiom,
    (((map_dB_dB @ (^[X : dB]: X)) = (^[Xs2 : list_dB]: Xs2)))). % map_ident
thf(fact_23_listsp_ONil, axiom,
    ((![A : dB > $o]: (listsp_dB @ A @ nil_dB)))). % listsp.Nil
thf(fact_24_listsp_ONil, axiom,
    ((![A : nat > $o]: (listsp_nat @ A @ nil_nat)))). % listsp.Nil
thf(fact_25_list_Osimps_I8_J, axiom,
    ((![F : nat > dB]: ((map_nat_dB @ F @ nil_nat) = nil_dB)))). % list.simps(8)
thf(fact_26_list_Osimps_I8_J, axiom,
    ((![F : nat > nat]: ((map_nat_nat @ F @ nil_nat) = nil_nat)))). % list.simps(8)
thf(fact_27_list_Osimps_I8_J, axiom,
    ((![F : dB > dB]: ((map_dB_dB @ F @ nil_dB) = nil_dB)))). % list.simps(8)
thf(fact_28_list_Osimps_I8_J, axiom,
    ((![F : dB > nat]: ((map_dB_nat @ F @ nil_dB) = nil_nat)))). % list.simps(8)
thf(fact_29__092_060open_062IT_A_Iu_A_092_060degree_062_Aa_091u_Pi_093_J_092_060close_062, axiom,
    ((it @ (app @ u @ (subst @ a @ u @ i))))). % \<open>IT (u \<degree> a[u/i])\<close>
thf(fact_30_list_Omap__ident, axiom,
    ((![T : list_dB]: ((map_dB_dB @ (^[X : dB]: X) @ T) = T)))). % list.map_ident
thf(fact_31_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_32_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_33_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_34_zero__reorient, axiom,
    ((![X2 : nat]: ((zero_zero_nat = X2) = (X2 = zero_zero_nat))))). % zero_reorient
thf(fact_35__092_060open_062IT_A_I_Ilift_Au_A0_A_092_060degree_062_AVar_A0_J_091a_091u_Pi_093_P0_093_J_092_060close_062, axiom,
    ((it @ (subst @ (app @ (lift @ u @ zero_zero_nat) @ (var @ zero_zero_nat)) @ (subst @ a @ u @ i) @ zero_zero_nat)))). % \<open>IT ((lift u 0 \<degree> Var 0)[a[u/i]/0])\<close>
thf(fact_36_map__eq__map__tailrec, axiom,
    ((map_dB_dB = map_tailrec_dB_dB))). % map_eq_map_tailrec
thf(fact_37_map__eq__map__tailrec, axiom,
    ((map_dB_nat = map_tailrec_dB_nat))). % map_eq_map_tailrec
thf(fact_38_list__ex1__simps_I1_J, axiom,
    ((![P : dB > $o]: (~ ((list_ex1_dB @ P @ nil_dB)))))). % list_ex1_simps(1)
thf(fact_39_list__ex1__simps_I1_J, axiom,
    ((![P : nat > $o]: (~ ((list_ex1_nat @ P @ nil_nat)))))). % list_ex1_simps(1)
thf(fact_40_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_41_count__list_Osimps_I1_J, axiom,
    ((![Y : dB]: ((count_list_dB @ nil_dB @ Y) = zero_zero_nat)))). % count_list.simps(1)
thf(fact_42_count__list_Osimps_I1_J, axiom,
    ((![Y : nat]: ((count_list_nat @ nil_nat @ Y) = zero_zero_nat)))). % count_list.simps(1)
thf(fact_43_bind__simps_I1_J, axiom,
    ((![F : dB > list_dB]: ((bind_dB_dB @ nil_dB @ F) = nil_dB)))). % bind_simps(1)
thf(fact_44_bind__simps_I1_J, axiom,
    ((![F : dB > list_nat]: ((bind_dB_nat @ nil_dB @ F) = nil_nat)))). % bind_simps(1)
thf(fact_45_bind__simps_I1_J, axiom,
    ((![F : nat > list_dB]: ((bind_nat_dB @ nil_nat @ F) = nil_dB)))). % bind_simps(1)
thf(fact_46_bind__simps_I1_J, axiom,
    ((![F : nat > list_nat]: ((bind_nat_nat @ nil_nat @ F) = nil_nat)))). % bind_simps(1)
thf(fact_47_zero__natural_Orsp, axiom,
    ((zero_zero_nat = zero_zero_nat))). % zero_natural.rsp
thf(fact_48_dB_Oinject_I1_J, axiom,
    ((![X1 : nat, Y1 : nat]: (((var @ X1) = (var @ Y1)) = (X1 = Y1))))). % dB.inject(1)
thf(fact_49_subst__eq, axiom,
    ((![K : nat, U : dB]: ((subst @ (var @ K) @ U @ K) = U)))). % subst_eq
thf(fact_50_dB_Odistinct_I1_J, axiom,
    ((![X1 : nat, X21 : dB, X22 : dB]: (~ (((var @ X1) = (app @ X21 @ X22))))))). % dB.distinct(1)
thf(fact_51_Var__IT, axiom,
    ((![N : nat]: (it @ (var @ N))))). % Var_IT
thf(fact_52_app__Var__IT, axiom,
    ((![T : dB, I : nat]: ((it @ T) => (it @ (app @ T @ (var @ I))))))). % app_Var_IT
thf(fact_53_subst__Var__IT, axiom,
    ((![R : dB, I : nat, J : nat]: ((it @ R) => (it @ (subst @ R @ (var @ I) @ J)))))). % subst_Var_IT
thf(fact_54_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_55_substn__subst__n, axiom,
    ((substn = (^[T2 : dB]: (^[S2 : dB]: (^[N2 : nat]: (subst @ T2 @ (liftn @ N2 @ S2 @ zero_zero_nat) @ N2))))))). % substn_subst_n
thf(fact_56_dB_Osize__gen_I1_J, axiom,
    ((![X1 : nat]: ((size_dB @ (var @ X1)) = zero_zero_nat)))). % dB.size_gen(1)
thf(fact_57_dB_Osize_I4_J, axiom,
    ((![X1 : nat]: ((size_size_dB @ (var @ X1)) = zero_zero_nat)))). % dB.size(4)
thf(fact_58_member__rec_I2_J, axiom,
    ((![Y : dB]: (~ ((member_dB @ nil_dB @ Y)))))). % member_rec(2)
thf(fact_59_member__rec_I2_J, axiom,
    ((![Y : nat]: (~ ((member_nat @ nil_nat @ Y)))))). % member_rec(2)
thf(fact_60_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_61_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_62_IT_OVar, axiom,
    ((![Rs : list_dB, N : nat]: ((listsp_dB @ it @ Rs) => (it @ (foldl_dB_dB @ app @ (var @ N) @ Rs)))))). % IT.Var
thf(fact_63_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_64_liftn__0, axiom,
    ((![T : dB, K : nat]: ((liftn @ zero_zero_nat @ T @ K) = T)))). % liftn_0
thf(fact_65_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_66_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_67_size__neq__size__imp__neq, axiom,
    ((![X2 : dB, Y : dB]: ((~ (((size_size_dB @ X2) = (size_size_dB @ Y)))) => (~ ((X2 = Y))))))). % size_neq_size_imp_neq
thf(fact_68_size__neq__size__imp__neq, axiom,
    ((![X2 : list_dB, Y : list_dB]: ((~ (((size_size_list_dB @ X2) = (size_size_list_dB @ Y)))) => (~ ((X2 = Y))))))). % size_neq_size_imp_neq
thf(fact_69_foldl__Nil, axiom,
    ((![F : dB > dB > dB, A2 : dB]: ((foldl_dB_dB @ F @ A2 @ nil_dB) = A2)))). % foldl_Nil
thf(fact_70_foldl__Nil, axiom,
    ((![F : nat > nat > nat, A2 : nat]: ((foldl_nat_nat @ F @ A2 @ nil_nat) = A2)))). % foldl_Nil
thf(fact_71_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_72_foldl__map, axiom,
    ((![G : nat > nat > nat, A2 : nat, F : nat > nat, Xs : list_nat]: ((foldl_nat_nat @ G @ A2 @ (map_nat_nat @ F @ Xs)) = (foldl_nat_nat @ (^[A3 : nat]: (^[X : nat]: (G @ A3 @ (F @ X)))) @ A2 @ Xs))))). % foldl_map
thf(fact_73_foldl__map, axiom,
    ((![G : dB > dB > dB, A2 : dB, F : dB > dB, Xs : list_dB]: ((foldl_dB_dB @ G @ A2 @ (map_dB_dB @ F @ Xs)) = (foldl_dB_dB @ (^[A3 : dB]: (^[X : dB]: (G @ A3 @ (F @ X)))) @ A2 @ Xs))))). % foldl_map
thf(fact_74_foldl__map, axiom,
    ((![G : dB > nat > dB, A2 : dB, F : dB > nat, Xs : list_dB]: ((foldl_dB_nat @ G @ A2 @ (map_dB_nat @ F @ Xs)) = (foldl_dB_dB @ (^[A3 : dB]: (^[X : dB]: (G @ A3 @ (F @ X)))) @ A2 @ Xs))))). % foldl_map
thf(fact_75_foldl__map, axiom,
    ((![G : nat > nat > nat, A2 : nat, F : dB > nat, Xs : list_dB]: ((foldl_nat_nat @ G @ A2 @ (map_dB_nat @ F @ Xs)) = (foldl_nat_dB @ (^[A3 : nat]: (^[X : dB]: (G @ A3 @ (F @ X)))) @ A2 @ Xs))))). % foldl_map
thf(fact_76_IT_Oinducts, axiom,
    ((![X2 : dB, P : dB > $o]: ((it @ X2) => ((![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, S3 : dB, Ss2 : list_dB]: ((it @ (foldl_dB_dB @ app @ (subst @ R2 @ S3 @ zero_zero_nat) @ Ss2)) => ((P @ (foldl_dB_dB @ app @ (subst @ R2 @ S3 @ zero_zero_nat) @ Ss2)) => ((it @ S3) => ((P @ S3) => (P @ (foldl_dB_dB @ app @ (app @ (abs @ R2) @ S3) @ Ss2))))))) => (P @ X2)))))))). % IT.inducts
thf(fact_77_IT_Osimps, axiom,
    ((it = (^[A3 : dB]: (((?[Rs3 : list_dB]: (?[N2 : nat]: (((A3 = (foldl_dB_dB @ app @ (var @ N2) @ Rs3))) & ((listsp_dB @ it @ Rs3)))))) | ((((?[R3 : dB]: (((A3 = (abs @ R3))) & ((it @ R3))))) | ((?[R3 : dB]: (?[S2 : dB]: (?[Ss3 : list_dB]: (((A3 = (foldl_dB_dB @ app @ (app @ (abs @ R3) @ S2) @ Ss3))) & ((((it @ (foldl_dB_dB @ app @ (subst @ R3 @ S2 @ zero_zero_nat) @ Ss3))) & ((it @ S2)))))))))))))))). % IT.simps
thf(fact_78_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, S3 : dB, Ss2 : list_dB]: ((A2 = (foldl_dB_dB @ app @ (app @ (abs @ R2) @ S3) @ Ss2)) => ((it @ (foldl_dB_dB @ app @ (subst @ R2 @ S3 @ zero_zero_nat) @ Ss2)) => (~ ((it @ S3)))))))))))))). % IT.cases
thf(fact_79_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_80_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_81_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_82_length__map, axiom,
    ((![F : dB > nat, Xs : list_dB]: ((size_size_list_nat @ (map_dB_nat @ F @ Xs)) = (size_size_list_dB @ Xs))))). % length_map
thf(fact_83_length__map, axiom,
    ((![F : dB > dB, Xs : list_dB]: ((size_size_list_dB @ (map_dB_dB @ F @ Xs)) = (size_size_list_dB @ Xs))))). % length_map
thf(fact_84_dB_Oinject_I3_J, axiom,
    ((![X3 : dB, Y3 : dB]: (((abs @ X3) = (abs @ Y3)) = (X3 = Y3))))). % dB.inject(3)
thf(fact_85_length__0__conv, axiom,
    ((![Xs : list_nat]: (((size_size_list_nat @ Xs) = zero_zero_nat) = (Xs = nil_nat))))). % length_0_conv
thf(fact_86_length__0__conv, axiom,
    ((![Xs : list_dB]: (((size_size_list_dB @ Xs) = zero_zero_nat) = (Xs = nil_dB))))). % length_0_conv
thf(fact_87_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_88_map__eq__imp__length__eq, axiom,
    ((![F : dB > dB, Xs : list_dB, G : dB > dB, Ys : list_dB]: (((map_dB_dB @ F @ Xs) = (map_dB_dB @ G @ Ys)) => ((size_size_list_dB @ Xs) = (size_size_list_dB @ Ys)))))). % map_eq_imp_length_eq
thf(fact_89_map__eq__imp__length__eq, axiom,
    ((![F : dB > nat, Xs : list_dB, G : dB > nat, Ys : list_dB]: (((map_dB_nat @ F @ Xs) = (map_dB_nat @ G @ Ys)) => ((size_size_list_dB @ Xs) = (size_size_list_dB @ Ys)))))). % map_eq_imp_length_eq
thf(fact_90_dB_Odistinct_I5_J, axiom,
    ((![X21 : dB, X22 : dB, X3 : dB]: (~ (((app @ X21 @ X22) = (abs @ X3))))))). % dB.distinct(5)
thf(fact_91_dB_Odistinct_I3_J, axiom,
    ((![X1 : nat, X3 : dB]: (~ (((var @ X1) = (abs @ X3))))))). % dB.distinct(3)
thf(fact_92_Lambda, axiom,
    ((![R : dB]: ((it @ R) => (it @ (abs @ R)))))). % Lambda
thf(fact_93_list_Osize_I3_J, axiom,
    (((size_size_list_nat @ nil_nat) = zero_zero_nat))). % list.size(3)
thf(fact_94_list_Osize_I3_J, axiom,
    (((size_size_list_dB @ nil_dB) = zero_zero_nat))). % list.size(3)
thf(fact_95_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_96_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_97_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_98_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_99_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_100_lem, axiom,
    ((![P : dB > $o, T : dB, N : nat]: ((![N3 : nat, Ts2 : list_dB]: ((![X5 : dB]: ((member_dB2 @ X5 @ (set_dB2 @ Ts2)) => (P @ X5))) => (P @ (foldl_dB_dB @ app @ (var @ N3) @ Ts2)))) => ((![U2 : dB]: ((P @ U2) => (![Ts2 : list_dB]: ((![X5 : dB]: ((member_dB2 @ X5 @ (set_dB2 @ Ts2)) => (P @ X5))) => (P @ (foldl_dB_dB @ app @ (abs @ U2) @ Ts2)))))) => (((size_size_dB @ T) = N) => (P @ T))))))). % lem
thf(fact_101_size__apps, axiom,
    ((![R : dB, Rs : list_dB]: ((size_size_dB @ (foldl_dB_dB @ app @ R @ Rs)) = (plus_plus_nat @ (plus_plus_nat @ (size_size_dB @ R) @ (foldl_nat_nat @ plus_plus_nat @ zero_zero_nat @ (map_dB_nat @ size_size_dB @ Rs))) @ (size_size_list_dB @ Rs)))))). % size_apps
thf(fact_102_Apps__dB__induct, axiom,
    ((![P : dB > $o, T : dB]: ((![N3 : nat, Ts2 : list_dB]: ((![X5 : dB]: ((member_dB2 @ X5 @ (set_dB2 @ Ts2)) => (P @ X5))) => (P @ (foldl_dB_dB @ app @ (var @ N3) @ Ts2)))) => ((![U2 : dB]: ((P @ U2) => (![Ts2 : list_dB]: ((![X5 : dB]: ((member_dB2 @ X5 @ (set_dB2 @ Ts2)) => (P @ X5))) => (P @ (foldl_dB_dB @ app @ (abs @ U2) @ Ts2)))))) => (P @ T)))))). % Apps_dB_induct
thf(fact_103_beta__cases_I3_J, axiom,
    ((![S : dB, T : dB, U : dB]: ((beta @ (app @ S @ T) @ U) => ((![S3 : dB]: ((S = (abs @ S3)) => (~ ((U = (subst @ S3 @ T @ zero_zero_nat)))))) => ((![T3 : dB]: ((U = (app @ T3 @ T)) => (~ ((beta @ S @ T3))))) => (~ ((![T3 : dB]: ((U = (app @ S @ T3)) => (~ ((beta @ T @ T3))))))))))))). % beta_cases(3)
thf(fact_104_beta, axiom,
    ((![S : dB, T : dB]: (beta @ (app @ (abs @ S) @ T) @ (subst @ S @ T @ zero_zero_nat))))). % beta
thf(fact_105_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_106_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_107_zero__eq__add__iff__both__eq__0, axiom,
    ((![X2 : nat, Y : nat]: ((zero_zero_nat = (plus_plus_nat @ X2 @ Y)) = (((X2 = zero_zero_nat)) & ((Y = zero_zero_nat))))))). % zero_eq_add_iff_both_eq_0
thf(fact_108_add__eq__0__iff__both__eq__0, axiom,
    ((![X2 : nat, Y : nat]: (((plus_plus_nat @ X2 @ Y) = zero_zero_nat) = (((X2 = zero_zero_nat)) & ((Y = zero_zero_nat))))))). % add_eq_0_iff_both_eq_0
thf(fact_109_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_110_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_111_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_112_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_113_add_Oright__neutral, axiom,
    ((![A2 : nat]: ((plus_plus_nat @ A2 @ zero_zero_nat) = A2)))). % add.right_neutral
thf(fact_114_add_Oleft__neutral, axiom,
    ((![A2 : nat]: ((plus_plus_nat @ zero_zero_nat @ A2) = A2)))). % add.left_neutral
thf(fact_115_Nat_Oadd__0__right, axiom,
    ((![M : nat]: ((plus_plus_nat @ M @ zero_zero_nat) = M)))). % Nat.add_0_right
thf(fact_116_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_117_map__eq__conv, axiom,
    ((![F : dB > dB, Xs : list_dB, G : dB > dB]: (((map_dB_dB @ F @ Xs) = (map_dB_dB @ G @ Xs)) = (![X : dB]: (((member_dB2 @ X @ (set_dB2 @ Xs))) => (((F @ X) = (G @ X))))))))). % map_eq_conv
thf(fact_118_map__eq__conv, axiom,
    ((![F : dB > nat, Xs : list_dB, G : dB > nat]: (((map_dB_nat @ F @ Xs) = (map_dB_nat @ G @ Xs)) = (![X : dB]: (((member_dB2 @ X @ (set_dB2 @ Xs))) => (((F @ X) = (G @ X))))))))). % map_eq_conv
thf(fact_119_in__listspI, axiom,
    ((![Xs : list_dB, A : dB > $o]: ((![X4 : dB]: ((member_dB2 @ X4 @ (set_dB2 @ Xs)) => (A @ X4))) => (listsp_dB @ A @ Xs))))). % in_listspI
thf(fact_120_count__notin, axiom,
    ((![X2 : dB, Xs : list_dB]: ((~ ((member_dB2 @ X2 @ (set_dB2 @ Xs)))) => ((count_list_dB @ Xs @ X2) = zero_zero_nat))))). % count_notin
thf(fact_121_neq__if__length__neq, axiom,
    ((![Xs : list_dB, Ys : list_dB]: ((~ (((size_size_list_dB @ Xs) = (size_size_list_dB @ Ys)))) => (~ ((Xs = Ys))))))). % neq_if_length_neq
thf(fact_122_Ex__list__of__length, axiom,
    ((![N : nat]: (?[Xs3 : list_dB]: ((size_size_list_dB @ Xs3) = N))))). % Ex_list_of_length
thf(fact_123_comm__monoid__add__class_Oadd__0, axiom,
    ((![A2 : nat]: ((plus_plus_nat @ zero_zero_nat @ A2) = A2)))). % comm_monoid_add_class.add_0
thf(fact_124_add_Ocomm__neutral, axiom,
    ((![A2 : nat]: ((plus_plus_nat @ A2 @ zero_zero_nat) = A2)))). % add.comm_neutral
thf(fact_125_map__ext, axiom,
    ((![Xs : list_dB, F : dB > dB, G : dB > dB]: ((![X4 : dB]: ((member_dB2 @ X4 @ (set_dB2 @ Xs)) => ((F @ X4) = (G @ X4)))) => ((map_dB_dB @ F @ Xs) = (map_dB_dB @ G @ Xs)))))). % map_ext
thf(fact_126_map__ext, axiom,
    ((![Xs : list_dB, F : dB > nat, G : dB > nat]: ((![X4 : dB]: ((member_dB2 @ X4 @ (set_dB2 @ Xs)) => ((F @ X4) = (G @ X4)))) => ((map_dB_nat @ F @ Xs) = (map_dB_nat @ G @ Xs)))))). % map_ext
thf(fact_127_map__idI, axiom,
    ((![Xs : list_dB, F : dB > dB]: ((![X4 : dB]: ((member_dB2 @ X4 @ (set_dB2 @ Xs)) => ((F @ X4) = X4))) => ((map_dB_dB @ F @ Xs) = Xs))))). % map_idI
thf(fact_128_map__cong, axiom,
    ((![Xs : list_dB, Ys : list_dB, F : dB > dB, G : dB > dB]: ((Xs = Ys) => ((![X4 : dB]: ((member_dB2 @ X4 @ (set_dB2 @ Ys)) => ((F @ X4) = (G @ X4)))) => ((map_dB_dB @ F @ Xs) = (map_dB_dB @ G @ Ys))))))). % map_cong
thf(fact_129_map__cong, axiom,
    ((![Xs : list_dB, Ys : list_dB, F : dB > nat, G : dB > nat]: ((Xs = Ys) => ((![X4 : dB]: ((member_dB2 @ X4 @ (set_dB2 @ Ys)) => ((F @ X4) = (G @ X4)))) => ((map_dB_nat @ F @ Xs) = (map_dB_nat @ G @ Ys))))))). % map_cong
thf(fact_130_ex__map__conv, axiom,
    ((![Ys : list_nat, F : dB > nat]: ((?[Xs2 : list_dB]: (Ys = (map_dB_nat @ F @ Xs2))) = (![X : nat]: (((member_nat2 @ X @ (set_nat2 @ Ys))) => ((?[Y2 : dB]: (X = (F @ Y2)))))))))). % ex_map_conv
thf(fact_131_ex__map__conv, axiom,
    ((![Ys : list_dB, F : dB > dB]: ((?[Xs2 : list_dB]: (Ys = (map_dB_dB @ F @ Xs2))) = (![X : dB]: (((member_dB2 @ X @ (set_dB2 @ Ys))) => ((?[Y2 : dB]: (X = (F @ Y2)))))))))). % ex_map_conv
thf(fact_132_list_Omap__cong, axiom,
    ((![X2 : list_dB, Ya : list_dB, F : dB > dB, G : dB > dB]: ((X2 = Ya) => ((![Z : dB]: ((member_dB2 @ Z @ (set_dB2 @ Ya)) => ((F @ Z) = (G @ Z)))) => ((map_dB_dB @ F @ X2) = (map_dB_dB @ G @ Ya))))))). % list.map_cong
thf(fact_133_list_Omap__cong, axiom,
    ((![X2 : list_dB, Ya : list_dB, F : dB > nat, G : dB > nat]: ((X2 = Ya) => ((![Z : dB]: ((member_dB2 @ Z @ (set_dB2 @ Ya)) => ((F @ Z) = (G @ Z)))) => ((map_dB_nat @ F @ X2) = (map_dB_nat @ G @ Ya))))))). % list.map_cong
thf(fact_134_list_Omap__cong0, axiom,
    ((![X2 : list_dB, F : dB > dB, G : dB > dB]: ((![Z : dB]: ((member_dB2 @ Z @ (set_dB2 @ X2)) => ((F @ Z) = (G @ Z)))) => ((map_dB_dB @ F @ X2) = (map_dB_dB @ G @ X2)))))). % list.map_cong0
thf(fact_135_list_Omap__cong0, axiom,
    ((![X2 : list_dB, F : dB > nat, G : dB > nat]: ((![Z : dB]: ((member_dB2 @ Z @ (set_dB2 @ X2)) => ((F @ Z) = (G @ Z)))) => ((map_dB_nat @ F @ X2) = (map_dB_nat @ G @ X2)))))). % list.map_cong0
thf(fact_136_list_Oinj__map__strong, axiom,
    ((![X2 : list_dB, Xa : list_dB, F : dB > dB, Fa : dB > dB]: ((![Z : dB, Za : dB]: ((member_dB2 @ Z @ (set_dB2 @ X2)) => ((member_dB2 @ Za @ (set_dB2 @ Xa)) => (((F @ Z) = (Fa @ Za)) => (Z = Za))))) => (((map_dB_dB @ F @ X2) = (map_dB_dB @ Fa @ Xa)) => (X2 = Xa)))))). % list.inj_map_strong
thf(fact_137_list_Oinj__map__strong, axiom,
    ((![X2 : list_dB, Xa : list_dB, F : dB > nat, Fa : dB > nat]: ((![Z : dB, Za : dB]: ((member_dB2 @ Z @ (set_dB2 @ X2)) => ((member_dB2 @ Za @ (set_dB2 @ Xa)) => (((F @ Z) = (Fa @ Za)) => (Z = Za))))) => (((map_dB_nat @ F @ X2) = (map_dB_nat @ Fa @ Xa)) => (X2 = Xa)))))). % list.inj_map_strong
thf(fact_138_plus__nat_Oadd__0, axiom,
    ((![N : nat]: ((plus_plus_nat @ zero_zero_nat @ N) = N)))). % plus_nat.add_0
thf(fact_139_add__eq__self__zero, axiom,
    ((![M : nat, N : nat]: (((plus_plus_nat @ M @ N) = M) => (N = zero_zero_nat))))). % add_eq_self_zero
thf(fact_140_add__right__imp__eq, axiom,
    ((![B2 : nat, A2 : nat, C : nat]: (((plus_plus_nat @ B2 @ A2) = (plus_plus_nat @ C @ A2)) => (B2 = C))))). % add_right_imp_eq
thf(fact_141_add__left__imp__eq, axiom,
    ((![A2 : nat, B2 : nat, C : nat]: (((plus_plus_nat @ A2 @ B2) = (plus_plus_nat @ A2 @ C)) => (B2 = C))))). % add_left_imp_eq
thf(fact_142_add_Oleft__commute, axiom,
    ((![B2 : nat, A2 : nat, C : nat]: ((plus_plus_nat @ B2 @ (plus_plus_nat @ A2 @ C)) = (plus_plus_nat @ A2 @ (plus_plus_nat @ B2 @ C)))))). % add.left_commute
thf(fact_143_add_Ocommute, axiom,
    ((plus_plus_nat = (^[A3 : nat]: (^[B3 : nat]: (plus_plus_nat @ B3 @ A3)))))). % add.commute
thf(fact_144_add_Oassoc, axiom,
    ((![A2 : nat, B2 : nat, C : nat]: ((plus_plus_nat @ (plus_plus_nat @ A2 @ B2) @ C) = (plus_plus_nat @ A2 @ (plus_plus_nat @ B2 @ C)))))). % add.assoc
thf(fact_145_group__cancel_Oadd2, axiom,
    ((![B : nat, K : nat, B2 : nat, A2 : nat]: ((B = (plus_plus_nat @ K @ B2)) => ((plus_plus_nat @ A2 @ B) = (plus_plus_nat @ K @ (plus_plus_nat @ A2 @ B2))))))). % group_cancel.add2
thf(fact_146_group__cancel_Oadd1, axiom,
    ((![A : nat, K : nat, A2 : nat, B2 : nat]: ((A = (plus_plus_nat @ K @ A2)) => ((plus_plus_nat @ A @ B2) = (plus_plus_nat @ K @ (plus_plus_nat @ A2 @ B2))))))). % group_cancel.add1
thf(fact_147_add__mono__thms__linordered__semiring_I4_J, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: (((I = J) & (K = L)) => ((plus_plus_nat @ I @ K) = (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_semiring(4)
thf(fact_148_ab__semigroup__add__class_Oadd__ac_I1_J, axiom,
    ((![A2 : nat, B2 : nat, C : nat]: ((plus_plus_nat @ (plus_plus_nat @ A2 @ B2) @ C) = (plus_plus_nat @ A2 @ (plus_plus_nat @ B2 @ C)))))). % ab_semigroup_add_class.add_ac(1)
thf(fact_149_appL, axiom,
    ((![S : dB, T : dB, U : dB]: ((beta @ S @ T) => (beta @ (app @ S @ U) @ (app @ T @ U)))))). % appL
thf(fact_150_appR, axiom,
    ((![S : dB, T : dB, U : dB]: ((beta @ S @ T) => (beta @ (app @ U @ S) @ (app @ U @ T)))))). % appR
thf(fact_151_foldl__cong, axiom,
    ((![A2 : nat, B2 : nat, L : list_nat, K : list_nat, F : nat > nat > nat, G : nat > nat > nat]: ((A2 = B2) => ((L = K) => ((![A4 : nat, X4 : nat]: ((member_nat2 @ X4 @ (set_nat2 @ L)) => ((F @ A4 @ X4) = (G @ A4 @ X4)))) => ((foldl_nat_nat @ F @ A2 @ L) = (foldl_nat_nat @ G @ B2 @ K)))))))). % foldl_cong
thf(fact_152_foldl__cong, axiom,
    ((![A2 : dB, B2 : dB, L : list_dB, K : list_dB, F : dB > dB > dB, G : dB > dB > dB]: ((A2 = B2) => ((L = K) => ((![A4 : dB, X4 : dB]: ((member_dB2 @ X4 @ (set_dB2 @ L)) => ((F @ A4 @ X4) = (G @ A4 @ X4)))) => ((foldl_dB_dB @ F @ A2 @ L) = (foldl_dB_dB @ G @ B2 @ K)))))))). % foldl_cong
thf(fact_153_beta__cases_I1_J, axiom,
    ((![I : nat, T : dB]: (~ ((beta @ (var @ I) @ T)))))). % beta_cases(1)
thf(fact_154_abs, axiom,
    ((![S : dB, T : dB]: ((beta @ S @ T) => (beta @ (abs @ S) @ (abs @ T)))))). % abs
thf(fact_155_beta__cases_I2_J, axiom,
    ((![R : dB, S : dB]: ((beta @ (abs @ R) @ S) => (~ ((![T3 : dB]: ((S = (abs @ T3)) => (~ ((beta @ R @ T3))))))))))). % beta_cases(2)
thf(fact_156_in__listsp__conv__set, axiom,
    ((listsp_dB = (^[A5 : dB > $o]: (^[Xs2 : list_dB]: (![X : dB]: (((member_dB2 @ X @ (set_dB2 @ Xs2))) => ((A5 @ X))))))))). % in_listsp_conv_set
thf(fact_157_in__listspD, axiom,
    ((![A : dB > $o, Xs : list_dB]: ((listsp_dB @ A @ Xs) => (![X5 : dB]: ((member_dB2 @ X5 @ (set_dB2 @ Xs)) => (A @ X5))))))). % in_listspD
thf(fact_158_subst__preserves__beta, axiom,
    ((![R : dB, S : dB, T : dB, I : nat]: ((beta @ R @ S) => (beta @ (subst @ R @ T @ I) @ (subst @ S @ T @ I)))))). % subst_preserves_beta
thf(fact_159_lift__preserves__beta, axiom,
    ((![R : dB, S : dB, I : nat]: ((beta @ R @ S) => (beta @ (lift @ R @ I) @ (lift @ S @ I)))))). % lift_preserves_beta
thf(fact_160_list__ex1__iff, axiom,
    ((list_ex1_dB = (^[P2 : dB > $o]: (^[Xs2 : list_dB]: (?[X : dB]: (((((member_dB2 @ X @ (set_dB2 @ Xs2))) & ((P2 @ X)))) & ((![Y2 : dB]: (((((member_dB2 @ Y2 @ (set_dB2 @ Xs2))) & ((P2 @ Y2)))) => ((Y2 = X)))))))))))). % list_ex1_iff
thf(fact_161_in__set__member, axiom,
    ((![X2 : dB, Xs : list_dB]: ((member_dB2 @ X2 @ (set_dB2 @ Xs)) = (member_dB @ Xs @ X2))))). % in_set_member
thf(fact_162_beta_Oinducts, axiom,
    ((![X1 : dB, X24 : dB, P : dB > dB > $o]: ((beta @ X1 @ X24) => ((![S3 : dB, T3 : dB]: (P @ (app @ (abs @ S3) @ T3) @ (subst @ S3 @ T3 @ zero_zero_nat))) => ((![S3 : dB, T3 : dB, U2 : dB]: ((beta @ S3 @ T3) => ((P @ S3 @ T3) => (P @ (app @ S3 @ U2) @ (app @ T3 @ U2))))) => ((![S3 : dB, T3 : dB, U2 : dB]: ((beta @ S3 @ T3) => ((P @ S3 @ T3) => (P @ (app @ U2 @ S3) @ (app @ U2 @ T3))))) => ((![S3 : dB, T3 : dB]: ((beta @ S3 @ T3) => ((P @ S3 @ T3) => (P @ (abs @ S3) @ (abs @ T3))))) => (P @ X1 @ X24))))))))). % beta.inducts
thf(fact_163_beta_Osimps, axiom,
    ((beta = (^[A1 : dB]: (^[A22 : dB]: (((?[S2 : dB]: (?[T2 : dB]: (((A1 = (app @ (abs @ S2) @ T2))) & ((A22 = (subst @ S2 @ T2 @ zero_zero_nat))))))) | ((((?[S2 : dB]: (?[T2 : dB]: (?[U3 : dB]: (((A1 = (app @ S2 @ U3))) & ((((A22 = (app @ T2 @ U3))) & ((beta @ S2 @ T2))))))))) | ((((?[S2 : dB]: (?[T2 : dB]: (?[U3 : dB]: (((A1 = (app @ U3 @ S2))) & ((((A22 = (app @ U3 @ T2))) & ((beta @ S2 @ T2))))))))) | ((?[S2 : dB]: (?[T2 : dB]: (((A1 = (abs @ S2))) & ((((A22 = (abs @ T2))) & ((beta @ S2 @ T2)))))))))))))))))). % beta.simps
thf(fact_164_beta_Ocases, axiom,
    ((![A12 : dB, A23 : dB]: ((beta @ A12 @ A23) => ((![S3 : dB, T3 : dB]: ((A12 = (app @ (abs @ S3) @ T3)) => (~ ((A23 = (subst @ S3 @ T3 @ zero_zero_nat)))))) => ((![S3 : dB, T3 : dB, U2 : dB]: ((A12 = (app @ S3 @ U2)) => ((A23 = (app @ T3 @ U2)) => (~ ((beta @ S3 @ T3)))))) => ((![S3 : dB, T3 : dB, U2 : dB]: ((A12 = (app @ U2 @ S3)) => ((A23 = (app @ U2 @ T3)) => (~ ((beta @ S3 @ T3)))))) => (~ ((![S3 : dB]: ((A12 = (abs @ S3)) => (![T3 : dB]: ((A23 = (abs @ T3)) => (~ ((beta @ S3 @ T3)))))))))))))))). % beta.cases
thf(fact_165_apps__preserves__beta, axiom,
    ((![R : dB, S : dB, Ss : list_dB]: ((beta @ R @ S) => (beta @ (foldl_dB_dB @ app @ R @ Ss) @ (foldl_dB_dB @ app @ S @ Ss)))))). % apps_preserves_beta
thf(fact_166_can__select__set__list__ex1, axiom,
    ((![P : dB > $o, A : list_dB]: ((can_select_dB @ P @ (set_dB2 @ A)) = (list_ex1_dB @ P @ A))))). % can_select_set_list_ex1
thf(fact_167_Euclid__induct, axiom,
    ((![P : nat > nat > $o, A2 : nat, B2 : nat]: ((![A4 : nat, B4 : nat]: ((P @ A4 @ B4) = (P @ B4 @ A4))) => ((![A4 : nat]: (P @ A4 @ zero_zero_nat)) => ((![A4 : nat, B4 : nat]: ((P @ A4 @ B4) => (P @ A4 @ (plus_plus_nat @ A4 @ B4)))) => (P @ A2 @ B2))))))). % Euclid_induct
thf(fact_168_verit__sum__simplify, axiom,
    ((![A2 : nat]: ((plus_plus_nat @ A2 @ zero_zero_nat) = A2)))). % verit_sum_simplify
thf(fact_169_add__0__iff, axiom,
    ((![B2 : nat, A2 : nat]: ((B2 = (plus_plus_nat @ B2 @ A2)) = (A2 = zero_zero_nat))))). % add_0_iff
thf(fact_170_subst__Abs, axiom,
    ((![T : dB, S : dB, K : nat]: ((subst @ (abs @ T) @ S @ K) = (abs @ (subst @ T @ (lift @ S @ zero_zero_nat) @ (plus_plus_nat @ K @ one_one_nat))))))). % subst_Abs
thf(fact_171_one__natural_Orsp, axiom,
    ((one_one_nat = one_one_nat))). % one_natural.rsp

% Conjectures (1)
thf(conj_0, conjecture,
    ((listsp_dB @ it @ (map_dB_dB @ (^[T2 : dB]: (lift @ T2 @ zero_zero_nat)) @ (map_dB_dB @ (^[T2 : dB]: (subst @ T2 @ u @ i)) @ nil_dB))))).
