% 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_170__5211130_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:38:04.144

% 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 (40)
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_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_OCons_001t__Lambda__OdB, type,
    cons_dB : dB > list_dB > list_dB).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat, type,
    cons_nat : nat > list_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__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_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_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_Nat_OSuc, type,
    suc : nat > nat).
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_nat : nat > set_nat > $o).
thf(sy_v_a____, type,
    a : dB).
thf(sy_v_b____, type,
    b : dB).
thf(sy_v_bs____, type,
    bs : list_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 (156)
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__092_060open_062listsp_AIT_A_Ilift_A_Ib_091u_Pi_093_J_A0_A_D_Amap_A_I_092_060lambda_062t_O_Alift_At_A0_J_A_Imap_A_I_092_060lambda_062t_O_At_091u_Pi_093_J_Abs_J_J_092_060close_062, axiom,
    ((listsp_dB @ it @ (cons_dB @ (lift @ (subst @ b @ u @ i) @ zero_zero_nat) @ (map_dB_dB @ (^[T : dB]: (lift @ T @ zero_zero_nat)) @ (map_dB_dB @ (^[T : dB]: (subst @ T @ u @ i)) @ bs)))))). % \<open>listsp IT (lift (b[u/i]) 0 # map (\<lambda>t. lift t 0) (map (\<lambda>t. t[u/i]) bs))\<close>
thf(fact_5_Cons_Ohyps_I3_J, axiom,
    ((listsp_dB @ it @ (map_dB_dB @ (^[T : dB]: (lift @ T @ zero_zero_nat)) @ (map_dB_dB @ (^[T : dB]: (subst @ T @ u @ i)) @ bs))))). % Cons.hyps(3)
thf(fact_6__092_060open_062IT_A_Ib_091u_Pi_093_J_092_060close_062, axiom,
    ((it @ (subst @ b @ u @ i)))). % \<open>IT (b[u/i])\<close>
thf(fact_7_lift__IT, axiom,
    ((![T2 : dB, I : nat]: ((it @ T2) => (it @ (lift @ T2 @ I)))))). % lift_IT
thf(fact_8__092_060open_062IT_A_Ilift_A_Ib_091u_Pi_093_J_A0_J_092_060close_062, axiom,
    ((it @ (lift @ (subst @ b @ u @ i) @ zero_zero_nat)))). % \<open>IT (lift (b[u/i]) 0)\<close>
thf(fact_9_lifts__IT, axiom,
    ((![Ts : list_dB]: ((listsp_dB @ it @ Ts) => (listsp_dB @ it @ (map_dB_dB @ (^[T : dB]: (lift @ T @ zero_zero_nat)) @ Ts)))))). % lifts_IT
thf(fact_10_subst__lift, axiom,
    ((![T2 : dB, K : nat, S : dB]: ((subst @ (lift @ T2 @ K) @ S @ K) = T2)))). % subst_lift
thf(fact_11_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_12_map__ident, axiom,
    (((map_dB_dB @ (^[X : dB]: X)) = (^[Xs : list_dB]: Xs)))). % map_ident
thf(fact_13_list_Oinject, axiom,
    ((![X21 : dB, X22 : list_dB, Y21 : dB, Y22 : list_dB]: (((cons_dB @ X21 @ X22) = (cons_dB @ Y21 @ Y22)) = (((X21 = Y21)) & ((X22 = Y22))))))). % list.inject
thf(fact_14_listsp__simps_I2_J, axiom,
    ((![A : dB > $o, X2 : dB, Xs2 : list_dB]: ((listsp_dB @ A @ (cons_dB @ X2 @ Xs2)) = (((A @ X2)) & ((listsp_dB @ A @ Xs2))))))). % listsp_simps(2)
thf(fact_15_listspE, axiom,
    ((![A : dB > $o, X2 : dB, L : list_dB]: ((listsp_dB @ A @ (cons_dB @ X2 @ L)) => (~ (((A @ X2) => (~ ((listsp_dB @ A @ L)))))))))). % listspE
thf(fact_16_listsp_OCons, axiom,
    ((![A : dB > $o, A2 : dB, L : list_dB]: ((A @ A2) => ((listsp_dB @ A @ L) => (listsp_dB @ A @ (cons_dB @ A2 @ L))))))). % listsp.Cons
thf(fact_17_list_Osimps_I9_J, axiom,
    ((![F : dB > nat, X21 : dB, X22 : list_dB]: ((map_dB_nat @ F @ (cons_dB @ X21 @ X22)) = (cons_nat @ (F @ X21) @ (map_dB_nat @ F @ X22)))))). % list.simps(9)
thf(fact_18_list_Osimps_I9_J, axiom,
    ((![F : dB > dB, X21 : dB, X22 : list_dB]: ((map_dB_dB @ F @ (cons_dB @ X21 @ X22)) = (cons_dB @ (F @ X21) @ (map_dB_dB @ F @ X22)))))). % list.simps(9)
thf(fact_19_Cons__eq__map__D, axiom,
    ((![X2 : nat, Xs2 : list_nat, F : dB > nat, Ys : list_dB]: (((cons_nat @ X2 @ Xs2) = (map_dB_nat @ F @ Ys)) => (?[Z : dB, Zs : list_dB]: ((Ys = (cons_dB @ Z @ Zs)) & ((X2 = (F @ Z)) & (Xs2 = (map_dB_nat @ F @ Zs))))))))). % Cons_eq_map_D
thf(fact_20_Cons__eq__map__D, axiom,
    ((![X2 : dB, Xs2 : list_dB, F : dB > dB, Ys : list_dB]: (((cons_dB @ X2 @ Xs2) = (map_dB_dB @ F @ Ys)) => (?[Z : dB, Zs : list_dB]: ((Ys = (cons_dB @ Z @ Zs)) & ((X2 = (F @ Z)) & (Xs2 = (map_dB_dB @ F @ Zs))))))))). % Cons_eq_map_D
thf(fact_21_map__eq__Cons__D, axiom,
    ((![F : dB > nat, Xs2 : list_dB, Y : nat, Ys : list_nat]: (((map_dB_nat @ F @ Xs2) = (cons_nat @ Y @ Ys)) => (?[Z : dB, Zs : list_dB]: ((Xs2 = (cons_dB @ Z @ Zs)) & (((F @ Z) = Y) & ((map_dB_nat @ F @ Zs) = Ys)))))))). % map_eq_Cons_D
thf(fact_22_map__eq__Cons__D, axiom,
    ((![F : dB > dB, Xs2 : list_dB, Y : dB, Ys : list_dB]: (((map_dB_dB @ F @ Xs2) = (cons_dB @ Y @ Ys)) => (?[Z : dB, Zs : list_dB]: ((Xs2 = (cons_dB @ Z @ Zs)) & (((F @ Z) = Y) & ((map_dB_dB @ F @ Zs) = Ys)))))))). % map_eq_Cons_D
thf(fact_23_Cons__eq__map__conv, axiom,
    ((![X2 : nat, Xs2 : list_nat, F : dB > nat, Ys : list_dB]: (((cons_nat @ X2 @ Xs2) = (map_dB_nat @ F @ Ys)) = (?[Z2 : dB]: (?[Zs2 : list_dB]: (((Ys = (cons_dB @ Z2 @ Zs2))) & ((((X2 = (F @ Z2))) & ((Xs2 = (map_dB_nat @ F @ Zs2)))))))))))). % Cons_eq_map_conv
thf(fact_24_Cons__eq__map__conv, axiom,
    ((![X2 : dB, Xs2 : list_dB, F : dB > dB, Ys : list_dB]: (((cons_dB @ X2 @ Xs2) = (map_dB_dB @ F @ Ys)) = (?[Z2 : dB]: (?[Zs2 : list_dB]: (((Ys = (cons_dB @ Z2 @ Zs2))) & ((((X2 = (F @ Z2))) & ((Xs2 = (map_dB_dB @ F @ Zs2)))))))))))). % Cons_eq_map_conv
thf(fact_25_zero__reorient, axiom,
    ((![X2 : nat]: ((zero_zero_nat = X2) = (X2 = zero_zero_nat))))). % zero_reorient
thf(fact_26_not__Cons__self2, axiom,
    ((![X2 : dB, Xs2 : list_dB]: (~ (((cons_dB @ X2 @ Xs2) = Xs2)))))). % not_Cons_self2
thf(fact_27_list_Omap__ident, axiom,
    ((![T2 : list_dB]: ((map_dB_dB @ (^[X : dB]: X) @ T2) = T2)))). % list.map_ident
thf(fact_28_map__eq__Cons__conv, axiom,
    ((![F : dB > nat, Xs2 : list_dB, Y : nat, Ys : list_nat]: (((map_dB_nat @ F @ Xs2) = (cons_nat @ Y @ Ys)) = (?[Z2 : dB]: (?[Zs2 : list_dB]: (((Xs2 = (cons_dB @ Z2 @ Zs2))) & (((((F @ Z2) = Y)) & (((map_dB_nat @ F @ Zs2) = Ys))))))))))). % map_eq_Cons_conv
thf(fact_29_map__eq__Cons__conv, axiom,
    ((![F : dB > dB, Xs2 : list_dB, Y : dB, Ys : list_dB]: (((map_dB_dB @ F @ Xs2) = (cons_dB @ Y @ Ys)) = (?[Z2 : dB]: (?[Zs2 : list_dB]: (((Xs2 = (cons_dB @ Z2 @ Zs2))) & (((((F @ Z2) = Y)) & (((map_dB_dB @ F @ Zs2) = Ys))))))))))). % map_eq_Cons_conv
thf(fact_30__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_31_map__eq__map__tailrec, axiom,
    ((map_dB_dB = map_tailrec_dB_dB))). % map_eq_map_tailrec
thf(fact_32_map__eq__map__tailrec, axiom,
    ((map_dB_nat = map_tailrec_dB_nat))). % map_eq_map_tailrec
thf(fact_33_substn__subst__0, axiom,
    ((![T2 : dB, S : dB]: ((substn @ T2 @ S @ zero_zero_nat) = (subst @ T2 @ S @ zero_zero_nat))))). % substn_subst_0
thf(fact_34__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_35_zero__natural_Orsp, axiom,
    ((zero_zero_nat = zero_zero_nat))). % zero_natural.rsp
thf(fact_36_member__rec_I1_J, axiom,
    ((![X2 : dB, Xs2 : list_dB, Y : dB]: ((member_dB @ (cons_dB @ X2 @ Xs2) @ Y) = (((X2 = Y)) | ((member_dB @ Xs2 @ Y))))))). % member_rec(1)
thf(fact_37_liftn__0, axiom,
    ((![T2 : dB, K : nat]: ((liftn @ zero_zero_nat @ T2 @ K) = T2)))). % liftn_0
thf(fact_38_dB_Oinject_I2_J, axiom,
    ((![X21 : dB, X22 : dB, Y21 : dB, Y22 : dB]: (((app @ X21 @ X22) = (app @ Y21 @ Y22)) = (((X21 = Y21)) & ((X22 = Y22))))))). % dB.inject(2)
thf(fact_39_dB_Oinject_I1_J, axiom,
    ((![X1 : nat, Y1 : nat]: (((var @ X1) = (var @ Y1)) = (X1 = Y1))))). % dB.inject(1)
thf(fact_40_subst__eq, axiom,
    ((![K : nat, U : dB]: ((subst @ (var @ K) @ U @ K) = U)))). % subst_eq
thf(fact_41_substn__subst__n, axiom,
    ((substn = (^[T : dB]: (^[S2 : dB]: (^[N : nat]: (subst @ T @ (liftn @ N @ S2 @ zero_zero_nat) @ N))))))). % substn_subst_n
thf(fact_42_dB_Odistinct_I1_J, axiom,
    ((![X1 : nat, X21 : dB, X22 : dB]: (~ (((var @ X1) = (app @ X21 @ X22))))))). % dB.distinct(1)
thf(fact_43_liftn_Osimps_I2_J, axiom,
    ((![N2 : nat, S : dB, T2 : dB, K : nat]: ((liftn @ N2 @ (app @ S @ T2) @ K) = (app @ (liftn @ N2 @ S @ K) @ (liftn @ N2 @ T2 @ K)))))). % liftn.simps(2)
thf(fact_44_substn_Osimps_I2_J, axiom,
    ((![T2 : dB, U : dB, S : dB, K : nat]: ((substn @ (app @ T2 @ U) @ S @ K) = (app @ (substn @ T2 @ S @ K) @ (substn @ U @ S @ K)))))). % substn.simps(2)
thf(fact_45_app__Var__IT, axiom,
    ((![T2 : dB, I : nat]: ((it @ T2) => (it @ (app @ T2 @ (var @ I))))))). % app_Var_IT
thf(fact_46_subst__App, axiom,
    ((![T2 : dB, U : dB, S : dB, K : nat]: ((subst @ (app @ T2 @ U) @ S @ K) = (app @ (subst @ T2 @ S @ K) @ (subst @ U @ S @ K)))))). % subst_App
thf(fact_47_lift_Osimps_I2_J, axiom,
    ((![S : dB, T2 : dB, K : nat]: ((lift @ (app @ S @ T2) @ K) = (app @ (lift @ S @ K) @ (lift @ T2 @ K)))))). % lift.simps(2)
thf(fact_48_Var__IT, axiom,
    ((![N2 : nat]: (it @ (var @ N2))))). % Var_IT
thf(fact_49_subst__Var__IT, axiom,
    ((![R : dB, I : nat, J : nat]: ((it @ R) => (it @ (subst @ R @ (var @ I) @ J)))))). % subst_Var_IT
thf(fact_50_dB_Osize__gen_I1_J, axiom,
    ((![X1 : nat]: ((size_dB @ (var @ X1)) = zero_zero_nat)))). % dB.size_gen(1)
thf(fact_51_dB_Osize_I4_J, axiom,
    ((![X1 : nat]: ((size_size_dB @ (var @ X1)) = zero_zero_nat)))). % dB.size(4)
thf(fact_52_liftn__lift, axiom,
    ((![N2 : nat, T2 : dB, K : nat]: ((liftn @ (suc @ N2) @ T2 @ K) = (lift @ (liftn @ N2 @ T2 @ K) @ K))))). % liftn_lift
thf(fact_53_IT_OVar, axiom,
    ((![Rs : list_dB, N2 : nat]: ((listsp_dB @ it @ Rs) => (it @ (foldl_dB_dB @ app @ (var @ N2) @ Rs)))))). % IT.Var
thf(fact_54_lift__map, axiom,
    ((![T2 : dB, Ts : list_dB, I : nat]: ((lift @ (foldl_dB_dB @ app @ T2 @ Ts) @ I) = (foldl_dB_dB @ app @ (lift @ T2 @ I) @ (map_dB_dB @ (^[T : dB]: (lift @ T @ I)) @ Ts)))))). % lift_map
thf(fact_55_subst__map, axiom,
    ((![T2 : dB, Ts : list_dB, U : dB, I : nat]: ((subst @ (foldl_dB_dB @ app @ T2 @ Ts) @ U @ I) = (foldl_dB_dB @ app @ (subst @ T2 @ U @ I) @ (map_dB_dB @ (^[T : dB]: (subst @ T @ U @ I)) @ Ts)))))). % subst_map
thf(fact_56_old_Onat_Oinject, axiom,
    ((![Nat : nat, Nat2 : nat]: (((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))). % old.nat.inject
thf(fact_57_nat_Oinject, axiom,
    ((![X23 : nat, Y2 : nat]: (((suc @ X23) = (suc @ Y2)) = (X23 = Y2))))). % nat.inject
thf(fact_58_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_59_Var__apps__eq__Var__apps__conv, axiom,
    ((![M : nat, Rs : list_dB, N2 : nat, Ss : list_dB]: (((foldl_dB_dB @ app @ (var @ M) @ Rs) = (foldl_dB_dB @ app @ (var @ N2) @ Ss)) = (((M = N2)) & ((Rs = Ss))))))). % Var_apps_eq_Var_apps_conv
thf(fact_60_n__not__Suc__n, axiom,
    ((![N2 : nat]: (~ ((N2 = (suc @ N2))))))). % n_not_Suc_n
thf(fact_61_Suc__inject, axiom,
    ((![X2 : nat, Y : nat]: (((suc @ X2) = (suc @ Y)) => (X2 = Y))))). % Suc_inject
thf(fact_62_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_63_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_64_nat_Odistinct_I1_J, axiom,
    ((![X23 : nat]: (~ ((zero_zero_nat = (suc @ X23))))))). % nat.distinct(1)
thf(fact_65_old_Onat_Odistinct_I2_J, axiom,
    ((![Nat2 : nat]: (~ (((suc @ Nat2) = zero_zero_nat)))))). % old.nat.distinct(2)
thf(fact_66_old_Onat_Odistinct_I1_J, axiom,
    ((![Nat2 : nat]: (~ ((zero_zero_nat = (suc @ Nat2))))))). % old.nat.distinct(1)
thf(fact_67_nat_OdiscI, axiom,
    ((![Nat : nat, X23 : nat]: ((Nat = (suc @ X23)) => (~ ((Nat = zero_zero_nat))))))). % nat.discI
thf(fact_68_nat__induct, axiom,
    ((![P : nat > $o, N2 : nat]: ((P @ zero_zero_nat) => ((![N3 : nat]: ((P @ N3) => (P @ (suc @ N3)))) => (P @ N2)))))). % nat_induct
thf(fact_69_diff__induct, axiom,
    ((![P : nat > nat > $o, M : nat, N2 : nat]: ((![X3 : nat]: (P @ X3 @ zero_zero_nat)) => ((![Y3 : nat]: (P @ zero_zero_nat @ (suc @ Y3))) => ((![X3 : nat, Y3 : nat]: ((P @ X3 @ Y3) => (P @ (suc @ X3) @ (suc @ Y3)))) => (P @ M @ N2))))))). % diff_induct
thf(fact_70_zero__induct, axiom,
    ((![P : nat > $o, K : nat]: ((P @ K) => ((![N3 : nat]: ((P @ (suc @ N3)) => (P @ N3))) => (P @ zero_zero_nat)))))). % zero_induct
thf(fact_71_Suc__neq__Zero, axiom,
    ((![M : nat]: (~ (((suc @ M) = zero_zero_nat)))))). % Suc_neq_Zero
thf(fact_72_Zero__neq__Suc, axiom,
    ((![M : nat]: (~ ((zero_zero_nat = (suc @ M))))))). % Zero_neq_Suc
thf(fact_73_Zero__not__Suc, axiom,
    ((![M : nat]: (~ ((zero_zero_nat = (suc @ M))))))). % Zero_not_Suc
thf(fact_74_old_Onat_Oexhaust, axiom,
    ((![Y : nat]: ((~ ((Y = zero_zero_nat))) => (~ ((![Nat3 : nat]: (~ ((Y = (suc @ Nat3))))))))))). % old.nat.exhaust
thf(fact_75_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_76_not0__implies__Suc, axiom,
    ((![N2 : nat]: ((~ ((N2 = zero_zero_nat))) => (?[M2 : nat]: (N2 = (suc @ M2))))))). % not0_implies_Suc
thf(fact_77_foldl__Cons, axiom,
    ((![F : nat > nat > nat, A2 : nat, X2 : nat, Xs2 : list_nat]: ((foldl_nat_nat @ F @ A2 @ (cons_nat @ X2 @ Xs2)) = (foldl_nat_nat @ F @ (F @ A2 @ X2) @ Xs2))))). % foldl_Cons
thf(fact_78_foldl__Cons, axiom,
    ((![F : dB > dB > dB, A2 : dB, X2 : dB, Xs2 : list_dB]: ((foldl_dB_dB @ F @ A2 @ (cons_dB @ X2 @ Xs2)) = (foldl_dB_dB @ F @ (F @ A2 @ X2) @ Xs2))))). % foldl_Cons
thf(fact_79_foldl__map, axiom,
    ((![G : nat > nat > nat, A2 : nat, F : nat > nat, Xs2 : list_nat]: ((foldl_nat_nat @ G @ A2 @ (map_nat_nat @ F @ Xs2)) = (foldl_nat_nat @ (^[A3 : nat]: (^[X : nat]: (G @ A3 @ (F @ X)))) @ A2 @ Xs2))))). % foldl_map
thf(fact_80_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_81_foldl__map, axiom,
    ((![G : dB > nat > dB, A2 : dB, F : dB > nat, Xs2 : list_dB]: ((foldl_dB_nat @ G @ A2 @ (map_dB_nat @ F @ Xs2)) = (foldl_dB_dB @ (^[A3 : dB]: (^[X : dB]: (G @ A3 @ (F @ X)))) @ A2 @ Xs2))))). % foldl_map
thf(fact_82_foldl__map, axiom,
    ((![G : nat > nat > nat, A2 : nat, F : dB > nat, Xs2 : list_dB]: ((foldl_nat_nat @ G @ A2 @ (map_dB_nat @ F @ Xs2)) = (foldl_nat_dB @ (^[A3 : nat]: (^[X : dB]: (G @ A3 @ (F @ X)))) @ A2 @ Xs2))))). % foldl_map
thf(fact_83_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_84_IT_Osimps, axiom,
    ((it = (^[A3 : dB]: (((?[Rs3 : list_dB]: (?[N : nat]: (((A3 = (foldl_dB_dB @ app @ (var @ N) @ 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_85_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_86_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_87_length__map, axiom,
    ((![F : dB > nat, Xs2 : list_dB]: ((size_size_list_nat @ (map_dB_nat @ F @ Xs2)) = (size_size_list_dB @ Xs2))))). % length_map
thf(fact_88_length__map, axiom,
    ((![F : dB > dB, Xs2 : list_dB]: ((size_size_list_dB @ (map_dB_dB @ F @ Xs2)) = (size_size_list_dB @ Xs2))))). % length_map
thf(fact_89_dB_Oinject_I3_J, axiom,
    ((![X32 : dB, Y32 : dB]: (((abs @ X32) = (abs @ Y32)) = (X32 = Y32))))). % dB.inject(3)
thf(fact_90_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_91_map__eq__imp__length__eq, axiom,
    ((![F : dB > dB, Xs2 : list_dB, G : dB > dB, Ys : list_dB]: (((map_dB_dB @ F @ Xs2) = (map_dB_dB @ G @ Ys)) => ((size_size_list_dB @ Xs2) = (size_size_list_dB @ Ys)))))). % map_eq_imp_length_eq
thf(fact_92_map__eq__imp__length__eq, axiom,
    ((![F : dB > nat, Xs2 : list_dB, G : dB > nat, Ys : list_dB]: (((map_dB_nat @ F @ Xs2) = (map_dB_nat @ G @ Ys)) => ((size_size_list_dB @ Xs2) = (size_size_list_dB @ Ys)))))). % map_eq_imp_length_eq
thf(fact_93_dB_Odistinct_I5_J, axiom,
    ((![X21 : dB, X22 : dB, X32 : dB]: (~ (((app @ X21 @ X22) = (abs @ X32))))))). % dB.distinct(5)
thf(fact_94_dB_Odistinct_I3_J, axiom,
    ((![X1 : nat, X32 : dB]: (~ (((var @ X1) = (abs @ X32))))))). % dB.distinct(3)
thf(fact_95_Lambda, axiom,
    ((![R : dB]: ((it @ R) => (it @ (abs @ R)))))). % Lambda
thf(fact_96_Suc__length__conv, axiom,
    ((![N2 : nat, Xs2 : list_dB]: (((suc @ N2) = (size_size_list_dB @ Xs2)) = (?[Y4 : dB]: (?[Ys2 : list_dB]: (((Xs2 = (cons_dB @ Y4 @ Ys2))) & (((size_size_list_dB @ Ys2) = N2))))))))). % Suc_length_conv
thf(fact_97_length__Suc__conv, axiom,
    ((![Xs2 : list_dB, N2 : nat]: (((size_size_list_dB @ Xs2) = (suc @ N2)) = (?[Y4 : dB]: (?[Ys2 : list_dB]: (((Xs2 = (cons_dB @ Y4 @ Ys2))) & (((size_size_list_dB @ Ys2) = N2))))))))). % length_Suc_conv
thf(fact_98_dB_Oinduct, axiom,
    ((![P : dB > $o, DB : dB]: ((![X3 : nat]: (P @ (var @ X3))) => ((![X1a : dB, X24 : dB]: ((P @ X1a) => ((P @ X24) => (P @ (app @ X1a @ X24))))) => ((![X3 : dB]: ((P @ X3) => (P @ (abs @ X3)))) => (P @ DB))))))). % dB.induct
thf(fact_99_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_100_ex__head__tail, axiom,
    ((![T2 : dB]: (?[Ts2 : list_dB, H : dB]: ((T2 = (foldl_dB_dB @ app @ H @ Ts2)) & ((?[N3 : nat]: (H = (var @ N3))) | (?[U2 : dB]: (H = (abs @ U2))))))))). % ex_head_tail
thf(fact_101_Abs__App__neq__Var__apps, axiom,
    ((![S : dB, T2 : dB, N2 : nat, Ss : list_dB]: (~ (((app @ (abs @ S) @ T2) = (foldl_dB_dB @ app @ (var @ N2) @ Ss))))))). % Abs_App_neq_Var_apps
thf(fact_102_Var__apps__neq__Abs__apps, axiom,
    ((![N2 : nat, Ts : list_dB, R : dB, Ss : list_dB]: (~ (((foldl_dB_dB @ app @ (var @ N2) @ Ts) = (foldl_dB_dB @ app @ (abs @ R) @ Ss))))))). % Var_apps_neq_Abs_apps
thf(fact_103_length__Cons, axiom,
    ((![X2 : dB, Xs2 : list_dB]: ((size_size_list_dB @ (cons_dB @ X2 @ Xs2)) = (suc @ (size_size_list_dB @ Xs2)))))). % length_Cons
thf(fact_104_lem, axiom,
    ((![P : dB > $o, T2 : dB, N2 : nat]: ((![N3 : nat, Ts2 : list_dB]: ((![X4 : dB]: ((member_dB2 @ X4 @ (set_dB2 @ Ts2)) => (P @ X4))) => (P @ (foldl_dB_dB @ app @ (var @ N3) @ Ts2)))) => ((![U2 : dB]: ((P @ U2) => (![Ts2 : list_dB]: ((![X4 : dB]: ((member_dB2 @ X4 @ (set_dB2 @ Ts2)) => (P @ X4))) => (P @ (foldl_dB_dB @ app @ (abs @ U2) @ Ts2)))))) => (((size_size_dB @ T2) = N2) => (P @ T2))))))). % lem
thf(fact_105_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_106_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_107_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_108_add_Oleft__neutral, axiom,
    ((![A2 : nat]: ((plus_plus_nat @ zero_zero_nat @ A2) = A2)))). % add.left_neutral
thf(fact_109_add_Oright__neutral, axiom,
    ((![A2 : nat]: ((plus_plus_nat @ A2 @ zero_zero_nat) = A2)))). % add.right_neutral
thf(fact_110_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_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__right__left, axiom,
    ((![A2 : nat, B2 : nat]: ((A2 = (plus_plus_nat @ B2 @ A2)) = (B2 = zero_zero_nat))))). % add_cancel_right_left
thf(fact_113_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_114_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_115_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_116_add__is__0, axiom,
    ((![M : nat, N2 : nat]: (((plus_plus_nat @ M @ N2) = zero_zero_nat) = (((M = zero_zero_nat)) & ((N2 = zero_zero_nat))))))). % add_is_0
thf(fact_117_Nat_Oadd__0__right, axiom,
    ((![M : nat]: ((plus_plus_nat @ M @ zero_zero_nat) = M)))). % Nat.add_0_right
thf(fact_118_add__Suc__right, axiom,
    ((![M : nat, N2 : nat]: ((plus_plus_nat @ M @ (suc @ N2)) = (suc @ (plus_plus_nat @ M @ N2)))))). % add_Suc_right
thf(fact_119_map__eq__conv, axiom,
    ((![F : dB > dB, Xs2 : list_dB, G : dB > dB]: (((map_dB_dB @ F @ Xs2) = (map_dB_dB @ G @ Xs2)) = (![X : dB]: (((member_dB2 @ X @ (set_dB2 @ Xs2))) => (((F @ X) = (G @ X))))))))). % map_eq_conv
thf(fact_120_map__eq__conv, axiom,
    ((![F : dB > nat, Xs2 : list_dB, G : dB > nat]: (((map_dB_nat @ F @ Xs2) = (map_dB_nat @ G @ Xs2)) = (![X : dB]: (((member_dB2 @ X @ (set_dB2 @ Xs2))) => (((F @ X) = (G @ X))))))))). % map_eq_conv
thf(fact_121_in__listspI, axiom,
    ((![Xs2 : list_dB, A : dB > $o]: ((![X3 : dB]: ((member_dB2 @ X3 @ (set_dB2 @ Xs2)) => (A @ X3))) => (listsp_dB @ A @ Xs2))))). % in_listspI
thf(fact_122_neq__if__length__neq, axiom,
    ((![Xs2 : list_dB, Ys : list_dB]: ((~ (((size_size_list_dB @ Xs2) = (size_size_list_dB @ Ys)))) => (~ ((Xs2 = Ys))))))). % neq_if_length_neq
thf(fact_123_Ex__list__of__length, axiom,
    ((![N2 : nat]: (?[Xs3 : list_dB]: ((size_size_list_dB @ Xs3) = N2))))). % Ex_list_of_length
thf(fact_124_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_125_add_Ocomm__neutral, axiom,
    ((![A2 : nat]: ((plus_plus_nat @ A2 @ zero_zero_nat) = A2)))). % add.comm_neutral
thf(fact_126_list_Oset__intros_I2_J, axiom,
    ((![Y : dB, X22 : list_dB, X21 : dB]: ((member_dB2 @ Y @ (set_dB2 @ X22)) => (member_dB2 @ Y @ (set_dB2 @ (cons_dB @ X21 @ X22))))))). % list.set_intros(2)
thf(fact_127_list_Oset__intros_I1_J, axiom,
    ((![X21 : dB, X22 : list_dB]: (member_dB2 @ X21 @ (set_dB2 @ (cons_dB @ X21 @ X22)))))). % list.set_intros(1)
thf(fact_128_set__ConsD, axiom,
    ((![Y : dB, X2 : dB, Xs2 : list_dB]: ((member_dB2 @ Y @ (set_dB2 @ (cons_dB @ X2 @ Xs2))) => ((Y = X2) | (member_dB2 @ Y @ (set_dB2 @ Xs2))))))). % set_ConsD
thf(fact_129_list_Oset__cases, axiom,
    ((![E : dB, A2 : list_dB]: ((member_dB2 @ E @ (set_dB2 @ A2)) => ((![Z22 : list_dB]: (~ ((A2 = (cons_dB @ E @ Z22))))) => (~ ((![Z1 : dB, Z22 : list_dB]: ((A2 = (cons_dB @ Z1 @ Z22)) => (~ ((member_dB2 @ E @ (set_dB2 @ Z22))))))))))))). % list.set_cases
thf(fact_130_map__ext, axiom,
    ((![Xs2 : list_dB, F : dB > dB, G : dB > dB]: ((![X3 : dB]: ((member_dB2 @ X3 @ (set_dB2 @ Xs2)) => ((F @ X3) = (G @ X3)))) => ((map_dB_dB @ F @ Xs2) = (map_dB_dB @ G @ Xs2)))))). % map_ext
thf(fact_131_map__ext, axiom,
    ((![Xs2 : list_dB, F : dB > nat, G : dB > nat]: ((![X3 : dB]: ((member_dB2 @ X3 @ (set_dB2 @ Xs2)) => ((F @ X3) = (G @ X3)))) => ((map_dB_nat @ F @ Xs2) = (map_dB_nat @ G @ Xs2)))))). % map_ext
thf(fact_132_map__idI, axiom,
    ((![Xs2 : list_dB, F : dB > dB]: ((![X3 : dB]: ((member_dB2 @ X3 @ (set_dB2 @ Xs2)) => ((F @ X3) = X3))) => ((map_dB_dB @ F @ Xs2) = Xs2))))). % map_idI
thf(fact_133_map__cong, axiom,
    ((![Xs2 : list_dB, Ys : list_dB, F : dB > dB, G : dB > dB]: ((Xs2 = Ys) => ((![X3 : dB]: ((member_dB2 @ X3 @ (set_dB2 @ Ys)) => ((F @ X3) = (G @ X3)))) => ((map_dB_dB @ F @ Xs2) = (map_dB_dB @ G @ Ys))))))). % map_cong
thf(fact_134_map__cong, axiom,
    ((![Xs2 : list_dB, Ys : list_dB, F : dB > nat, G : dB > nat]: ((Xs2 = Ys) => ((![X3 : dB]: ((member_dB2 @ X3 @ (set_dB2 @ Ys)) => ((F @ X3) = (G @ X3)))) => ((map_dB_nat @ F @ Xs2) = (map_dB_nat @ G @ Ys))))))). % map_cong
thf(fact_135_ex__map__conv, axiom,
    ((![Ys : list_nat, F : dB > nat]: ((?[Xs : list_dB]: (Ys = (map_dB_nat @ F @ Xs))) = (![X : nat]: (((member_nat @ X @ (set_nat2 @ Ys))) => ((?[Y4 : dB]: (X = (F @ Y4)))))))))). % ex_map_conv
thf(fact_136_ex__map__conv, axiom,
    ((![Ys : list_dB, F : dB > dB]: ((?[Xs : list_dB]: (Ys = (map_dB_dB @ F @ Xs))) = (![X : dB]: (((member_dB2 @ X @ (set_dB2 @ Ys))) => ((?[Y4 : dB]: (X = (F @ Y4)))))))))). % ex_map_conv
thf(fact_137_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_138_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_139_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_140_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_141_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_142_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_143_add__eq__self__zero, axiom,
    ((![M : nat, N2 : nat]: (((plus_plus_nat @ M @ N2) = M) => (N2 = zero_zero_nat))))). % add_eq_self_zero
thf(fact_144_plus__nat_Oadd__0, axiom,
    ((![N2 : nat]: ((plus_plus_nat @ zero_zero_nat @ N2) = N2)))). % plus_nat.add_0
thf(fact_145_add__Suc__shift, axiom,
    ((![M : nat, N2 : nat]: ((plus_plus_nat @ (suc @ M) @ N2) = (plus_plus_nat @ M @ (suc @ N2)))))). % add_Suc_shift
thf(fact_146_nat__arith_Osuc1, axiom,
    ((![A : nat, K : nat, A2 : nat]: ((A = (plus_plus_nat @ K @ A2)) => ((suc @ A) = (plus_plus_nat @ K @ (suc @ A2))))))). % nat_arith.suc1
thf(fact_147_add__Suc, axiom,
    ((![M : nat, N2 : nat]: ((plus_plus_nat @ (suc @ M) @ N2) = (suc @ (plus_plus_nat @ M @ N2)))))). % add_Suc
thf(fact_148_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, X3 : nat]: ((member_nat @ X3 @ (set_nat2 @ L)) => ((F @ A4 @ X3) = (G @ A4 @ X3)))) => ((foldl_nat_nat @ F @ A2 @ L) = (foldl_nat_nat @ G @ B2 @ K)))))))). % foldl_cong
thf(fact_149_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, X3 : dB]: ((member_dB2 @ X3 @ (set_dB2 @ L)) => ((F @ A4 @ X3) = (G @ A4 @ X3)))) => ((foldl_dB_dB @ F @ A2 @ L) = (foldl_dB_dB @ G @ B2 @ K)))))))). % foldl_cong
thf(fact_150_in__listspD, axiom,
    ((![A : dB > $o, Xs2 : list_dB]: ((listsp_dB @ A @ Xs2) => (![X4 : dB]: ((member_dB2 @ X4 @ (set_dB2 @ Xs2)) => (A @ X4))))))). % in_listspD
thf(fact_151_in__listsp__conv__set, axiom,
    ((listsp_dB = (^[A5 : dB > $o]: (^[Xs : list_dB]: (![X : dB]: (((member_dB2 @ X @ (set_dB2 @ Xs))) => ((A5 @ X))))))))). % in_listsp_conv_set
thf(fact_152_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_153_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_154_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_155_add_Ocommute, axiom,
    ((plus_plus_nat = (^[A3 : nat]: (^[B3 : nat]: (plus_plus_nat @ B3 @ A3)))))). % add.commute

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