% 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_172__5211146_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:38:05.091

% Could-be-implicit typings (5)
thf(ty_n_t__List__Olist_It__List__Olist_It__Lambda__OdB_J_J, type,
    list_list_dB : $tType).
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__Lambda__OdB, type,
    dB : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).

% Explicit typings (35)
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_If_001t__List__Olist_It__Lambda__OdB_J, type,
    if_list_dB : $o > list_dB > list_dB > list_dB).
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_ListOrder_Ostep1_001t__Lambda__OdB, type,
    step1_dB : (dB > dB > $o) > list_dB > list_dB > $o).
thf(sy_c_List_Ocount__list_001t__Lambda__OdB, type,
    count_list_dB : list_dB > dB > 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_Oinsert_001t__Lambda__OdB, type,
    insert_dB : dB > list_dB > list_dB).
thf(sy_c_List_Olist_OCons_001t__Lambda__OdB, type,
    cons_dB : dB > list_dB > list_dB).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Lambda__OdB_J, type,
    cons_list_dB : list_dB > list_list_dB > list_list_dB).
thf(sy_c_List_Olist_ONil_001t__Lambda__OdB, type,
    nil_dB : list_dB).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Lambda__OdB_J, type,
    nil_list_dB : list_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_Olist_Oset_001t__Lambda__OdB, type,
    set_dB2 : list_dB > set_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_Osize__class_Osize_001t__Lambda__OdB, type,
    size_size_dB : dB > nat).
thf(sy_c_Set_Othe__elem_001t__Lambda__OdB, type,
    the_elem_dB : set_dB > dB).
thf(sy_c_member_001t__Lambda__OdB, type,
    member_dB : dB > set_dB > $o).
thf(sy_v_a____, type,
    a : dB).
thf(sy_v_as____, type,
    as : list_dB).
thf(sy_v_i____, type,
    i : nat).
thf(sy_v_n____, type,
    n : nat).
thf(sy_v_rs____, type,
    rs : list_dB).
thf(sy_v_t____, type,
    t : dB).
thf(sy_v_u1____, type,
    u1 : dB).
thf(sy_v_u____, type,
    u : dB).

% Relevant facts (138)
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_Imap_A_I_092_060lambda_062t_O_Alift_At_A0_J_A_Imap_A_I_092_060lambda_062t_O_At_091u_Pi_093_J_Aas_J_J_092_060close_062, axiom,
    ((listsp_dB @ it @ (map_dB_dB @ (^[T : dB]: (lift @ T @ zero_zero_nat)) @ (map_dB_dB @ (^[T : dB]: (subst @ T @ u @ i)) @ as))))). % \<open>listsp IT (map (\<lambda>t. lift t 0) (map (\<lambda>t. t[u/i]) as))\<close>
thf(fact_5_lift__IT, axiom,
    ((![T2 : dB, I : nat]: ((it @ T2) => (it @ (lift @ T2 @ I)))))). % lift_IT
thf(fact_6_Var__IT, axiom,
    ((![N : nat]: (it @ (var @ N))))). % Var_IT
thf(fact_7_app__Var__IT, axiom,
    ((![T2 : dB, I : nat]: ((it @ T2) => (it @ (app @ T2 @ (var @ I))))))). % app_Var_IT
thf(fact_8_subst__Var__IT, axiom,
    ((![R : dB, I : nat, J : nat]: ((it @ R) => (it @ (subst @ R @ (var @ I) @ J)))))). % subst_Var_IT
thf(fact_9__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_10__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_11_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_12_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_13_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_14_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_15_subst__lift, axiom,
    ((![T2 : dB, K : nat, S : dB]: ((subst @ (lift @ T2 @ K) @ S @ K) = T2)))). % subst_lift
thf(fact_16_subst__eq, axiom,
    ((![K : nat, U : dB]: ((subst @ (var @ K) @ U @ K) = U)))). % subst_eq
thf(fact_17_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_18_map__ident, axiom,
    (((map_dB_dB @ (^[X : dB]: X)) = (^[Xs : list_dB]: Xs)))). % map_ident
thf(fact_19_dB_Oinject_I1_J, axiom,
    ((![X1 : nat, Y1 : nat]: (((var @ X1) = (var @ Y1)) = (X1 = Y1))))). % dB.inject(1)
thf(fact_20_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_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_list_Omap__ident, axiom,
    ((![T2 : list_dB]: ((map_dB_dB @ (^[X : dB]: X) @ T2) = T2)))). % list.map_ident
thf(fact_23_dB_Odistinct_I1_J, axiom,
    ((![X1 : nat, X21 : dB, X22 : dB]: (~ (((var @ X1) = (app @ X21 @ X22))))))). % dB.distinct(1)
thf(fact_24_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_25_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_26_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_27_IT_OVar, axiom,
    ((![Rs : list_dB, N : nat]: ((listsp_dB @ it @ Rs) => (it @ (foldl_dB_dB @ app @ (var @ N) @ Rs)))))). % IT.Var
thf(fact_28_local_OCons, axiom,
    ((rs = (cons_dB @ a @ as)))). % local.Cons
thf(fact_29_IT_Oinducts, axiom,
    ((![X2 : dB, P : dB > $o]: ((it @ X2) => ((![Rs2 : list_dB, N2 : nat]: ((listsp_dB @ (^[X : dB]: (((it @ X)) & ((P @ X)))) @ Rs2) => (P @ (foldl_dB_dB @ app @ (var @ N2) @ Rs2)))) => ((![R2 : dB]: ((it @ R2) => ((P @ R2) => (P @ (abs @ R2))))) => ((![R2 : dB, S2 : dB, Ss2 : list_dB]: ((it @ (foldl_dB_dB @ app @ (subst @ R2 @ S2 @ zero_zero_nat) @ Ss2)) => ((P @ (foldl_dB_dB @ app @ (subst @ R2 @ S2 @ zero_zero_nat) @ Ss2)) => ((it @ S2) => ((P @ S2) => (P @ (foldl_dB_dB @ app @ (app @ (abs @ R2) @ S2) @ Ss2))))))) => (P @ X2)))))))). % IT.inducts
thf(fact_30_IT_Osimps, axiom,
    ((it = (^[A3 : dB]: (((?[Rs3 : list_dB]: (?[N3 : nat]: (((A3 = (foldl_dB_dB @ app @ (var @ N3) @ 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_31_IT_Ocases, axiom,
    ((![A2 : dB]: ((it @ A2) => ((![Rs2 : list_dB]: ((?[N2 : nat]: (A2 = (foldl_dB_dB @ app @ (var @ N2) @ 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_32_dB_Osize__gen_I1_J, axiom,
    ((![X1 : nat]: ((size_dB @ (var @ X1)) = zero_zero_nat)))). % dB.size_gen(1)
thf(fact_33_map__eq__map__tailrec, axiom,
    ((map_dB_dB = map_tailrec_dB_dB))). % map_eq_map_tailrec
thf(fact_34_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_35_dB_Oinject_I3_J, axiom,
    ((![X3 : dB, Y3 : dB]: (((abs @ X3) = (abs @ Y3)) = (X3 = Y3))))). % dB.inject(3)
thf(fact_36_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_37_not__Cons__self2, axiom,
    ((![X2 : dB, Xs2 : list_dB]: (~ (((cons_dB @ X2 @ Xs2) = Xs2)))))). % not_Cons_self2
thf(fact_38_Lambda, axiom,
    ((![R : dB]: ((it @ R) => (it @ (abs @ R)))))). % Lambda
thf(fact_39_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_40_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_41_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_42_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_43_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_44_dB_Odistinct_I5_J, axiom,
    ((![X21 : dB, X22 : dB, X3 : dB]: (~ (((app @ X21 @ X22) = (abs @ X3))))))). % dB.distinct(5)
thf(fact_45_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_46_dB_Odistinct_I3_J, axiom,
    ((![X1 : nat, X3 : dB]: (~ (((var @ X1) = (abs @ X3))))))). % dB.distinct(3)
thf(fact_47_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_48_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_49_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_50_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_51_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_52_zero__reorient, axiom,
    ((![X2 : nat]: ((zero_zero_nat = X2) = (X2 = zero_zero_nat))))). % zero_reorient
thf(fact_53_ex__head__tail, axiom,
    ((![T2 : dB]: (?[Ts2 : list_dB, H : dB]: ((T2 = (foldl_dB_dB @ app @ H @ Ts2)) & ((?[N2 : nat]: (H = (var @ N2))) | (?[U2 : dB]: (H = (abs @ U2))))))))). % ex_head_tail
thf(fact_54_Abs__App__neq__Var__apps, axiom,
    ((![S : dB, T2 : dB, N : nat, Ss : list_dB]: (~ (((app @ (abs @ S) @ T2) = (foldl_dB_dB @ app @ (var @ N) @ Ss))))))). % Abs_App_neq_Var_apps
thf(fact_55_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_56_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_57_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_58_Apps__dB__induct, axiom,
    ((![P : dB > $o, T2 : dB]: ((![N2 : nat, Ts2 : list_dB]: ((![X5 : dB]: ((member_dB @ X5 @ (set_dB2 @ Ts2)) => (P @ X5))) => (P @ (foldl_dB_dB @ app @ (var @ N2) @ Ts2)))) => ((![U2 : dB]: ((P @ U2) => (![Ts2 : list_dB]: ((![X5 : dB]: ((member_dB @ X5 @ (set_dB2 @ Ts2)) => (P @ X5))) => (P @ (foldl_dB_dB @ app @ (abs @ U2) @ Ts2)))))) => (P @ T2)))))). % Apps_dB_induct
thf(fact_59_beta__cases_I3_J, axiom,
    ((![S : dB, T2 : dB, U : dB]: ((beta @ (app @ S @ T2) @ U) => ((![S2 : dB]: ((S = (abs @ S2)) => (~ ((U = (subst @ S2 @ T2 @ zero_zero_nat)))))) => ((![T3 : dB]: ((U = (app @ T3 @ T2)) => (~ ((beta @ S @ T3))))) => (~ ((![T3 : dB]: ((U = (app @ S @ T3)) => (~ ((beta @ T2 @ T3))))))))))))). % beta_cases(3)
thf(fact_60_beta, axiom,
    ((![S : dB, T2 : dB]: (beta @ (app @ (abs @ S) @ T2) @ (subst @ S @ T2 @ zero_zero_nat))))). % beta
thf(fact_61_beta_Ocases, axiom,
    ((![A1 : dB, A22 : dB]: ((beta @ A1 @ A22) => ((![S2 : dB, T3 : dB]: ((A1 = (app @ (abs @ S2) @ T3)) => (~ ((A22 = (subst @ S2 @ T3 @ zero_zero_nat)))))) => ((![S2 : dB, T3 : dB, U2 : dB]: ((A1 = (app @ S2 @ U2)) => ((A22 = (app @ T3 @ U2)) => (~ ((beta @ S2 @ T3)))))) => ((![S2 : dB, T3 : dB, U2 : dB]: ((A1 = (app @ U2 @ S2)) => ((A22 = (app @ U2 @ T3)) => (~ ((beta @ S2 @ T3)))))) => (~ ((![S2 : dB]: ((A1 = (abs @ S2)) => (![T3 : dB]: ((A22 = (abs @ T3)) => (~ ((beta @ S2 @ T3)))))))))))))))). % beta.cases
thf(fact_62_beta_Osimps, axiom,
    ((beta = (^[A12 : dB]: (^[A23 : dB]: (((?[S3 : dB]: (?[T : dB]: (((A12 = (app @ (abs @ S3) @ T))) & ((A23 = (subst @ S3 @ T @ zero_zero_nat))))))) | ((((?[S3 : dB]: (?[T : dB]: (?[U3 : dB]: (((A12 = (app @ S3 @ U3))) & ((((A23 = (app @ T @ U3))) & ((beta @ S3 @ T))))))))) | ((((?[S3 : dB]: (?[T : dB]: (?[U3 : dB]: (((A12 = (app @ U3 @ S3))) & ((((A23 = (app @ U3 @ T))) & ((beta @ S3 @ T))))))))) | ((?[S3 : dB]: (?[T : dB]: (((A12 = (abs @ S3))) & ((((A23 = (abs @ T))) & ((beta @ S3 @ T)))))))))))))))))). % beta.simps
thf(fact_63_beta_Oinducts, axiom,
    ((![X1 : dB, X24 : dB, P : dB > dB > $o]: ((beta @ X1 @ X24) => ((![S2 : dB, T3 : dB]: (P @ (app @ (abs @ S2) @ T3) @ (subst @ S2 @ T3 @ zero_zero_nat))) => ((![S2 : dB, T3 : dB, U2 : dB]: ((beta @ S2 @ T3) => ((P @ S2 @ T3) => (P @ (app @ S2 @ U2) @ (app @ T3 @ U2))))) => ((![S2 : dB, T3 : dB, U2 : dB]: ((beta @ S2 @ T3) => ((P @ S2 @ T3) => (P @ (app @ U2 @ S2) @ (app @ U2 @ T3))))) => ((![S2 : dB, T3 : dB]: ((beta @ S2 @ T3) => ((P @ S2 @ T3) => (P @ (abs @ S2) @ (abs @ T3))))) => (P @ X1 @ X24))))))))). % beta.inducts
thf(fact_64_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_dB @ X @ (set_dB2 @ Xs2))) => (((F @ X) = (G @ X))))))))). % map_eq_conv
thf(fact_65_in__listspI, axiom,
    ((![Xs2 : list_dB, A : dB > $o]: ((![X4 : dB]: ((member_dB @ X4 @ (set_dB2 @ Xs2)) => (A @ X4))) => (listsp_dB @ A @ Xs2))))). % in_listspI
thf(fact_66_list_Oset__cases, axiom,
    ((![E : dB, A2 : list_dB]: ((member_dB @ E @ (set_dB2 @ A2)) => ((![Z22 : list_dB]: (~ ((A2 = (cons_dB @ E @ Z22))))) => (~ ((![Z1 : dB, Z22 : list_dB]: ((A2 = (cons_dB @ Z1 @ Z22)) => (~ ((member_dB @ E @ (set_dB2 @ Z22))))))))))))). % list.set_cases
thf(fact_67_set__ConsD, axiom,
    ((![Y : dB, X2 : dB, Xs2 : list_dB]: ((member_dB @ Y @ (set_dB2 @ (cons_dB @ X2 @ Xs2))) => ((Y = X2) | (member_dB @ Y @ (set_dB2 @ Xs2))))))). % set_ConsD
thf(fact_68_list_Oset__intros_I1_J, axiom,
    ((![X21 : dB, X22 : list_dB]: (member_dB @ X21 @ (set_dB2 @ (cons_dB @ X21 @ X22)))))). % list.set_intros(1)
thf(fact_69_list_Oset__intros_I2_J, axiom,
    ((![Y : dB, X22 : list_dB, X21 : dB]: ((member_dB @ Y @ (set_dB2 @ X22)) => (member_dB @ Y @ (set_dB2 @ (cons_dB @ X21 @ X22))))))). % list.set_intros(2)
thf(fact_70_list_Oinj__map__strong, axiom,
    ((![X2 : list_dB, Xa : list_dB, F : dB > dB, Fa : dB > dB]: ((![Z : dB, Za : dB]: ((member_dB @ Z @ (set_dB2 @ X2)) => ((member_dB @ 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_71_list_Omap__cong0, axiom,
    ((![X2 : list_dB, F : dB > dB, G : dB > dB]: ((![Z : dB]: ((member_dB @ Z @ (set_dB2 @ X2)) => ((F @ Z) = (G @ Z)))) => ((map_dB_dB @ F @ X2) = (map_dB_dB @ G @ X2)))))). % list.map_cong0
thf(fact_72_list_Omap__cong, axiom,
    ((![X2 : list_dB, Ya : list_dB, F : dB > dB, G : dB > dB]: ((X2 = Ya) => ((![Z : dB]: ((member_dB @ Z @ (set_dB2 @ Ya)) => ((F @ Z) = (G @ Z)))) => ((map_dB_dB @ F @ X2) = (map_dB_dB @ G @ Ya))))))). % list.map_cong
thf(fact_73_ex__map__conv, axiom,
    ((![Ys : list_dB, F : dB > dB]: ((?[Xs : list_dB]: (Ys = (map_dB_dB @ F @ Xs))) = (![X : dB]: (((member_dB @ X @ (set_dB2 @ Ys))) => ((?[Y2 : dB]: (X = (F @ Y2)))))))))). % ex_map_conv
thf(fact_74_map__cong, axiom,
    ((![Xs2 : list_dB, Ys : list_dB, F : dB > dB, G : dB > dB]: ((Xs2 = Ys) => ((![X4 : dB]: ((member_dB @ X4 @ (set_dB2 @ Ys)) => ((F @ X4) = (G @ X4)))) => ((map_dB_dB @ F @ Xs2) = (map_dB_dB @ G @ Ys))))))). % map_cong
thf(fact_75_map__idI, axiom,
    ((![Xs2 : list_dB, F : dB > dB]: ((![X4 : dB]: ((member_dB @ X4 @ (set_dB2 @ Xs2)) => ((F @ X4) = X4))) => ((map_dB_dB @ F @ Xs2) = Xs2))))). % map_idI
thf(fact_76_map__ext, axiom,
    ((![Xs2 : list_dB, F : dB > dB, G : dB > dB]: ((![X4 : dB]: ((member_dB @ X4 @ (set_dB2 @ Xs2)) => ((F @ X4) = (G @ X4)))) => ((map_dB_dB @ F @ Xs2) = (map_dB_dB @ G @ Xs2)))))). % map_ext
thf(fact_77_appR, axiom,
    ((![S : dB, T2 : dB, U : dB]: ((beta @ S @ T2) => (beta @ (app @ U @ S) @ (app @ U @ T2)))))). % appR
thf(fact_78_appL, axiom,
    ((![S : dB, T2 : dB, U : dB]: ((beta @ S @ T2) => (beta @ (app @ S @ U) @ (app @ T2 @ U)))))). % appL
thf(fact_79_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_dB @ 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_80_beta__cases_I1_J, axiom,
    ((![I : nat, T2 : dB]: (~ ((beta @ (var @ I) @ T2)))))). % beta_cases(1)
thf(fact_81_in__listspD, axiom,
    ((![A : dB > $o, Xs2 : list_dB]: ((listsp_dB @ A @ Xs2) => (![X5 : dB]: ((member_dB @ X5 @ (set_dB2 @ Xs2)) => (A @ X5))))))). % in_listspD
thf(fact_82_in__listsp__conv__set, axiom,
    ((listsp_dB = (^[A5 : dB > $o]: (^[Xs : list_dB]: (![X : dB]: (((member_dB @ X @ (set_dB2 @ Xs))) => ((A5 @ X))))))))). % in_listsp_conv_set
thf(fact_83_abs, axiom,
    ((![S : dB, T2 : dB]: ((beta @ S @ T2) => (beta @ (abs @ S) @ (abs @ T2)))))). % abs
thf(fact_84_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_85_subst__preserves__beta, axiom,
    ((![R : dB, S : dB, T2 : dB, I : nat]: ((beta @ R @ S) => (beta @ (subst @ R @ T2 @ I) @ (subst @ S @ T2 @ I)))))). % subst_preserves_beta
thf(fact_86_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_87_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_88_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_89_lem, axiom,
    ((![P : dB > $o, T2 : dB, N : nat]: ((![N2 : nat, Ts2 : list_dB]: ((![X5 : dB]: ((member_dB @ X5 @ (set_dB2 @ Ts2)) => (P @ X5))) => (P @ (foldl_dB_dB @ app @ (var @ N2) @ Ts2)))) => ((![U2 : dB]: ((P @ U2) => (![Ts2 : list_dB]: ((![X5 : dB]: ((member_dB @ X5 @ (set_dB2 @ Ts2)) => (P @ X5))) => (P @ (foldl_dB_dB @ app @ (abs @ U2) @ Ts2)))))) => (((size_size_dB @ T2) = N) => (P @ T2))))))). % lem
thf(fact_90_substn__subst__n, axiom,
    ((substn = (^[T : dB]: (^[S3 : dB]: (^[N3 : nat]: (subst @ T @ (liftn @ N3 @ S3 @ zero_zero_nat) @ N3))))))). % substn_subst_n
thf(fact_91_apps__betasE, axiom,
    ((![R : dB, Rs : list_dB, S : dB]: ((beta @ (foldl_dB_dB @ app @ R @ Rs) @ S) => ((![R4 : dB]: ((beta @ R @ R4) => (~ ((S = (foldl_dB_dB @ app @ R4 @ Rs)))))) => ((![Rs4 : list_dB]: ((step1_dB @ beta @ Rs @ Rs4) => (~ ((S = (foldl_dB_dB @ app @ R @ Rs4)))))) => (~ ((![T3 : dB]: ((R = (abs @ T3)) => (![U2 : dB, Us : list_dB]: ((Rs = (cons_dB @ U2 @ Us)) => (~ ((S = (foldl_dB_dB @ app @ (subst @ T3 @ U2 @ zero_zero_nat) @ Us)))))))))))))))). % apps_betasE
thf(fact_92_liftn__0, axiom,
    ((![T2 : dB, K : nat]: ((liftn @ zero_zero_nat @ T2 @ K) = T2)))). % liftn_0
thf(fact_93_liftn_Osimps_I2_J, axiom,
    ((![N : nat, S : dB, T2 : dB, K : nat]: ((liftn @ N @ (app @ S @ T2) @ K) = (app @ (liftn @ N @ S @ K) @ (liftn @ N @ T2 @ K)))))). % liftn.simps(2)
thf(fact_94_apps__preserves__betas, axiom,
    ((![Rs : list_dB, Ss : list_dB, R : dB]: ((step1_dB @ beta @ Rs @ Ss) => (beta @ (foldl_dB_dB @ app @ R @ Rs) @ (foldl_dB_dB @ app @ R @ Ss)))))). % apps_preserves_betas
thf(fact_95_dB_Osize_I4_J, axiom,
    ((![X1 : nat]: ((size_size_dB @ (var @ X1)) = zero_zero_nat)))). % dB.size(4)
thf(fact_96_head__Var__reduction, axiom,
    ((![N : nat, Rs : list_dB, V : dB]: ((beta @ (foldl_dB_dB @ app @ (var @ N) @ Rs) @ V) => (?[Ss2 : list_dB]: ((step1_dB @ beta @ Rs @ Ss2) & (V = (foldl_dB_dB @ app @ (var @ N) @ Ss2)))))))). % head_Var_reduction
thf(fact_97_Cons__step1__Cons, axiom,
    ((![R : dB > dB > $o, Y : dB, Ys : list_dB, X2 : dB, Xs2 : list_dB]: ((step1_dB @ R @ (cons_dB @ Y @ Ys) @ (cons_dB @ X2 @ Xs2)) = (((((R @ Y @ X2)) & ((Xs2 = Ys)))) | ((((X2 = Y)) & ((step1_dB @ R @ Ys @ Xs2))))))))). % Cons_step1_Cons
thf(fact_98_ex__step1I, axiom,
    ((![X2 : dB, Xs2 : list_dB, R : dB > dB > $o, Y : dB]: ((member_dB @ X2 @ (set_dB2 @ Xs2)) => ((R @ Y @ X2) => (?[Ys2 : list_dB]: ((step1_dB @ R @ Ys2 @ Xs2) & (member_dB @ Y @ (set_dB2 @ Ys2))))))))). % ex_step1I
thf(fact_99_Cons__step1E, axiom,
    ((![R : dB > dB > $o, Ys : list_dB, X2 : dB, Xs2 : list_dB]: ((step1_dB @ R @ Ys @ (cons_dB @ X2 @ Xs2)) => ((![Y4 : dB]: ((Ys = (cons_dB @ Y4 @ Xs2)) => (~ ((R @ Y4 @ X2))))) => (~ ((![Zs : list_dB]: ((Ys = (cons_dB @ X2 @ Zs)) => (~ ((step1_dB @ R @ Zs @ Xs2)))))))))))). % Cons_step1E
thf(fact_100_count__notin, axiom,
    ((![X2 : dB, Xs2 : list_dB]: ((~ ((member_dB @ X2 @ (set_dB2 @ Xs2)))) => ((count_list_dB @ Xs2 @ X2) = zero_zero_nat))))). % count_notin
thf(fact_101_not__in__set__insert, axiom,
    ((![X2 : dB, Xs2 : list_dB]: ((~ ((member_dB @ X2 @ (set_dB2 @ Xs2)))) => ((insert_dB @ X2 @ Xs2) = (cons_dB @ X2 @ Xs2)))))). % not_in_set_insert
thf(fact_102_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_103_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_104_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_105_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_106_listsp__simps_I1_J, axiom,
    ((![A : dB > $o]: (listsp_dB @ A @ nil_dB)))). % listsp_simps(1)
thf(fact_107_in__set__insert, axiom,
    ((![X2 : dB, Xs2 : list_dB]: ((member_dB @ X2 @ (set_dB2 @ Xs2)) => ((insert_dB @ X2 @ Xs2) = Xs2))))). % in_set_insert
thf(fact_108_insert__Nil, axiom,
    ((![X2 : dB]: ((insert_dB @ X2 @ nil_dB) = (cons_dB @ X2 @ nil_dB))))). % insert_Nil
thf(fact_109_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_110_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_111_not__step1__Nil, axiom,
    ((![R : dB > dB > $o, Xs2 : list_dB]: (~ ((step1_dB @ R @ Xs2 @ nil_dB)))))). % not_step1_Nil
thf(fact_112_not__Nil__step1, axiom,
    ((![R : dB > dB > $o, Xs2 : list_dB]: (~ ((step1_dB @ R @ nil_dB @ Xs2)))))). % not_Nil_step1
thf(fact_113_transpose_Ocases, axiom,
    ((![X2 : list_list_dB]: ((~ ((X2 = nil_list_dB))) => ((![Xss : list_list_dB]: (~ ((X2 = (cons_list_dB @ nil_dB @ Xss))))) => (~ ((![X4 : dB, Xs3 : list_dB, Xss : list_list_dB]: (~ ((X2 = (cons_list_dB @ (cons_dB @ X4 @ Xs3) @ Xss)))))))))))). % transpose.cases
thf(fact_114_map__tailrec__rev_Oinduct, axiom,
    ((![P : (dB > dB) > list_dB > list_dB > $o, A0 : dB > dB, A1 : list_dB, A22 : list_dB]: ((![F2 : dB > dB, X_1 : list_dB]: (P @ F2 @ nil_dB @ X_1)) => ((![F2 : dB > dB, A4 : dB, As : list_dB, Bs : list_dB]: ((P @ F2 @ As @ (cons_dB @ (F2 @ A4) @ Bs)) => (P @ F2 @ (cons_dB @ A4 @ As) @ Bs))) => (P @ A0 @ A1 @ A22)))))). % map_tailrec_rev.induct
thf(fact_115_list__nonempty__induct, axiom,
    ((![Xs2 : list_dB, P : list_dB > $o]: ((~ ((Xs2 = nil_dB))) => ((![X4 : dB]: (P @ (cons_dB @ X4 @ nil_dB))) => ((![X4 : dB, Xs3 : list_dB]: ((~ ((Xs3 = nil_dB))) => ((P @ Xs3) => (P @ (cons_dB @ X4 @ Xs3))))) => (P @ Xs2))))))). % list_nonempty_induct
thf(fact_116_successively_Oinduct, axiom,
    ((![P : (dB > dB > $o) > list_dB > $o, A0 : dB > dB > $o, A1 : list_dB]: ((![P2 : dB > dB > $o]: (P @ P2 @ nil_dB)) => ((![P2 : dB > dB > $o, X4 : dB]: (P @ P2 @ (cons_dB @ X4 @ nil_dB))) => ((![P2 : dB > dB > $o, X4 : dB, Y4 : dB, Xs3 : list_dB]: ((P @ P2 @ (cons_dB @ Y4 @ Xs3)) => (P @ P2 @ (cons_dB @ X4 @ (cons_dB @ Y4 @ Xs3))))) => (P @ A0 @ A1))))))). % successively.induct
thf(fact_117_remdups__adj_Oinduct, axiom,
    ((![P : list_dB > $o, A0 : list_dB]: ((P @ nil_dB) => ((![X4 : dB]: (P @ (cons_dB @ X4 @ nil_dB))) => ((![X4 : dB, Y4 : dB, Xs3 : list_dB]: (((X4 = Y4) => (P @ (cons_dB @ X4 @ Xs3))) => (((~ ((X4 = Y4))) => (P @ (cons_dB @ Y4 @ Xs3))) => (P @ (cons_dB @ X4 @ (cons_dB @ Y4 @ Xs3)))))) => (P @ A0))))))). % remdups_adj.induct
thf(fact_118_sorted__wrt_Oinduct, axiom,
    ((![P : (dB > dB > $o) > list_dB > $o, A0 : dB > dB > $o, A1 : list_dB]: ((![P2 : dB > dB > $o]: (P @ P2 @ nil_dB)) => ((![P2 : dB > dB > $o, X4 : dB, Ys2 : list_dB]: ((P @ P2 @ Ys2) => (P @ P2 @ (cons_dB @ X4 @ Ys2)))) => (P @ A0 @ A1)))))). % sorted_wrt.induct
thf(fact_119_remdups__adj_Ocases, axiom,
    ((![X2 : list_dB]: ((~ ((X2 = nil_dB))) => ((![X4 : dB]: (~ ((X2 = (cons_dB @ X4 @ nil_dB))))) => (~ ((![X4 : dB, Y4 : dB, Xs3 : list_dB]: (~ ((X2 = (cons_dB @ X4 @ (cons_dB @ Y4 @ Xs3))))))))))))). % remdups_adj.cases
thf(fact_120_shuffles_Oinduct, axiom,
    ((![P : list_dB > list_dB > $o, A0 : list_dB, A1 : list_dB]: ((![X_1 : list_dB]: (P @ nil_dB @ X_1)) => ((![Xs3 : list_dB]: (P @ Xs3 @ nil_dB)) => ((![X4 : dB, Xs3 : list_dB, Y4 : dB, Ys2 : list_dB]: ((P @ Xs3 @ (cons_dB @ Y4 @ Ys2)) => ((P @ (cons_dB @ X4 @ Xs3) @ Ys2) => (P @ (cons_dB @ X4 @ Xs3) @ (cons_dB @ Y4 @ Ys2))))) => (P @ A0 @ A1))))))). % shuffles.induct
thf(fact_121_induct__list012, axiom,
    ((![P : list_dB > $o, Xs2 : list_dB]: ((P @ nil_dB) => ((![X4 : dB]: (P @ (cons_dB @ X4 @ nil_dB))) => ((![X4 : dB, Y4 : dB, Zs : list_dB]: ((P @ Zs) => ((P @ (cons_dB @ Y4 @ Zs)) => (P @ (cons_dB @ X4 @ (cons_dB @ Y4 @ Zs)))))) => (P @ Xs2))))))). % induct_list012
thf(fact_122_splice_Oinduct, axiom,
    ((![P : list_dB > list_dB > $o, A0 : list_dB, A1 : list_dB]: ((![X_1 : list_dB]: (P @ nil_dB @ X_1)) => ((![X4 : dB, Xs3 : list_dB, Ys2 : list_dB]: ((P @ Ys2 @ Xs3) => (P @ (cons_dB @ X4 @ Xs3) @ Ys2))) => (P @ A0 @ A1)))))). % splice.induct
thf(fact_123_list__induct2_H, axiom,
    ((![P : list_dB > list_dB > $o, Xs2 : list_dB, Ys : list_dB]: ((P @ nil_dB @ nil_dB) => ((![X4 : dB, Xs3 : list_dB]: (P @ (cons_dB @ X4 @ Xs3) @ nil_dB)) => ((![Y4 : dB, Ys2 : list_dB]: (P @ nil_dB @ (cons_dB @ Y4 @ Ys2))) => ((![X4 : dB, Xs3 : list_dB, Y4 : dB, Ys2 : list_dB]: ((P @ Xs3 @ Ys2) => (P @ (cons_dB @ X4 @ Xs3) @ (cons_dB @ Y4 @ Ys2)))) => (P @ Xs2 @ Ys)))))))). % list_induct2'
thf(fact_124_neq__Nil__conv, axiom,
    ((![Xs2 : list_dB]: ((~ ((Xs2 = nil_dB))) = (?[Y2 : dB]: (?[Ys3 : list_dB]: (Xs2 = (cons_dB @ Y2 @ Ys3)))))))). % neq_Nil_conv
thf(fact_125_list_Oinducts, axiom,
    ((![P : list_dB > $o, List : list_dB]: ((P @ nil_dB) => ((![X12 : dB, X23 : list_dB]: ((P @ X23) => (P @ (cons_dB @ X12 @ X23)))) => (P @ List)))))). % list.inducts
thf(fact_126_list_Oexhaust, axiom,
    ((![Y : list_dB]: ((~ ((Y = nil_dB))) => (~ ((![X212 : dB, X222 : list_dB]: (~ ((Y = (cons_dB @ X212 @ X222))))))))))). % list.exhaust
thf(fact_127_list_OdiscI, axiom,
    ((![List : list_dB, X21 : dB, X22 : list_dB]: ((List = (cons_dB @ X21 @ X22)) => (~ ((List = nil_dB))))))). % list.discI
thf(fact_128_list_Odistinct_I1_J, axiom,
    ((![X21 : dB, X22 : list_dB]: (~ ((nil_dB = (cons_dB @ X21 @ X22))))))). % list.distinct(1)
thf(fact_129_list_Osimps_I8_J, axiom,
    ((![F : dB > dB]: ((map_dB_dB @ F @ nil_dB) = nil_dB)))). % list.simps(8)
thf(fact_130_count__list_Osimps_I1_J, axiom,
    ((![Y : dB]: ((count_list_dB @ nil_dB @ Y) = zero_zero_nat)))). % count_list.simps(1)
thf(fact_131_listsp_ONil, axiom,
    ((![A : dB > $o]: (listsp_dB @ A @ nil_dB)))). % listsp.Nil
thf(fact_132_foldl__Nil, axiom,
    ((![F : dB > dB > dB, A2 : dB]: ((foldl_dB_dB @ F @ A2 @ nil_dB) = A2)))). % foldl_Nil
thf(fact_133_listsp_Ocases, axiom,
    ((![A : dB > $o, A2 : list_dB]: ((listsp_dB @ A @ A2) => ((~ ((A2 = nil_dB))) => (~ ((![A4 : dB, L2 : list_dB]: ((A2 = (cons_dB @ A4 @ L2)) => ((A @ A4) => (~ ((listsp_dB @ A @ L2))))))))))))). % listsp.cases
thf(fact_134_listsp_Osimps, axiom,
    ((listsp_dB = (^[A5 : dB > $o]: (^[A3 : list_dB]: (((A3 = nil_dB)) | ((?[B3 : dB]: (?[L3 : list_dB]: (((A3 = (cons_dB @ B3 @ L3))) & ((((A5 @ B3)) & ((listsp_dB @ A5 @ L3)))))))))))))). % listsp.simps
thf(fact_135_listsp_Oinducts, axiom,
    ((![A : dB > $o, X2 : list_dB, P : list_dB > $o]: ((listsp_dB @ A @ X2) => ((P @ nil_dB) => ((![A4 : dB, L2 : list_dB]: ((A @ A4) => ((listsp_dB @ A @ L2) => ((P @ L2) => (P @ (cons_dB @ A4 @ L2)))))) => (P @ X2))))))). % listsp.inducts
thf(fact_136_List_Oinsert__def, axiom,
    ((insert_dB = (^[X : dB]: (^[Xs : list_dB]: (if_list_dB @ (member_dB @ X @ (set_dB2 @ Xs)) @ Xs @ (cons_dB @ X @ Xs))))))). % List.insert_def
thf(fact_137_the__elem__set, axiom,
    ((![X2 : dB]: ((the_elem_dB @ (set_dB2 @ (cons_dB @ X2 @ nil_dB))) = X2)))). % the_elem_set

% Helper facts (3)
thf(help_If_3_1_If_001t__List__Olist_It__Lambda__OdB_J_T, axiom,
    ((![P : $o]: ((P = $true) | (P = $false))))).
thf(help_If_2_1_If_001t__List__Olist_It__Lambda__OdB_J_T, axiom,
    ((![X2 : list_dB, Y : list_dB]: ((if_list_dB @ $false @ X2 @ Y) = Y)))).
thf(help_If_1_1_If_001t__List__Olist_It__Lambda__OdB_J_T, axiom,
    ((![X2 : list_dB, Y : list_dB]: ((if_list_dB @ $true @ X2 @ Y) = X2)))).

% Conjectures (1)
thf(conj_0, conjecture,
    ((it @ (foldl_dB_dB @ app @ (var @ zero_zero_nat) @ (map_dB_dB @ (^[T : dB]: (lift @ T @ zero_zero_nat)) @ (map_dB_dB @ (^[T : dB]: (subst @ T @ u @ i)) @ as)))))).
