% 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_22__5209596_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:36:55.752

% Could-be-implicit typings (7)
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).
thf(ty_n_t__Int__Oint, type,
    int : $tType).

% Explicit typings (45)
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__Int__Oint, type,
    zero_zero_int : int).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_If_001t__Nat__Onat, type,
    if_nat : $o > nat > nat > nat).
thf(sy_c_InductTermi_OIT, type,
    it : dB > $o).
thf(sy_c_Lambda_OdB_OAbs, type,
    abs : dB > dB).
thf(sy_c_Lambda_OdB_OApp, type,
    app : dB > dB > dB).
thf(sy_c_Lambda_OdB_OVar, type,
    var : nat > dB).
thf(sy_c_Lambda_OdB_Osize__dB, type,
    size_dB : dB > nat).
thf(sy_c_Lambda_Olift, type,
    lift : dB > nat > dB).
thf(sy_c_Lambda_Osubst, type,
    subst : dB > dB > nat > dB).
thf(sy_c_List_Obind_001t__Lambda__OdB_001t__Lambda__OdB, type,
    bind_dB_dB : list_dB > (dB > list_dB) > list_dB).
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_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_Ogen__length_001t__Lambda__OdB, type,
    gen_length_dB : nat > list_dB > 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_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_Omap__tailrec_001t__Nat__Onat_001t__Nat__Onat, type,
    map_tailrec_nat_nat : (nat > nat) > list_nat > list_nat).
thf(sy_c_List_Omember_001t__Lambda__OdB, type,
    member_dB : list_dB > dB > $o).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint, type,
    semiri2019852685at_int : nat > int).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat, type,
    semiri1382578993at_nat : nat > nat).
thf(sy_c_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_Orderings_Oord__class_Oless_001t__Int__Oint, type,
    ord_less_int : int > int > $o).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat, type,
    ord_less_nat : nat > nat > $o).
thf(sy_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_ia, type,
    ia : nat).
thf(sy_v_n, type,
    n : nat).
thf(sy_v_rs, type,
    rs : list_dB).

% Relevant facts (205)
thf(fact_0_lift__map, axiom,
    ((![T : dB, Ts : list_dB, I : nat]: ((lift @ (foldl_dB_dB @ app @ T @ Ts) @ I) = (foldl_dB_dB @ app @ (lift @ T @ I) @ (map_dB_dB @ (^[T2 : dB]: (lift @ T2 @ I)) @ Ts)))))). % lift_map
thf(fact_1_Var__apps__eq__Var__apps__conv, axiom,
    ((![M : nat, Rs : list_dB, N : nat, Ss : list_dB]: (((foldl_dB_dB @ app @ (var @ M) @ Rs) = (foldl_dB_dB @ app @ (var @ N) @ Ss)) = (((M = N)) & ((Rs = Ss))))))). % Var_apps_eq_Var_apps_conv
thf(fact_2_IT_OVar, axiom,
    ((![Rs : list_dB, N : nat]: ((listsp_dB @ it @ Rs) => (it @ (foldl_dB_dB @ app @ (var @ N) @ Rs)))))). % IT.Var
thf(fact_3_apps__eq__tail__conv, axiom,
    ((![R : dB, Ts : list_dB, S : dB]: (((foldl_dB_dB @ app @ R @ Ts) = (foldl_dB_dB @ app @ S @ Ts)) = (R = S))))). % apps_eq_tail_conv
thf(fact_4_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_5_map__ident, axiom,
    (((map_nat_nat @ (^[X : nat]: X)) = (^[Xs : list_nat]: Xs)))). % map_ident
thf(fact_6_map__ident, axiom,
    (((map_dB_dB @ (^[X : dB]: X)) = (^[Xs : list_dB]: Xs)))). % map_ident
thf(fact_7_dB_Oinject_I1_J, axiom,
    ((![X1 : nat, Y1 : nat]: (((var @ X1) = (var @ Y1)) = (X1 = Y1))))). % dB.inject(1)
thf(fact_8_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_9_foldl__map, axiom,
    ((![G : dB > dB > dB, A2 : dB, F : nat > dB, Xs2 : list_nat]: ((foldl_dB_dB @ G @ A2 @ (map_nat_dB @ F @ Xs2)) = (foldl_dB_nat @ (^[A3 : dB]: (^[X : nat]: (G @ A3 @ (F @ X)))) @ A2 @ Xs2))))). % foldl_map
thf(fact_10_foldl__map, axiom,
    ((![G : nat > dB > nat, A2 : nat, F : nat > dB, Xs2 : list_nat]: ((foldl_nat_dB @ G @ A2 @ (map_nat_dB @ F @ Xs2)) = (foldl_nat_nat @ (^[A3 : nat]: (^[X : nat]: (G @ A3 @ (F @ X)))) @ A2 @ Xs2))))). % foldl_map
thf(fact_11_foldl__map, axiom,
    ((![G : nat > dB > nat, A2 : nat, F : dB > dB, Xs2 : list_dB]: ((foldl_nat_dB @ G @ A2 @ (map_dB_dB @ F @ Xs2)) = (foldl_nat_dB @ (^[A3 : nat]: (^[X : dB]: (G @ A3 @ (F @ X)))) @ A2 @ Xs2))))). % foldl_map
thf(fact_12_foldl__map, axiom,
    ((![G : dB > nat > dB, A2 : dB, F : nat > nat, Xs2 : list_nat]: ((foldl_dB_nat @ G @ A2 @ (map_nat_nat @ F @ Xs2)) = (foldl_dB_nat @ (^[A3 : dB]: (^[X : nat]: (G @ A3 @ (F @ X)))) @ A2 @ Xs2))))). % foldl_map
thf(fact_13_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_14_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_15_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_16_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_17_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_18_dB_Odistinct_I1_J, axiom,
    ((![X1 : nat, X21 : dB, X22 : dB]: (~ (((var @ X1) = (app @ X21 @ X22))))))). % dB.distinct(1)
thf(fact_19_list_Omap__ident, axiom,
    ((![T : list_dB]: ((map_dB_dB @ (^[X : dB]: X) @ T) = T)))). % list.map_ident
thf(fact_20_list_Omap__ident, axiom,
    ((![T : list_nat]: ((map_nat_nat @ (^[X : nat]: X) @ T) = T)))). % list.map_ident
thf(fact_21_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_22_map__eq__map__tailrec, axiom,
    ((map_dB_dB = map_tailrec_dB_dB))). % map_eq_map_tailrec
thf(fact_23_map__eq__map__tailrec, axiom,
    ((map_dB_nat = map_tailrec_dB_nat))). % map_eq_map_tailrec
thf(fact_24_map__eq__map__tailrec, axiom,
    ((map_nat_nat = map_tailrec_nat_nat))). % map_eq_map_tailrec
thf(fact_25_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_26_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_27_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_28_ex__head__tail, axiom,
    ((![T : dB]: (?[Ts2 : list_dB, H : dB]: ((T = (foldl_dB_dB @ app @ H @ Ts2)) & ((?[N2 : nat]: (H = (var @ N2))) | (?[U2 : dB]: (H = (abs @ U2))))))))). % ex_head_tail
thf(fact_29_linorder__neqE__nat, axiom,
    ((![X2 : nat, Y : nat]: ((~ ((X2 = Y))) => ((~ ((ord_less_nat @ X2 @ Y))) => (ord_less_nat @ Y @ X2)))))). % linorder_neqE_nat
thf(fact_30_infinite__descent, axiom,
    ((![P : nat > $o, N : nat]: ((![N2 : nat]: ((~ ((P @ N2))) => (?[M2 : nat]: ((ord_less_nat @ M2 @ N2) & (~ ((P @ M2))))))) => (P @ N))))). % infinite_descent
thf(fact_31_nat__less__induct, axiom,
    ((![P : nat > $o, N : nat]: ((![N2 : nat]: ((![M2 : nat]: ((ord_less_nat @ M2 @ N2) => (P @ M2))) => (P @ N2))) => (P @ N))))). % nat_less_induct
thf(fact_32_dB_Oinject_I3_J, axiom,
    ((![X3 : dB, Y3 : dB]: (((abs @ X3) = (abs @ Y3)) = (X3 = Y3))))). % dB.inject(3)
thf(fact_33_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_34_Nil__is__map__conv, axiom,
    ((![F : dB > nat, Xs2 : list_dB]: ((nil_nat = (map_dB_nat @ F @ Xs2)) = (Xs2 = nil_dB))))). % Nil_is_map_conv
thf(fact_35_Nil__is__map__conv, axiom,
    ((![F : nat > nat, Xs2 : list_nat]: ((nil_nat = (map_nat_nat @ F @ Xs2)) = (Xs2 = nil_nat))))). % Nil_is_map_conv
thf(fact_36_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_37_map__is__Nil__conv, axiom,
    ((![F : dB > nat, Xs2 : list_dB]: (((map_dB_nat @ F @ Xs2) = nil_nat) = (Xs2 = nil_dB))))). % map_is_Nil_conv
thf(fact_38_map__is__Nil__conv, axiom,
    ((![F : nat > nat, Xs2 : list_nat]: (((map_nat_nat @ F @ Xs2) = nil_nat) = (Xs2 = nil_nat))))). % map_is_Nil_conv
thf(fact_39_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_40_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_41_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_42_listsp__simps_I1_J, axiom,
    ((![A : dB > $o]: (listsp_dB @ A @ nil_dB)))). % listsp_simps(1)
thf(fact_43_subst__eq, axiom,
    ((![K : nat, U : dB]: ((subst @ (var @ K) @ U @ K) = U)))). % subst_eq
thf(fact_44_subst__lift, axiom,
    ((![T : dB, K : nat, S : dB]: ((subst @ (lift @ T @ K) @ S @ K) = T)))). % subst_lift
thf(fact_45_subst__lt, axiom,
    ((![J : nat, I : nat, U : dB]: ((ord_less_nat @ J @ I) => ((subst @ (var @ J) @ U @ I) = (var @ J)))))). % subst_lt
thf(fact_46_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_47_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_48_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_49_list_Osimps_I8_J, axiom,
    ((![F : dB > dB]: ((map_dB_dB @ F @ nil_dB) = nil_dB)))). % list.simps(8)
thf(fact_50_list_Osimps_I8_J, axiom,
    ((![F : dB > nat]: ((map_dB_nat @ F @ nil_dB) = nil_nat)))). % list.simps(8)
thf(fact_51_list_Osimps_I8_J, axiom,
    ((![F : nat > nat]: ((map_nat_nat @ F @ nil_nat) = nil_nat)))). % list.simps(8)
thf(fact_52_dB_Odistinct_I5_J, axiom,
    ((![X21 : dB, X22 : dB, X3 : dB]: (~ (((app @ X21 @ X22) = (abs @ X3))))))). % dB.distinct(5)
thf(fact_53_dB_Odistinct_I3_J, axiom,
    ((![X1 : nat, X3 : dB]: (~ (((var @ X1) = (abs @ X3))))))). % dB.distinct(3)
thf(fact_54_foldl__Nil, axiom,
    ((![F : nat > nat > nat, A2 : nat]: ((foldl_nat_nat @ F @ A2 @ nil_nat) = A2)))). % foldl_Nil
thf(fact_55_foldl__Nil, axiom,
    ((![F : dB > nat > dB, A2 : dB]: ((foldl_dB_nat @ F @ A2 @ nil_nat) = A2)))). % foldl_Nil
thf(fact_56_foldl__Nil, axiom,
    ((![F : dB > dB > dB, A2 : dB]: ((foldl_dB_dB @ F @ A2 @ nil_dB) = A2)))). % foldl_Nil
thf(fact_57_foldl__Nil, axiom,
    ((![F : nat > dB > nat, A2 : nat]: ((foldl_nat_dB @ F @ A2 @ nil_dB) = A2)))). % foldl_Nil
thf(fact_58_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_59_listsp_ONil, axiom,
    ((![A : dB > $o]: (listsp_dB @ A @ nil_dB)))). % listsp.Nil
thf(fact_60_Lambda, axiom,
    ((![R : dB]: ((it @ R) => (it @ (abs @ R)))))). % Lambda
thf(fact_61_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_62_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_63_nat__neq__iff, axiom,
    ((![M : nat, N : nat]: ((~ ((M = N))) = (((ord_less_nat @ M @ N)) | ((ord_less_nat @ N @ M))))))). % nat_neq_iff
thf(fact_64_less__not__refl, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_not_refl
thf(fact_65_less__not__refl2, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ N @ M) => (~ ((M = N))))))). % less_not_refl2
thf(fact_66_less__not__refl3, axiom,
    ((![S : nat, T : nat]: ((ord_less_nat @ S @ T) => (~ ((S = T))))))). % less_not_refl3
thf(fact_67_less__irrefl__nat, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_irrefl_nat
thf(fact_68_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_69_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_70_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_71_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_72_list__ex1__simps_I1_J, axiom,
    ((![P : dB > $o]: (~ ((list_ex1_dB @ P @ nil_dB)))))). % list_ex1_simps(1)
thf(fact_73_Apps__dB__induct, axiom,
    ((![P : dB > $o, T : dB]: ((![N2 : nat, Ts2 : list_dB]: ((![X5 : dB]: ((member_dB2 @ X5 @ (set_dB2 @ Ts2)) => (P @ X5))) => (P @ (foldl_dB_dB @ app @ (var @ N2) @ 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_74_bind__simps_I1_J, axiom,
    ((![F : dB > list_dB]: ((bind_dB_dB @ nil_dB @ F) = nil_dB)))). % bind_simps(1)
thf(fact_75_member__rec_I2_J, axiom,
    ((![Y : dB]: (~ ((member_dB @ nil_dB @ Y)))))). % member_rec(2)
thf(fact_76_bot__nat__0_Onot__eq__extremum, axiom,
    ((![A2 : nat]: ((~ ((A2 = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ A2))))). % bot_nat_0.not_eq_extremum
thf(fact_77_less__nat__zero__code, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_78_neq0__conv, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N))))). % neq0_conv
thf(fact_79_map__eq__conv, axiom,
    ((![F : nat > nat, Xs2 : list_nat, G : nat > nat]: (((map_nat_nat @ F @ Xs2) = (map_nat_nat @ G @ Xs2)) = (![X : nat]: (((member_nat @ X @ (set_nat2 @ Xs2))) => (((F @ X) = (G @ X))))))))). % map_eq_conv
thf(fact_80_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_81_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_82_in__listspI, axiom,
    ((![Xs2 : list_dB, A : dB > $o]: ((![X4 : dB]: ((member_dB2 @ X4 @ (set_dB2 @ Xs2)) => (A @ X4))) => (listsp_dB @ A @ Xs2))))). % in_listspI
thf(fact_83_in__set__member, axiom,
    ((![X2 : dB, Xs2 : list_dB]: ((member_dB2 @ X2 @ (set_dB2 @ Xs2)) = (member_dB @ Xs2 @ X2))))). % in_set_member
thf(fact_84_list__ex1__iff, axiom,
    ((list_ex1_dB = (^[P2 : dB > $o]: (^[Xs : list_dB]: (?[X : dB]: (((((member_dB2 @ X @ (set_dB2 @ Xs))) & ((P2 @ X)))) & ((![Y2 : dB]: (((((member_dB2 @ Y2 @ (set_dB2 @ Xs))) & ((P2 @ Y2)))) => ((Y2 = X)))))))))))). % list_ex1_iff
thf(fact_85_list_Oinj__map__strong, axiom,
    ((![X2 : list_nat, Xa : list_nat, F : nat > nat, Fa : nat > nat]: ((![Z : nat, Za : nat]: ((member_nat @ Z @ (set_nat2 @ X2)) => ((member_nat @ Za @ (set_nat2 @ Xa)) => (((F @ Z) = (Fa @ Za)) => (Z = Za))))) => (((map_nat_nat @ F @ X2) = (map_nat_nat @ Fa @ Xa)) => (X2 = Xa)))))). % list.inj_map_strong
thf(fact_86_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_87_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_88_list_Omap__cong0, axiom,
    ((![X2 : list_nat, F : nat > nat, G : nat > nat]: ((![Z : nat]: ((member_nat @ Z @ (set_nat2 @ X2)) => ((F @ Z) = (G @ Z)))) => ((map_nat_nat @ F @ X2) = (map_nat_nat @ G @ X2)))))). % list.map_cong0
thf(fact_89_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_90_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_91_list_Omap__cong, axiom,
    ((![X2 : list_nat, Ya : list_nat, F : nat > nat, G : nat > nat]: ((X2 = Ya) => ((![Z : nat]: ((member_nat @ Z @ (set_nat2 @ Ya)) => ((F @ Z) = (G @ Z)))) => ((map_nat_nat @ F @ X2) = (map_nat_nat @ G @ Ya))))))). % list.map_cong
thf(fact_92_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_93_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_94_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))) => ((?[Y2 : dB]: (X = (F @ Y2)))))))))). % ex_map_conv
thf(fact_95_ex__map__conv, axiom,
    ((![Ys : list_nat, F : nat > nat]: ((?[Xs : list_nat]: (Ys = (map_nat_nat @ F @ Xs))) = (![X : nat]: (((member_nat @ X @ (set_nat2 @ Ys))) => ((?[Y2 : nat]: (X = (F @ Y2)))))))))). % ex_map_conv
thf(fact_96_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))) => ((?[Y2 : dB]: (X = (F @ Y2)))))))))). % ex_map_conv
thf(fact_97_map__cong, axiom,
    ((![Xs2 : list_nat, Ys : list_nat, F : nat > nat, G : nat > nat]: ((Xs2 = Ys) => ((![X4 : nat]: ((member_nat @ X4 @ (set_nat2 @ Ys)) => ((F @ X4) = (G @ X4)))) => ((map_nat_nat @ F @ Xs2) = (map_nat_nat @ G @ Ys))))))). % map_cong
thf(fact_98_map__cong, axiom,
    ((![Xs2 : list_dB, Ys : list_dB, F : dB > dB, G : dB > dB]: ((Xs2 = Ys) => ((![X4 : dB]: ((member_dB2 @ X4 @ (set_dB2 @ Ys)) => ((F @ X4) = (G @ X4)))) => ((map_dB_dB @ F @ Xs2) = (map_dB_dB @ G @ Ys))))))). % map_cong
thf(fact_99_map__cong, axiom,
    ((![Xs2 : list_dB, Ys : list_dB, F : dB > nat, G : dB > nat]: ((Xs2 = Ys) => ((![X4 : dB]: ((member_dB2 @ X4 @ (set_dB2 @ Ys)) => ((F @ X4) = (G @ X4)))) => ((map_dB_nat @ F @ Xs2) = (map_dB_nat @ G @ Ys))))))). % map_cong
thf(fact_100_map__idI, axiom,
    ((![Xs2 : list_nat, F : nat > nat]: ((![X4 : nat]: ((member_nat @ X4 @ (set_nat2 @ Xs2)) => ((F @ X4) = X4))) => ((map_nat_nat @ F @ Xs2) = Xs2))))). % map_idI
thf(fact_101_map__idI, axiom,
    ((![Xs2 : list_dB, F : dB > dB]: ((![X4 : dB]: ((member_dB2 @ X4 @ (set_dB2 @ Xs2)) => ((F @ X4) = X4))) => ((map_dB_dB @ F @ Xs2) = Xs2))))). % map_idI
thf(fact_102_map__ext, axiom,
    ((![Xs2 : list_nat, F : nat > nat, G : nat > nat]: ((![X4 : nat]: ((member_nat @ X4 @ (set_nat2 @ Xs2)) => ((F @ X4) = (G @ X4)))) => ((map_nat_nat @ F @ Xs2) = (map_nat_nat @ G @ Xs2)))))). % map_ext
thf(fact_103_map__ext, axiom,
    ((![Xs2 : list_dB, F : dB > dB, G : dB > dB]: ((![X4 : dB]: ((member_dB2 @ X4 @ (set_dB2 @ Xs2)) => ((F @ X4) = (G @ X4)))) => ((map_dB_dB @ F @ Xs2) = (map_dB_dB @ G @ Xs2)))))). % map_ext
thf(fact_104_map__ext, axiom,
    ((![Xs2 : list_dB, F : dB > nat, G : dB > nat]: ((![X4 : dB]: ((member_dB2 @ X4 @ (set_dB2 @ Xs2)) => ((F @ X4) = (G @ X4)))) => ((map_dB_nat @ F @ Xs2) = (map_dB_nat @ G @ Xs2)))))). % map_ext
thf(fact_105_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_nat @ 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_106_foldl__cong, axiom,
    ((![A2 : dB, B2 : dB, L : list_nat, K : list_nat, F : dB > nat > dB, G : dB > nat > dB]: ((A2 = B2) => ((L = K) => ((![A4 : dB, X4 : nat]: ((member_nat @ X4 @ (set_nat2 @ L)) => ((F @ A4 @ X4) = (G @ A4 @ X4)))) => ((foldl_dB_nat @ F @ A2 @ L) = (foldl_dB_nat @ G @ B2 @ K)))))))). % foldl_cong
thf(fact_107_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_108_foldl__cong, axiom,
    ((![A2 : nat, B2 : nat, L : list_dB, K : list_dB, F : nat > dB > nat, G : nat > dB > nat]: ((A2 = B2) => ((L = K) => ((![A4 : nat, X4 : dB]: ((member_dB2 @ X4 @ (set_dB2 @ L)) => ((F @ A4 @ X4) = (G @ A4 @ X4)))) => ((foldl_nat_dB @ F @ A2 @ L) = (foldl_nat_dB @ G @ B2 @ K)))))))). % foldl_cong
thf(fact_109_bot__nat__0_Oextremum__strict, axiom,
    ((![A2 : nat]: (~ ((ord_less_nat @ A2 @ zero_zero_nat)))))). % bot_nat_0.extremum_strict
thf(fact_110_infinite__descent0, axiom,
    ((![P : nat > $o, N : nat]: ((P @ zero_zero_nat) => ((![N2 : nat]: ((ord_less_nat @ zero_zero_nat @ N2) => ((~ ((P @ N2))) => (?[M2 : nat]: ((ord_less_nat @ M2 @ N2) & (~ ((P @ M2)))))))) => (P @ N)))))). % infinite_descent0
thf(fact_111_gr__implies__not0, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not0
thf(fact_112_less__zeroE, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_zeroE
thf(fact_113_not__less0, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less0
thf(fact_114_not__gr0, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr0
thf(fact_115_gr0I, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr0I
thf(fact_116_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_117_in__listspD, axiom,
    ((![A : dB > $o, Xs2 : list_dB]: ((listsp_dB @ A @ Xs2) => (![X5 : dB]: ((member_dB2 @ X5 @ (set_dB2 @ Xs2)) => (A @ X5))))))). % in_listspD
thf(fact_118_pinf_I1_J, axiom,
    ((![P : nat > $o, P3 : nat > $o, Q : nat > $o, Q2 : nat > $o]: ((?[Z2 : nat]: (![X4 : nat]: ((ord_less_nat @ Z2 @ X4) => ((P @ X4) = (P3 @ X4))))) => ((?[Z2 : nat]: (![X4 : nat]: ((ord_less_nat @ Z2 @ X4) => ((Q @ X4) = (Q2 @ X4))))) => (?[Z : nat]: (![X5 : nat]: ((ord_less_nat @ Z @ X5) => ((((P @ X5)) & ((Q @ X5))) = (((P3 @ X5)) & ((Q2 @ X5)))))))))))). % pinf(1)
thf(fact_119_pinf_I1_J, axiom,
    ((![P : int > $o, P3 : int > $o, Q : int > $o, Q2 : int > $o]: ((?[Z2 : int]: (![X4 : int]: ((ord_less_int @ Z2 @ X4) => ((P @ X4) = (P3 @ X4))))) => ((?[Z2 : int]: (![X4 : int]: ((ord_less_int @ Z2 @ X4) => ((Q @ X4) = (Q2 @ X4))))) => (?[Z : int]: (![X5 : int]: ((ord_less_int @ Z @ X5) => ((((P @ X5)) & ((Q @ X5))) = (((P3 @ X5)) & ((Q2 @ X5)))))))))))). % pinf(1)
thf(fact_120_pinf_I2_J, axiom,
    ((![P : nat > $o, P3 : nat > $o, Q : nat > $o, Q2 : nat > $o]: ((?[Z2 : nat]: (![X4 : nat]: ((ord_less_nat @ Z2 @ X4) => ((P @ X4) = (P3 @ X4))))) => ((?[Z2 : nat]: (![X4 : nat]: ((ord_less_nat @ Z2 @ X4) => ((Q @ X4) = (Q2 @ X4))))) => (?[Z : nat]: (![X5 : nat]: ((ord_less_nat @ Z @ X5) => ((((P @ X5)) | ((Q @ X5))) = (((P3 @ X5)) | ((Q2 @ X5)))))))))))). % pinf(2)
thf(fact_121_pinf_I2_J, axiom,
    ((![P : int > $o, P3 : int > $o, Q : int > $o, Q2 : int > $o]: ((?[Z2 : int]: (![X4 : int]: ((ord_less_int @ Z2 @ X4) => ((P @ X4) = (P3 @ X4))))) => ((?[Z2 : int]: (![X4 : int]: ((ord_less_int @ Z2 @ X4) => ((Q @ X4) = (Q2 @ X4))))) => (?[Z : int]: (![X5 : int]: ((ord_less_int @ Z @ X5) => ((((P @ X5)) | ((Q @ X5))) = (((P3 @ X5)) | ((Q2 @ X5)))))))))))). % pinf(2)
thf(fact_122_pinf_I3_J, axiom,
    ((![T : nat]: (?[Z : nat]: (![X5 : nat]: ((ord_less_nat @ Z @ X5) => (~ ((X5 = T))))))))). % pinf(3)
thf(fact_123_pinf_I3_J, axiom,
    ((![T : int]: (?[Z : int]: (![X5 : int]: ((ord_less_int @ Z @ X5) => (~ ((X5 = T))))))))). % pinf(3)
thf(fact_124_pinf_I4_J, axiom,
    ((![T : nat]: (?[Z : nat]: (![X5 : nat]: ((ord_less_nat @ Z @ X5) => (~ ((X5 = T))))))))). % pinf(4)
thf(fact_125_pinf_I4_J, axiom,
    ((![T : int]: (?[Z : int]: (![X5 : int]: ((ord_less_int @ Z @ X5) => (~ ((X5 = T))))))))). % pinf(4)
thf(fact_126_pinf_I5_J, axiom,
    ((![T : nat]: (?[Z : nat]: (![X5 : nat]: ((ord_less_nat @ Z @ X5) => (~ ((ord_less_nat @ X5 @ T))))))))). % pinf(5)
thf(fact_127_pinf_I5_J, axiom,
    ((![T : int]: (?[Z : int]: (![X5 : int]: ((ord_less_int @ Z @ X5) => (~ ((ord_less_int @ X5 @ T))))))))). % pinf(5)
thf(fact_128_pinf_I7_J, axiom,
    ((![T : nat]: (?[Z : nat]: (![X5 : nat]: ((ord_less_nat @ Z @ X5) => (ord_less_nat @ T @ X5))))))). % pinf(7)
thf(fact_129_pinf_I7_J, axiom,
    ((![T : int]: (?[Z : int]: (![X5 : int]: ((ord_less_int @ Z @ X5) => (ord_less_int @ T @ X5))))))). % pinf(7)
thf(fact_130_minf_I1_J, axiom,
    ((![P : nat > $o, P3 : nat > $o, Q : nat > $o, Q2 : nat > $o]: ((?[Z2 : nat]: (![X4 : nat]: ((ord_less_nat @ X4 @ Z2) => ((P @ X4) = (P3 @ X4))))) => ((?[Z2 : nat]: (![X4 : nat]: ((ord_less_nat @ X4 @ Z2) => ((Q @ X4) = (Q2 @ X4))))) => (?[Z : nat]: (![X5 : nat]: ((ord_less_nat @ X5 @ Z) => ((((P @ X5)) & ((Q @ X5))) = (((P3 @ X5)) & ((Q2 @ X5)))))))))))). % minf(1)
thf(fact_131_minf_I1_J, axiom,
    ((![P : int > $o, P3 : int > $o, Q : int > $o, Q2 : int > $o]: ((?[Z2 : int]: (![X4 : int]: ((ord_less_int @ X4 @ Z2) => ((P @ X4) = (P3 @ X4))))) => ((?[Z2 : int]: (![X4 : int]: ((ord_less_int @ X4 @ Z2) => ((Q @ X4) = (Q2 @ X4))))) => (?[Z : int]: (![X5 : int]: ((ord_less_int @ X5 @ Z) => ((((P @ X5)) & ((Q @ X5))) = (((P3 @ X5)) & ((Q2 @ X5)))))))))))). % minf(1)
thf(fact_132_minf_I2_J, axiom,
    ((![P : nat > $o, P3 : nat > $o, Q : nat > $o, Q2 : nat > $o]: ((?[Z2 : nat]: (![X4 : nat]: ((ord_less_nat @ X4 @ Z2) => ((P @ X4) = (P3 @ X4))))) => ((?[Z2 : nat]: (![X4 : nat]: ((ord_less_nat @ X4 @ Z2) => ((Q @ X4) = (Q2 @ X4))))) => (?[Z : nat]: (![X5 : nat]: ((ord_less_nat @ X5 @ Z) => ((((P @ X5)) | ((Q @ X5))) = (((P3 @ X5)) | ((Q2 @ X5)))))))))))). % minf(2)
thf(fact_133_minf_I2_J, axiom,
    ((![P : int > $o, P3 : int > $o, Q : int > $o, Q2 : int > $o]: ((?[Z2 : int]: (![X4 : int]: ((ord_less_int @ X4 @ Z2) => ((P @ X4) = (P3 @ X4))))) => ((?[Z2 : int]: (![X4 : int]: ((ord_less_int @ X4 @ Z2) => ((Q @ X4) = (Q2 @ X4))))) => (?[Z : int]: (![X5 : int]: ((ord_less_int @ X5 @ Z) => ((((P @ X5)) | ((Q @ X5))) = (((P3 @ X5)) | ((Q2 @ X5)))))))))))). % minf(2)
thf(fact_134_minf_I3_J, axiom,
    ((![T : nat]: (?[Z : nat]: (![X5 : nat]: ((ord_less_nat @ X5 @ Z) => (~ ((X5 = T))))))))). % minf(3)
thf(fact_135_minf_I3_J, axiom,
    ((![T : int]: (?[Z : int]: (![X5 : int]: ((ord_less_int @ X5 @ Z) => (~ ((X5 = T))))))))). % minf(3)
thf(fact_136_minf_I4_J, axiom,
    ((![T : nat]: (?[Z : nat]: (![X5 : nat]: ((ord_less_nat @ X5 @ Z) => (~ ((X5 = T))))))))). % minf(4)
thf(fact_137_minf_I4_J, axiom,
    ((![T : int]: (?[Z : int]: (![X5 : int]: ((ord_less_int @ X5 @ Z) => (~ ((X5 = T))))))))). % minf(4)
thf(fact_138_minf_I5_J, axiom,
    ((![T : nat]: (?[Z : nat]: (![X5 : nat]: ((ord_less_nat @ X5 @ Z) => (ord_less_nat @ X5 @ T))))))). % minf(5)
thf(fact_139_minf_I5_J, axiom,
    ((![T : int]: (?[Z : int]: (![X5 : int]: ((ord_less_int @ X5 @ Z) => (ord_less_int @ X5 @ T))))))). % minf(5)
thf(fact_140_minf_I7_J, axiom,
    ((![T : nat]: (?[Z : nat]: (![X5 : nat]: ((ord_less_nat @ X5 @ Z) => (~ ((ord_less_nat @ T @ X5))))))))). % minf(7)
thf(fact_141_minf_I7_J, axiom,
    ((![T : int]: (?[Z : int]: (![X5 : int]: ((ord_less_int @ X5 @ Z) => (~ ((ord_less_int @ T @ X5))))))))). % minf(7)
thf(fact_142_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_143_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_144_lem, axiom,
    ((![P : dB > $o, T : dB, N : nat]: ((![N2 : nat, Ts2 : list_dB]: ((![X5 : dB]: ((member_dB2 @ X5 @ (set_dB2 @ Ts2)) => (P @ X5))) => (P @ (foldl_dB_dB @ app @ (var @ N2) @ 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_145_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_nat @ zero_zero_nat @ zero_zero_nat))))). % less_numeral_extra(3)
thf(fact_146_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_int @ zero_zero_int @ zero_zero_int))))). % less_numeral_extra(3)
thf(fact_147_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_148_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_149_zero__reorient, axiom,
    ((![X2 : nat]: ((zero_zero_nat = X2) = (X2 = zero_zero_nat))))). % zero_reorient
thf(fact_150_zero__reorient, axiom,
    ((![X2 : int]: ((zero_zero_int = X2) = (X2 = zero_zero_int))))). % zero_reorient
thf(fact_151_dB_Osize_I4_J, axiom,
    ((![X1 : nat]: ((size_size_dB @ (var @ X1)) = zero_zero_nat)))). % dB.size(4)
thf(fact_152_zero__less__iff__neq__zero, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) = (~ ((N = zero_zero_nat))))))). % zero_less_iff_neq_zero
thf(fact_153_gr__implies__not__zero, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not_zero
thf(fact_154_not__less__zero, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less_zero
thf(fact_155_gr__zeroI, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr_zeroI
thf(fact_156_count__notin, axiom,
    ((![X2 : dB, Xs2 : list_dB]: ((~ ((member_dB2 @ X2 @ (set_dB2 @ Xs2)))) => ((count_list_dB @ Xs2 @ X2) = zero_zero_nat))))). % count_notin
thf(fact_157_dB_Osize__gen_I1_J, axiom,
    ((![X1 : nat]: ((size_dB @ (var @ X1)) = zero_zero_nat)))). % dB.size_gen(1)
thf(fact_158_of__nat__0__less__iff, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ (semiri1382578993at_nat @ N)) = (ord_less_nat @ zero_zero_nat @ N))))). % of_nat_0_less_iff
thf(fact_159_of__nat__0__less__iff, axiom,
    ((![N : nat]: ((ord_less_int @ zero_zero_int @ (semiri2019852685at_int @ N)) = (ord_less_nat @ zero_zero_nat @ N))))). % of_nat_0_less_iff
thf(fact_160_of__nat__eq__iff, axiom,
    ((![M : nat, N : nat]: (((semiri2019852685at_int @ M) = (semiri2019852685at_int @ N)) = (M = N))))). % of_nat_eq_iff
thf(fact_161_length__map, axiom,
    ((![F : nat > nat, Xs2 : list_nat]: ((size_size_list_nat @ (map_nat_nat @ F @ Xs2)) = (size_size_list_nat @ Xs2))))). % length_map
thf(fact_162_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_163_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_164_length__0__conv, axiom,
    ((![Xs2 : list_dB]: (((size_size_list_dB @ Xs2) = zero_zero_nat) = (Xs2 = nil_dB))))). % length_0_conv
thf(fact_165_of__nat__eq__0__iff, axiom,
    ((![M : nat]: (((semiri1382578993at_nat @ M) = zero_zero_nat) = (M = zero_zero_nat))))). % of_nat_eq_0_iff
thf(fact_166_of__nat__eq__0__iff, axiom,
    ((![M : nat]: (((semiri2019852685at_int @ M) = zero_zero_int) = (M = zero_zero_nat))))). % of_nat_eq_0_iff
thf(fact_167_of__nat__0__eq__iff, axiom,
    ((![N : nat]: ((zero_zero_nat = (semiri1382578993at_nat @ N)) = (zero_zero_nat = N))))). % of_nat_0_eq_iff
thf(fact_168_of__nat__0__eq__iff, axiom,
    ((![N : nat]: ((zero_zero_int = (semiri2019852685at_int @ N)) = (zero_zero_nat = N))))). % of_nat_0_eq_iff
thf(fact_169_of__nat__0, axiom,
    (((semiri1382578993at_nat @ zero_zero_nat) = zero_zero_nat))). % of_nat_0
thf(fact_170_of__nat__0, axiom,
    (((semiri2019852685at_int @ zero_zero_nat) = zero_zero_int))). % of_nat_0
thf(fact_171_of__nat__less__iff, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ (semiri1382578993at_nat @ M) @ (semiri1382578993at_nat @ N)) = (ord_less_nat @ M @ N))))). % of_nat_less_iff
thf(fact_172_of__nat__less__iff, axiom,
    ((![M : nat, N : nat]: ((ord_less_int @ (semiri2019852685at_int @ M) @ (semiri2019852685at_int @ N)) = (ord_less_nat @ M @ N))))). % of_nat_less_iff
thf(fact_173_length__greater__0__conv, axiom,
    ((![Xs2 : list_dB]: ((ord_less_nat @ zero_zero_nat @ (size_size_list_dB @ Xs2)) = (~ ((Xs2 = nil_dB))))))). % length_greater_0_conv
thf(fact_174_length__induct, axiom,
    ((![P : list_dB > $o, Xs2 : list_dB]: ((![Xs3 : list_dB]: ((![Ys2 : list_dB]: ((ord_less_nat @ (size_size_list_dB @ Ys2) @ (size_size_list_dB @ Xs3)) => (P @ Ys2))) => (P @ Xs3))) => (P @ Xs2))))). % length_induct
thf(fact_175_map__eq__imp__length__eq, axiom,
    ((![F : nat > nat, Xs2 : list_nat, G : nat > nat, Ys : list_nat]: (((map_nat_nat @ F @ Xs2) = (map_nat_nat @ G @ Ys)) => ((size_size_list_nat @ Xs2) = (size_size_list_nat @ Ys)))))). % map_eq_imp_length_eq
thf(fact_176_map__eq__imp__length__eq, axiom,
    ((![F : nat > nat, Xs2 : list_nat, G : dB > nat, Ys : list_dB]: (((map_nat_nat @ F @ Xs2) = (map_dB_nat @ G @ Ys)) => ((size_size_list_nat @ Xs2) = (size_size_list_dB @ Ys)))))). % map_eq_imp_length_eq
thf(fact_177_map__eq__imp__length__eq, axiom,
    ((![F : dB > nat, Xs2 : list_dB, G : nat > nat, Ys : list_nat]: (((map_dB_nat @ F @ Xs2) = (map_nat_nat @ G @ Ys)) => ((size_size_list_dB @ Xs2) = (size_size_list_nat @ Ys)))))). % map_eq_imp_length_eq
thf(fact_178_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_179_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_180_of__nat__less__0__iff, axiom,
    ((![M : nat]: (~ ((ord_less_nat @ (semiri1382578993at_nat @ M) @ zero_zero_nat)))))). % of_nat_less_0_iff
thf(fact_181_of__nat__less__0__iff, axiom,
    ((![M : nat]: (~ ((ord_less_int @ (semiri2019852685at_int @ M) @ zero_zero_int)))))). % of_nat_less_0_iff
thf(fact_182_less__imp__of__nat__less, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (ord_less_nat @ (semiri1382578993at_nat @ M) @ (semiri1382578993at_nat @ N)))))). % less_imp_of_nat_less
thf(fact_183_less__imp__of__nat__less, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (ord_less_int @ (semiri2019852685at_int @ M) @ (semiri2019852685at_int @ N)))))). % less_imp_of_nat_less
thf(fact_184_of__nat__less__imp__less, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ (semiri1382578993at_nat @ M) @ (semiri1382578993at_nat @ N)) => (ord_less_nat @ M @ N))))). % of_nat_less_imp_less
thf(fact_185_of__nat__less__imp__less, axiom,
    ((![M : nat, N : nat]: ((ord_less_int @ (semiri2019852685at_int @ M) @ (semiri2019852685at_int @ N)) => (ord_less_nat @ M @ N))))). % of_nat_less_imp_less
thf(fact_186_list_Osize_I3_J, axiom,
    (((size_size_list_dB @ nil_dB) = zero_zero_nat))). % list.size(3)
thf(fact_187_length__pos__if__in__set, axiom,
    ((![X2 : dB, Xs2 : list_dB]: ((member_dB2 @ X2 @ (set_dB2 @ Xs2)) => (ord_less_nat @ zero_zero_nat @ (size_size_list_dB @ Xs2)))))). % length_pos_if_in_set
thf(fact_188_count__list_Osimps_I1_J, axiom,
    ((![Y : dB]: ((count_list_dB @ nil_dB @ Y) = zero_zero_nat)))). % count_list.simps(1)
thf(fact_189_pos__int__cases, axiom,
    ((![K : int]: ((ord_less_int @ zero_zero_int @ K) => (~ ((![N2 : nat]: ((K = (semiri2019852685at_int @ N2)) => (~ ((ord_less_nat @ zero_zero_nat @ N2))))))))))). % pos_int_cases
thf(fact_190_zero__less__imp__eq__int, axiom,
    ((![K : int]: ((ord_less_int @ zero_zero_int @ K) => (?[N2 : nat]: ((ord_less_nat @ zero_zero_nat @ N2) & (K = (semiri2019852685at_int @ N2)))))))). % zero_less_imp_eq_int
thf(fact_191_Ex__list__of__length, axiom,
    ((![N : nat]: (?[Xs3 : list_dB]: ((size_size_list_dB @ Xs3) = N))))). % Ex_list_of_length
thf(fact_192_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_193_int__int__eq, axiom,
    ((![M : nat, N : nat]: (((semiri2019852685at_int @ M) = (semiri2019852685at_int @ N)) = (M = N))))). % int_int_eq
thf(fact_194_nat__less__as__int, axiom,
    ((ord_less_nat = (^[A3 : nat]: (^[B3 : nat]: (ord_less_int @ (semiri2019852685at_int @ A3) @ (semiri2019852685at_int @ B3))))))). % nat_less_as_int
thf(fact_195_nat__int__comparison_I2_J, axiom,
    ((ord_less_nat = (^[A3 : nat]: (^[B3 : nat]: (ord_less_int @ (semiri2019852685at_int @ A3) @ (semiri2019852685at_int @ B3))))))). % nat_int_comparison(2)
thf(fact_196_verit__comp__simplify1_I1_J, axiom,
    ((![A2 : nat]: (~ ((ord_less_nat @ A2 @ A2)))))). % verit_comp_simplify1(1)
thf(fact_197_verit__comp__simplify1_I1_J, axiom,
    ((![A2 : int]: (~ ((ord_less_int @ A2 @ A2)))))). % verit_comp_simplify1(1)
thf(fact_198_int__if, axiom,
    ((![P : $o, A2 : nat, B2 : nat]: ((P => ((semiri2019852685at_int @ (if_nat @ P @ A2 @ B2)) = (semiri2019852685at_int @ A2))) & ((~ (P)) => ((semiri2019852685at_int @ (if_nat @ P @ A2 @ B2)) = (semiri2019852685at_int @ B2))))))). % int_if
thf(fact_199_nat__int__comparison_I1_J, axiom,
    (((^[Y4 : nat]: (^[Z3 : nat]: (Y4 = Z3))) = (^[A3 : nat]: (^[B3 : nat]: ((semiri2019852685at_int @ A3) = (semiri2019852685at_int @ B3))))))). % nat_int_comparison(1)
thf(fact_200_int__ops_I1_J, axiom,
    (((semiri2019852685at_int @ zero_zero_nat) = zero_zero_int))). % int_ops(1)
thf(fact_201_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_202_length__code, axiom,
    ((size_size_list_dB = (gen_length_dB @ zero_zero_nat)))). % length_code
thf(fact_203_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_204_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

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

% Conjectures (2)
thf(conj_0, hypothesis,
    ((listsp_dB @ (^[X : dB]: (((it @ X)) & ((![Y2 : nat]: (it @ (lift @ X @ Y2)))))) @ rs))).
thf(conj_1, conjecture,
    (((~ ((ord_less_nat @ n @ ia))) | (it @ (foldl_dB_dB @ app @ (var @ n) @ (map_dB_dB @ (^[T2 : dB]: (lift @ T2 @ ia)) @ rs)))))).
