% 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_261__5212222_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:38:42.450

% Could-be-implicit typings (9)
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__List__Olist_It__Nat__Onat_J, type,
    list_nat : $tType).
thf(ty_n_t__List__Olist_It__Int__Oint_J, type,
    list_int : $tType).
thf(ty_n_t__Set__Oset_It__Nat__Onat_J, type,
    set_nat : $tType).
thf(ty_n_t__LambdaType__Otype, type,
    type : $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_Ouminus__class_Ouminus_001t__Int__Oint, type,
    uminus_uminus_int : int > int).
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_InductTermi_OIT, type,
    it : dB > $o).
thf(sy_c_Int_Onat, type,
    nat2 : int > nat).
thf(sy_c_LambdaType_Oshift_001t__LambdaType__Otype, type,
    shift_type : (nat > type) > nat > type > nat > type).
thf(sy_c_LambdaType_Otype_OFun, type,
    fun : type > type > type).
thf(sy_c_LambdaType_Otyping, type,
    typing : (nat > type) > dB > type > $o).
thf(sy_c_Lambda_OdB_OAbs, type,
    abs : dB > dB).
thf(sy_c_Lambda_Olift, type,
    lift : 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_Ocount__list_001t__Lambda__OdB, type,
    count_list_dB : list_dB > dB > nat).
thf(sy_c_List_Oinsert_001t__Lambda__OdB, type,
    insert_dB : dB > list_dB > list_dB).
thf(sy_c_List_Olist_OCons_001t__Int__Oint, type,
    cons_int : int > list_int > list_int).
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_OCons_001t__Nat__Onat, type,
    cons_nat : nat > list_nat > list_nat).
thf(sy_c_List_Olist_ONil_001t__Int__Oint, type,
    nil_int : list_int).
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_ONil_001t__Nat__Onat, type,
    nil_nat : list_nat).
thf(sy_c_List_Olist__ex1_001t__Lambda__OdB, type,
    list_ex1_dB : (dB > $o) > list_dB > $o).
thf(sy_c_List_Olistrelp_001t__Lambda__OdB_001t__Lambda__OdB, type,
    listrelp_dB_dB : (dB > dB > $o) > list_dB > 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__rev_001t__Lambda__OdB_001t__Lambda__OdB, type,
    map_ta277069629_dB_dB : (dB > dB) > list_dB > list_dB > list_dB).
thf(sy_c_List_On__lists_001t__Lambda__OdB, type,
    n_lists_dB : nat > list_dB > list_list_dB).
thf(sy_c_List_Onths_001t__Lambda__OdB, type,
    nths_dB : list_dB > set_nat > list_dB).
thf(sy_c_List_Oord_Olexordp__eq_001t__Lambda__OdB, type,
    lexordp_eq_dB : (dB > dB > $o) > list_dB > list_dB > $o).
thf(sy_c_List_Oord__class_Olexordp__eq_001t__Int__Oint, type,
    ord_lexordp_eq_int : list_int > list_int > $o).
thf(sy_c_List_Oord__class_Olexordp__eq_001t__Nat__Onat, type,
    ord_lexordp_eq_nat : list_nat > list_nat > $o).
thf(sy_c_List_Oproduct__lists_001t__Lambda__OdB, type,
    product_lists_dB : list_list_dB > list_list_dB).
thf(sy_c_List_Osubseqs_001t__Lambda__OdB, type,
    subseqs_dB : list_dB > list_list_dB).
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_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__Nat__Onat, type,
    member_nat : nat > set_nat > $o).
thf(sy_v_T, type,
    t : type).
thf(sy_v_Ta____, type,
    ta : type).
thf(sy_v_U____, type,
    u : type).
thf(sy_v_e, type,
    e : nat > type).
thf(sy_v_ea____, type,
    ea : nat > type).
thf(sy_v_s____, type,
    s : dB).
thf(sy_v_t, type,
    t2 : dB).
thf(sy_v_ta____, type,
    ta2 : dB).

% Relevant facts (139)
thf(fact_0_App_Ohyps_I2_J, axiom,
    ((it @ s))). % App.hyps(2)
thf(fact_1_App_Ohyps_I4_J, axiom,
    ((it @ ta2))). % App.hyps(4)
thf(fact_2__092_060open_062IT_A_Ilift_At_A0_J_092_060close_062, axiom,
    ((it @ (lift @ ta2 @ zero_zero_nat)))). % \<open>IT (lift t 0)\<close>
thf(fact_3_lift__IT, axiom,
    ((![T : dB, I : nat]: ((it @ T) => (it @ (lift @ T @ I)))))). % lift_IT
thf(fact_4_listsp__simps_I1_J, axiom,
    ((![A : dB > $o]: (listsp_dB @ A @ nil_dB)))). % listsp_simps(1)
thf(fact_5_listsp_Ocases, axiom,
    ((![A : dB > $o, A2 : list_dB]: ((listsp_dB @ A @ A2) => ((~ ((A2 = nil_dB))) => (~ ((![A3 : dB, L : list_dB]: ((A2 = (cons_dB @ A3 @ L)) => ((A @ A3) => (~ ((listsp_dB @ A @ L))))))))))))). % listsp.cases
thf(fact_6_listsp_Osimps, axiom,
    ((listsp_dB = (^[A4 : dB > $o]: (^[A5 : list_dB]: (((A5 = nil_dB)) | ((?[B : dB]: (?[L2 : list_dB]: (((A5 = (cons_dB @ B @ L2))) & ((((A4 @ B)) & ((listsp_dB @ A4 @ L2)))))))))))))). % listsp.simps
thf(fact_7_listsp_Oinducts, axiom,
    ((![A : dB > $o, X : list_dB, P : list_dB > $o]: ((listsp_dB @ A @ X) => ((P @ nil_dB) => ((![A3 : dB, L : list_dB]: ((A @ A3) => ((listsp_dB @ A @ L) => ((P @ L) => (P @ (cons_dB @ A3 @ L)))))) => (P @ X))))))). % listsp.inducts
thf(fact_8_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_9_listsp_ONil, axiom,
    ((![A : dB > $o]: (listsp_dB @ A @ nil_dB)))). % listsp.Nil
thf(fact_10_listsp__simps_I2_J, axiom,
    ((![A : dB > $o, X : dB, Xs : list_dB]: ((listsp_dB @ A @ (cons_dB @ X @ Xs)) = (((A @ X)) & ((listsp_dB @ A @ Xs))))))). % listsp_simps(2)
thf(fact_11_listspE, axiom,
    ((![A : dB > $o, X : dB, L3 : list_dB]: ((listsp_dB @ A @ (cons_dB @ X @ L3)) => (~ (((A @ X) => (~ ((listsp_dB @ A @ L3)))))))))). % listspE
thf(fact_12_listsp_OCons, axiom,
    ((![A : dB > $o, A2 : dB, L3 : list_dB]: ((A @ A2) => ((listsp_dB @ A @ L3) => (listsp_dB @ A @ (cons_dB @ A2 @ L3))))))). % listsp.Cons
thf(fact_13_list_Odistinct_I1_J, axiom,
    ((![X21 : dB, X22 : list_dB]: (~ ((nil_dB = (cons_dB @ X21 @ X22))))))). % list.distinct(1)
thf(fact_14_list_OdiscI, axiom,
    ((![List : list_dB, X21 : dB, X22 : list_dB]: ((List = (cons_dB @ X21 @ X22)) => (~ ((List = nil_dB))))))). % list.discI
thf(fact_15_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_16_zero__reorient, axiom,
    ((![X : int]: ((zero_zero_int = X) = (X = zero_zero_int))))). % zero_reorient
thf(fact_17_not__Cons__self2, axiom,
    ((![X : dB, Xs : list_dB]: (~ (((cons_dB @ X @ Xs) = Xs)))))). % not_Cons_self2
thf(fact_18_map__tailrec__rev_Oinduct, axiom,
    ((![P : (dB > dB) > list_dB > list_dB > $o, A0 : dB > dB, A1 : list_dB, A22 : list_dB]: ((![F : dB > dB, X_1 : list_dB]: (P @ F @ nil_dB @ X_1)) => ((![F : dB > dB, A3 : dB, As : list_dB, Bs : list_dB]: ((P @ F @ As @ (cons_dB @ (F @ A3) @ Bs)) => (P @ F @ (cons_dB @ A3 @ As) @ Bs))) => (P @ A0 @ A1 @ A22)))))). % map_tailrec_rev.induct
thf(fact_19_list__nonempty__induct, axiom,
    ((![Xs : list_dB, P : list_dB > $o]: ((~ ((Xs = nil_dB))) => ((![X2 : dB]: (P @ (cons_dB @ X2 @ nil_dB))) => ((![X2 : dB, Xs2 : list_dB]: ((~ ((Xs2 = nil_dB))) => ((P @ Xs2) => (P @ (cons_dB @ X2 @ Xs2))))) => (P @ Xs))))))). % list_nonempty_induct
thf(fact_20_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, X2 : dB]: (P @ P2 @ (cons_dB @ X2 @ nil_dB))) => ((![P2 : dB > dB > $o, X2 : dB, Y : dB, Xs2 : list_dB]: ((P @ P2 @ (cons_dB @ Y @ Xs2)) => (P @ P2 @ (cons_dB @ X2 @ (cons_dB @ Y @ Xs2))))) => (P @ A0 @ A1))))))). % successively.induct
thf(fact_21_remdups__adj_Oinduct, axiom,
    ((![P : list_dB > $o, A0 : list_dB]: ((P @ nil_dB) => ((![X2 : dB]: (P @ (cons_dB @ X2 @ nil_dB))) => ((![X2 : dB, Y : dB, Xs2 : list_dB]: (((X2 = Y) => (P @ (cons_dB @ X2 @ Xs2))) => (((~ ((X2 = Y))) => (P @ (cons_dB @ Y @ Xs2))) => (P @ (cons_dB @ X2 @ (cons_dB @ Y @ Xs2)))))) => (P @ A0))))))). % remdups_adj.induct
thf(fact_22_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, X2 : dB, Ys : list_dB]: ((P @ P2 @ Ys) => (P @ P2 @ (cons_dB @ X2 @ Ys)))) => (P @ A0 @ A1)))))). % sorted_wrt.induct
thf(fact_23_remdups__adj_Ocases, axiom,
    ((![X : list_dB]: ((~ ((X = nil_dB))) => ((![X2 : dB]: (~ ((X = (cons_dB @ X2 @ nil_dB))))) => (~ ((![X2 : dB, Y : dB, Xs2 : list_dB]: (~ ((X = (cons_dB @ X2 @ (cons_dB @ Y @ Xs2))))))))))))). % remdups_adj.cases
thf(fact_24_transpose_Ocases, axiom,
    ((![X : list_list_dB]: ((~ ((X = nil_list_dB))) => ((![Xss : list_list_dB]: (~ ((X = (cons_list_dB @ nil_dB @ Xss))))) => (~ ((![X2 : dB, Xs2 : list_dB, Xss : list_list_dB]: (~ ((X = (cons_list_dB @ (cons_dB @ X2 @ Xs2) @ Xss)))))))))))). % transpose.cases
thf(fact_25_shuffles_Oinduct, axiom,
    ((![P : list_dB > list_dB > $o, A0 : list_dB, A1 : list_dB]: ((![X_1 : list_dB]: (P @ nil_dB @ X_1)) => ((![Xs2 : list_dB]: (P @ Xs2 @ nil_dB)) => ((![X2 : dB, Xs2 : list_dB, Y : dB, Ys : list_dB]: ((P @ Xs2 @ (cons_dB @ Y @ Ys)) => ((P @ (cons_dB @ X2 @ Xs2) @ Ys) => (P @ (cons_dB @ X2 @ Xs2) @ (cons_dB @ Y @ Ys))))) => (P @ A0 @ A1))))))). % shuffles.induct
thf(fact_26_induct__list012, axiom,
    ((![P : list_dB > $o, Xs : list_dB]: ((P @ nil_dB) => ((![X2 : dB]: (P @ (cons_dB @ X2 @ nil_dB))) => ((![X2 : dB, Y : dB, Zs : list_dB]: ((P @ Zs) => ((P @ (cons_dB @ Y @ Zs)) => (P @ (cons_dB @ X2 @ (cons_dB @ Y @ Zs)))))) => (P @ Xs))))))). % induct_list012
thf(fact_27_splice_Oinduct, axiom,
    ((![P : list_dB > list_dB > $o, A0 : list_dB, A1 : list_dB]: ((![X_1 : list_dB]: (P @ nil_dB @ X_1)) => ((![X2 : dB, Xs2 : list_dB, Ys : list_dB]: ((P @ Ys @ Xs2) => (P @ (cons_dB @ X2 @ Xs2) @ Ys))) => (P @ A0 @ A1)))))). % splice.induct
thf(fact_28_list__induct2_H, axiom,
    ((![P : list_dB > list_dB > $o, Xs : list_dB, Ys2 : list_dB]: ((P @ nil_dB @ nil_dB) => ((![X2 : dB, Xs2 : list_dB]: (P @ (cons_dB @ X2 @ Xs2) @ nil_dB)) => ((![Y : dB, Ys : list_dB]: (P @ nil_dB @ (cons_dB @ Y @ Ys))) => ((![X2 : dB, Xs2 : list_dB, Y : dB, Ys : list_dB]: ((P @ Xs2 @ Ys) => (P @ (cons_dB @ X2 @ Xs2) @ (cons_dB @ Y @ Ys)))) => (P @ Xs @ Ys2)))))))). % list_induct2'
thf(fact_29_neq__Nil__conv, axiom,
    ((![Xs : list_dB]: ((~ ((Xs = nil_dB))) = (?[Y2 : dB]: (?[Ys3 : list_dB]: (Xs = (cons_dB @ Y2 @ Ys3)))))))). % neq_Nil_conv
thf(fact_30_list_Oinducts, axiom,
    ((![P : list_dB > $o, List : list_dB]: ((P @ nil_dB) => ((![X1 : dB, X23 : list_dB]: ((P @ X23) => (P @ (cons_dB @ X1 @ X23)))) => (P @ List)))))). % list.inducts
thf(fact_31_list_Oexhaust, axiom,
    ((![Y3 : list_dB]: ((~ ((Y3 = nil_dB))) => (~ ((![X212 : dB, X222 : list_dB]: (~ ((Y3 = (cons_dB @ X212 @ X222))))))))))). % list.exhaust
thf(fact_32_n__lists__Nil, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((n_lists_dB @ N @ nil_dB) = (cons_list_dB @ nil_dB @ nil_list_dB))) & ((~ ((N = zero_zero_nat))) => ((n_lists_dB @ N @ nil_dB) = nil_list_dB)))))). % n_lists_Nil
thf(fact_33_insert__Nil, axiom,
    ((![X : dB]: ((insert_dB @ X @ nil_dB) = (cons_dB @ X @ nil_dB))))). % insert_Nil
thf(fact_34_nths__singleton, axiom,
    ((![A : set_nat, X : dB]: (((member_nat @ zero_zero_nat @ A) => ((nths_dB @ (cons_dB @ X @ nil_dB) @ A) = (cons_dB @ X @ nil_dB))) & ((~ ((member_nat @ zero_zero_nat @ A))) => ((nths_dB @ (cons_dB @ X @ nil_dB) @ A) = nil_dB)))))). % nths_singleton
thf(fact_35_App_Ohyps_I3_J, axiom,
    ((typing @ ea @ ta2 @ ta))). % App.hyps(3)
thf(fact_36_list__ex1__simps_I1_J, axiom,
    ((![P : dB > $o]: (~ ((list_ex1_dB @ P @ nil_dB)))))). % list_ex1_simps(1)
thf(fact_37_n__lists_Osimps_I1_J, axiom,
    ((![Xs : list_dB]: ((n_lists_dB @ zero_zero_nat @ Xs) = (cons_list_dB @ nil_dB @ nil_list_dB))))). % n_lists.simps(1)
thf(fact_38_count__list_Osimps_I1_J, axiom,
    ((![Y3 : dB]: ((count_list_dB @ nil_dB @ Y3) = zero_zero_nat)))). % count_list.simps(1)
thf(fact_39_map__tailrec__rev_Oelims, axiom,
    ((![X : dB > dB, Xa : list_dB, Xb : list_dB, Y3 : list_dB]: (((map_ta277069629_dB_dB @ X @ Xa @ Xb) = Y3) => (((Xa = nil_dB) => (~ ((Y3 = Xb)))) => (~ ((![A3 : dB, As : list_dB]: ((Xa = (cons_dB @ A3 @ As)) => (~ ((Y3 = (map_ta277069629_dB_dB @ X @ As @ (cons_dB @ (X @ A3) @ Xb)))))))))))))). % map_tailrec_rev.elims
thf(fact_40_assms, axiom,
    ((typing @ e @ t2 @ t))). % assms
thf(fact_41_nths__nil, axiom,
    ((![A : set_nat]: ((nths_dB @ nil_dB @ A) = nil_dB)))). % nths_nil
thf(fact_42_App_Ohyps_I1_J, axiom,
    ((typing @ ea @ s @ (fun @ ta @ u)))). % App.hyps(1)
thf(fact_43_map__tailrec__rev_Osimps_I2_J, axiom,
    ((![F2 : dB > dB, A2 : dB, As2 : list_dB, Bs2 : list_dB]: ((map_ta277069629_dB_dB @ F2 @ (cons_dB @ A2 @ As2) @ Bs2) = (map_ta277069629_dB_dB @ F2 @ As2 @ (cons_dB @ (F2 @ A2) @ Bs2)))))). % map_tailrec_rev.simps(2)
thf(fact_44_product__lists_Osimps_I1_J, axiom,
    (((product_lists_dB @ nil_list_dB) = (cons_list_dB @ nil_dB @ nil_list_dB)))). % product_lists.simps(1)
thf(fact_45_subseqs_Osimps_I1_J, axiom,
    (((subseqs_dB @ nil_dB) = (cons_list_dB @ nil_dB @ nil_list_dB)))). % subseqs.simps(1)
thf(fact_46_ord_Olexordp__eq__simps_I3_J, axiom,
    ((![Less : dB > dB > $o, X : dB, Xs : list_dB]: (~ ((lexordp_eq_dB @ Less @ (cons_dB @ X @ Xs) @ nil_dB)))))). % ord.lexordp_eq_simps(3)
thf(fact_47_bind__simps_I1_J, axiom,
    ((![F2 : dB > list_dB]: ((bind_dB_dB @ nil_dB @ F2) = nil_dB)))). % bind_simps(1)
thf(fact_48_listrelp_Ocases, axiom,
    ((![R : dB > dB > $o, A1 : list_dB, A22 : list_dB]: ((listrelp_dB_dB @ R @ A1 @ A22) => (((A1 = nil_dB) => (~ ((A22 = nil_dB)))) => (~ ((![X2 : dB, Y : dB, Xs2 : list_dB]: ((A1 = (cons_dB @ X2 @ Xs2)) => (![Ys : list_dB]: ((A22 = (cons_dB @ Y @ Ys)) => ((R @ X2 @ Y) => (~ ((listrelp_dB_dB @ R @ Xs2 @ Ys))))))))))))))). % listrelp.cases
thf(fact_49_listrelp_Osimps, axiom,
    ((listrelp_dB_dB = (^[R2 : dB > dB > $o]: (^[A12 : list_dB]: (^[A23 : list_dB]: (((((A12 = nil_dB)) & ((A23 = nil_dB)))) | ((?[X3 : dB]: (?[Y2 : dB]: (?[Xs3 : list_dB]: (?[Ys3 : list_dB]: (((A12 = (cons_dB @ X3 @ Xs3))) & ((((A23 = (cons_dB @ Y2 @ Ys3))) & ((((R2 @ X3 @ Y2)) & ((listrelp_dB_dB @ R2 @ Xs3 @ Ys3))))))))))))))))))). % listrelp.simps
thf(fact_50_ord_Olexordp__eq__simps_I4_J, axiom,
    ((![Less : dB > dB > $o, X : dB, Xs : list_dB, Y3 : dB, Ys2 : list_dB]: ((lexordp_eq_dB @ Less @ (cons_dB @ X @ Xs) @ (cons_dB @ Y3 @ Ys2)) = (((Less @ X @ Y3)) | ((((~ ((Less @ Y3 @ X)))) & ((lexordp_eq_dB @ Less @ Xs @ Ys2))))))))). % ord.lexordp_eq_simps(4)
thf(fact_51_ord_Olexordp__eq__simps_I1_J, axiom,
    ((![Less : dB > dB > $o, Ys2 : list_dB]: (lexordp_eq_dB @ Less @ nil_dB @ Ys2)))). % ord.lexordp_eq_simps(1)
thf(fact_52_ord_Olexordp__eq__simps_I2_J, axiom,
    ((![Less : dB > dB > $o, Xs : list_dB]: ((lexordp_eq_dB @ Less @ Xs @ nil_dB) = (Xs = nil_dB))))). % ord.lexordp_eq_simps(2)
thf(fact_53_ord_Olexordp__eq_OCons__eq, axiom,
    ((![Less : dB > dB > $o, X : dB, Y3 : dB, Xs : list_dB, Ys2 : list_dB]: ((~ ((Less @ X @ Y3))) => ((~ ((Less @ Y3 @ X))) => ((lexordp_eq_dB @ Less @ Xs @ Ys2) => (lexordp_eq_dB @ Less @ (cons_dB @ X @ Xs) @ (cons_dB @ Y3 @ Ys2)))))))). % ord.lexordp_eq.Cons_eq
thf(fact_54_ord_Olexordp__eq_OCons, axiom,
    ((![Less : dB > dB > $o, X : dB, Y3 : dB, Xs : list_dB, Ys2 : list_dB]: ((Less @ X @ Y3) => (lexordp_eq_dB @ Less @ (cons_dB @ X @ Xs) @ (cons_dB @ Y3 @ Ys2)))))). % ord.lexordp_eq.Cons
thf(fact_55_ord_Olexordp__eq_ONil, axiom,
    ((![Less : dB > dB > $o, Ys2 : list_dB]: (lexordp_eq_dB @ Less @ nil_dB @ Ys2)))). % ord.lexordp_eq.Nil
thf(fact_56_listrelp_OCons, axiom,
    ((![R : dB > dB > $o, X : dB, Y3 : dB, Xs : list_dB, Ys2 : list_dB]: ((R @ X @ Y3) => ((listrelp_dB_dB @ R @ Xs @ Ys2) => (listrelp_dB_dB @ R @ (cons_dB @ X @ Xs) @ (cons_dB @ Y3 @ Ys2))))))). % listrelp.Cons
thf(fact_57_listrelp_ONil, axiom,
    ((![R : dB > dB > $o]: (listrelp_dB_dB @ R @ nil_dB @ nil_dB)))). % listrelp.Nil
thf(fact_58_ord_Olexordp__eq_Oinducts, axiom,
    ((![Less : dB > dB > $o, X12 : list_dB, X24 : list_dB, P : list_dB > list_dB > $o]: ((lexordp_eq_dB @ Less @ X12 @ X24) => ((![X_1 : list_dB]: (P @ nil_dB @ X_1)) => ((![X2 : dB, Y : dB, Xs2 : list_dB, Ys : list_dB]: ((Less @ X2 @ Y) => (P @ (cons_dB @ X2 @ Xs2) @ (cons_dB @ Y @ Ys)))) => ((![X2 : dB, Y : dB, Xs2 : list_dB, Ys : list_dB]: ((~ ((Less @ X2 @ Y))) => ((~ ((Less @ Y @ X2))) => ((lexordp_eq_dB @ Less @ Xs2 @ Ys) => ((P @ Xs2 @ Ys) => (P @ (cons_dB @ X2 @ Xs2) @ (cons_dB @ Y @ Ys))))))) => (P @ X12 @ X24)))))))). % ord.lexordp_eq.inducts
thf(fact_59_ord_Olexordp__eq_Osimps, axiom,
    ((lexordp_eq_dB = (^[Less2 : dB > dB > $o]: (^[A12 : list_dB]: (^[A23 : list_dB]: (((?[Ys3 : list_dB]: (((A12 = nil_dB)) & ((A23 = Ys3))))) | ((((?[X3 : dB]: (?[Y2 : dB]: (?[Xs3 : list_dB]: (?[Ys3 : list_dB]: (((A12 = (cons_dB @ X3 @ Xs3))) & ((((A23 = (cons_dB @ Y2 @ Ys3))) & ((Less2 @ X3 @ Y2)))))))))) | ((?[X3 : dB]: (?[Y2 : dB]: (?[Xs3 : list_dB]: (?[Ys3 : list_dB]: (((A12 = (cons_dB @ X3 @ Xs3))) & ((((A23 = (cons_dB @ Y2 @ Ys3))) & ((((~ ((Less2 @ X3 @ Y2)))) & ((((~ ((Less2 @ Y2 @ X3)))) & ((lexordp_eq_dB @ Less2 @ Xs3 @ Ys3))))))))))))))))))))))). % ord.lexordp_eq.simps
thf(fact_60_ord_Olexordp__eq_Ocases, axiom,
    ((![Less : dB > dB > $o, A1 : list_dB, A22 : list_dB]: ((lexordp_eq_dB @ Less @ A1 @ A22) => ((~ ((A1 = nil_dB))) => ((![X2 : dB]: ((?[Xs2 : list_dB]: (A1 = (cons_dB @ X2 @ Xs2))) => (![Y : dB]: ((?[Ys : list_dB]: (A22 = (cons_dB @ Y @ Ys))) => (~ ((Less @ X2 @ Y))))))) => (~ ((![X2 : dB, Y : dB, Xs2 : list_dB]: ((A1 = (cons_dB @ X2 @ Xs2)) => (![Ys : list_dB]: ((A22 = (cons_dB @ Y @ Ys)) => ((~ ((Less @ X2 @ Y))) => ((~ ((Less @ Y @ X2))) => (~ ((lexordp_eq_dB @ Less @ Xs2 @ Ys))))))))))))))))). % ord.lexordp_eq.cases
thf(fact_61_listrelp_Oinducts, axiom,
    ((![R : dB > dB > $o, X12 : list_dB, X24 : list_dB, P : list_dB > list_dB > $o]: ((listrelp_dB_dB @ R @ X12 @ X24) => ((P @ nil_dB @ nil_dB) => ((![X2 : dB, Y : dB, Xs2 : list_dB, Ys : list_dB]: ((R @ X2 @ Y) => ((listrelp_dB_dB @ R @ Xs2 @ Ys) => ((P @ Xs2 @ Ys) => (P @ (cons_dB @ X2 @ Xs2) @ (cons_dB @ Y @ Ys)))))) => (P @ X12 @ X24))))))). % listrelp.inducts
thf(fact_62_type_Oinject_I2_J, axiom,
    ((![X21 : type, X22 : type, Y21 : type, Y22 : type]: (((fun @ X21 @ X22) = (fun @ Y21 @ Y22)) = (((X21 = Y21)) & ((X22 = Y22))))))). % type.inject(2)
thf(fact_63_type__induct, axiom,
    ((![P : type > $o, T2 : type]: ((![T3 : type]: ((![T1 : type, T22 : type]: ((T3 = (fun @ T1 @ T22)) => (P @ T1))) => ((![T1 : type, T22 : type]: ((T3 = (fun @ T1 @ T22)) => (P @ T22))) => (P @ T3)))) => (P @ T2))))). % type_induct
thf(fact_64_zero__natural_Orsp, axiom,
    ((zero_zero_nat = zero_zero_nat))). % zero_natural.rsp
thf(fact_65_lift__type, axiom,
    ((![E : nat > type, T : dB, T2 : type, I : nat, U : type]: ((typing @ E @ T @ T2) => (typing @ (shift_type @ E @ I @ U) @ (lift @ T @ I) @ T2))))). % lift_type
thf(fact_66_lexordp__eq_Oinducts, axiom,
    ((![X12 : list_nat, X24 : list_nat, P : list_nat > list_nat > $o]: ((ord_lexordp_eq_nat @ X12 @ X24) => ((![X_1 : list_nat]: (P @ nil_nat @ X_1)) => ((![X2 : nat, Y : nat, Xs2 : list_nat, Ys : list_nat]: ((ord_less_nat @ X2 @ Y) => (P @ (cons_nat @ X2 @ Xs2) @ (cons_nat @ Y @ Ys)))) => ((![X2 : nat, Y : nat, Xs2 : list_nat, Ys : list_nat]: ((~ ((ord_less_nat @ X2 @ Y))) => ((~ ((ord_less_nat @ Y @ X2))) => ((ord_lexordp_eq_nat @ Xs2 @ Ys) => ((P @ Xs2 @ Ys) => (P @ (cons_nat @ X2 @ Xs2) @ (cons_nat @ Y @ Ys))))))) => (P @ X12 @ X24)))))))). % lexordp_eq.inducts
thf(fact_67_lexordp__eq_Oinducts, axiom,
    ((![X12 : list_int, X24 : list_int, P : list_int > list_int > $o]: ((ord_lexordp_eq_int @ X12 @ X24) => ((![X_1 : list_int]: (P @ nil_int @ X_1)) => ((![X2 : int, Y : int, Xs2 : list_int, Ys : list_int]: ((ord_less_int @ X2 @ Y) => (P @ (cons_int @ X2 @ Xs2) @ (cons_int @ Y @ Ys)))) => ((![X2 : int, Y : int, Xs2 : list_int, Ys : list_int]: ((~ ((ord_less_int @ X2 @ Y))) => ((~ ((ord_less_int @ Y @ X2))) => ((ord_lexordp_eq_int @ Xs2 @ Ys) => ((P @ Xs2 @ Ys) => (P @ (cons_int @ X2 @ Xs2) @ (cons_int @ Y @ Ys))))))) => (P @ X12 @ X24)))))))). % lexordp_eq.inducts
thf(fact_68_lexordp__eq_Osimps, axiom,
    ((ord_lexordp_eq_nat = (^[A12 : list_nat]: (^[A23 : list_nat]: (((?[Ys3 : list_nat]: (((A12 = nil_nat)) & ((A23 = Ys3))))) | ((((?[X3 : nat]: (?[Y2 : nat]: (?[Xs3 : list_nat]: (?[Ys3 : list_nat]: (((A12 = (cons_nat @ X3 @ Xs3))) & ((((A23 = (cons_nat @ Y2 @ Ys3))) & ((ord_less_nat @ X3 @ Y2)))))))))) | ((?[X3 : nat]: (?[Y2 : nat]: (?[Xs3 : list_nat]: (?[Ys3 : list_nat]: (((A12 = (cons_nat @ X3 @ Xs3))) & ((((A23 = (cons_nat @ Y2 @ Ys3))) & ((((~ ((ord_less_nat @ X3 @ Y2)))) & ((((~ ((ord_less_nat @ Y2 @ X3)))) & ((ord_lexordp_eq_nat @ Xs3 @ Ys3)))))))))))))))))))))). % lexordp_eq.simps
thf(fact_69_lexordp__eq_Osimps, axiom,
    ((ord_lexordp_eq_int = (^[A12 : list_int]: (^[A23 : list_int]: (((?[Ys3 : list_int]: (((A12 = nil_int)) & ((A23 = Ys3))))) | ((((?[X3 : int]: (?[Y2 : int]: (?[Xs3 : list_int]: (?[Ys3 : list_int]: (((A12 = (cons_int @ X3 @ Xs3))) & ((((A23 = (cons_int @ Y2 @ Ys3))) & ((ord_less_int @ X3 @ Y2)))))))))) | ((?[X3 : int]: (?[Y2 : int]: (?[Xs3 : list_int]: (?[Ys3 : list_int]: (((A12 = (cons_int @ X3 @ Xs3))) & ((((A23 = (cons_int @ Y2 @ Ys3))) & ((((~ ((ord_less_int @ X3 @ Y2)))) & ((((~ ((ord_less_int @ Y2 @ X3)))) & ((ord_lexordp_eq_int @ Xs3 @ Ys3)))))))))))))))))))))). % lexordp_eq.simps
thf(fact_70_lexordp__eq_Ocases, axiom,
    ((![A1 : list_nat, A22 : list_nat]: ((ord_lexordp_eq_nat @ A1 @ A22) => ((~ ((A1 = nil_nat))) => ((![X2 : nat]: ((?[Xs2 : list_nat]: (A1 = (cons_nat @ X2 @ Xs2))) => (![Y : nat]: ((?[Ys : list_nat]: (A22 = (cons_nat @ Y @ Ys))) => (~ ((ord_less_nat @ X2 @ Y))))))) => (~ ((![X2 : nat, Y : nat, Xs2 : list_nat]: ((A1 = (cons_nat @ X2 @ Xs2)) => (![Ys : list_nat]: ((A22 = (cons_nat @ Y @ Ys)) => ((~ ((ord_less_nat @ X2 @ Y))) => ((~ ((ord_less_nat @ Y @ X2))) => (~ ((ord_lexordp_eq_nat @ Xs2 @ Ys))))))))))))))))). % lexordp_eq.cases
thf(fact_71_lexordp__eq_Ocases, axiom,
    ((![A1 : list_int, A22 : list_int]: ((ord_lexordp_eq_int @ A1 @ A22) => ((~ ((A1 = nil_int))) => ((![X2 : int]: ((?[Xs2 : list_int]: (A1 = (cons_int @ X2 @ Xs2))) => (![Y : int]: ((?[Ys : list_int]: (A22 = (cons_int @ Y @ Ys))) => (~ ((ord_less_int @ X2 @ Y))))))) => (~ ((![X2 : int, Y : int, Xs2 : list_int]: ((A1 = (cons_int @ X2 @ Xs2)) => (![Ys : list_int]: ((A22 = (cons_int @ Y @ Ys)) => ((~ ((ord_less_int @ X2 @ Y))) => ((~ ((ord_less_int @ Y @ X2))) => (~ ((ord_lexordp_eq_int @ Xs2 @ Ys))))))))))))))))). % lexordp_eq.cases
thf(fact_72_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_73_less__nat__zero__code, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_74_neq0__conv, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N))))). % neq0_conv
thf(fact_75_shift__gt, axiom,
    ((![J : nat, I : nat, E : nat > type, T2 : type]: ((ord_less_nat @ J @ I) => ((shift_type @ E @ I @ T2 @ J) = (E @ J)))))). % shift_gt
thf(fact_76_shift__eq, axiom,
    ((![I : nat, J : nat, E : nat > type, T2 : type]: ((I = J) => ((shift_type @ E @ I @ T2 @ J) = T2))))). % shift_eq
thf(fact_77_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_78_lexordp__eq__simps_I4_J, axiom,
    ((![X : nat, Xs : list_nat, Y3 : nat, Ys2 : list_nat]: ((ord_lexordp_eq_nat @ (cons_nat @ X @ Xs) @ (cons_nat @ Y3 @ Ys2)) = (((ord_less_nat @ X @ Y3)) | ((((~ ((ord_less_nat @ Y3 @ X)))) & ((ord_lexordp_eq_nat @ Xs @ Ys2))))))))). % lexordp_eq_simps(4)
thf(fact_79_lexordp__eq__simps_I4_J, axiom,
    ((![X : int, Xs : list_int, Y3 : int, Ys2 : list_int]: ((ord_lexordp_eq_int @ (cons_int @ X @ Xs) @ (cons_int @ Y3 @ Ys2)) = (((ord_less_int @ X @ Y3)) | ((((~ ((ord_less_int @ Y3 @ X)))) & ((ord_lexordp_eq_int @ Xs @ Ys2))))))))). % lexordp_eq_simps(4)
thf(fact_80_bot__nat__0_Oextremum__strict, axiom,
    ((![A2 : nat]: (~ ((ord_less_nat @ A2 @ zero_zero_nat)))))). % bot_nat_0.extremum_strict
thf(fact_81_infinite__descent0, axiom,
    ((![P : nat > $o, N : nat]: ((P @ zero_zero_nat) => ((![N2 : nat]: ((ord_less_nat @ zero_zero_nat @ N2) => ((~ ((P @ N2))) => (?[M : nat]: ((ord_less_nat @ M @ N2) & (~ ((P @ M)))))))) => (P @ N)))))). % infinite_descent0
thf(fact_82_gr__implies__not0, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_nat @ M2 @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not0
thf(fact_83_less__zeroE, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_zeroE
thf(fact_84_not__less0, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less0
thf(fact_85_not__gr0, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr0
thf(fact_86_gr0I, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr0I
thf(fact_87_gr__zeroI, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr_zeroI
thf(fact_88_not__less__zero, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less_zero
thf(fact_89_gr__implies__not__zero, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_nat @ M2 @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not_zero
thf(fact_90_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_91_lexordp__eq_OCons__eq, axiom,
    ((![X : nat, Y3 : nat, Xs : list_nat, Ys2 : list_nat]: ((~ ((ord_less_nat @ X @ Y3))) => ((~ ((ord_less_nat @ Y3 @ X))) => ((ord_lexordp_eq_nat @ Xs @ Ys2) => (ord_lexordp_eq_nat @ (cons_nat @ X @ Xs) @ (cons_nat @ Y3 @ Ys2)))))))). % lexordp_eq.Cons_eq
thf(fact_92_lexordp__eq_OCons__eq, axiom,
    ((![X : int, Y3 : int, Xs : list_int, Ys2 : list_int]: ((~ ((ord_less_int @ X @ Y3))) => ((~ ((ord_less_int @ Y3 @ X))) => ((ord_lexordp_eq_int @ Xs @ Ys2) => (ord_lexordp_eq_int @ (cons_int @ X @ Xs) @ (cons_int @ Y3 @ Ys2)))))))). % lexordp_eq.Cons_eq
thf(fact_93_lexordp__eq_OCons, axiom,
    ((![X : nat, Y3 : nat, Xs : list_nat, Ys2 : list_nat]: ((ord_less_nat @ X @ Y3) => (ord_lexordp_eq_nat @ (cons_nat @ X @ Xs) @ (cons_nat @ Y3 @ Ys2)))))). % lexordp_eq.Cons
thf(fact_94_lexordp__eq_OCons, axiom,
    ((![X : int, Y3 : int, Xs : list_int, Ys2 : list_int]: ((ord_less_int @ X @ Y3) => (ord_lexordp_eq_int @ (cons_int @ X @ Xs) @ (cons_int @ Y3 @ Ys2)))))). % lexordp_eq.Cons
thf(fact_95_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_nat @ zero_zero_nat @ zero_zero_nat))))). % less_numeral_extra(3)
thf(fact_96_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_int @ zero_zero_int @ zero_zero_int))))). % less_numeral_extra(3)
thf(fact_97_typing__elims_I3_J, axiom,
    ((![E : nat > type, T : dB, T2 : type]: ((typing @ E @ (abs @ T) @ T2) => (~ ((![T3 : type, U2 : type]: ((T2 = (fun @ T3 @ U2)) => (~ ((typing @ (shift_type @ E @ zero_zero_nat @ T3) @ T @ U2))))))))))). % typing_elims(3)
thf(fact_98_Abs, axiom,
    ((![Env : nat > type, T2 : type, T : dB, U : type]: ((typing @ (shift_type @ Env @ zero_zero_nat @ T2) @ T @ U) => (typing @ Env @ (abs @ T) @ (fun @ T2 @ U)))))). % Abs
thf(fact_99_linorder__neqE__nat, axiom,
    ((![X : nat, Y3 : nat]: ((~ ((X = Y3))) => ((~ ((ord_less_nat @ X @ Y3))) => (ord_less_nat @ Y3 @ X)))))). % linorder_neqE_nat
thf(fact_100_infinite__descent, axiom,
    ((![P : nat > $o, N : nat]: ((![N2 : nat]: ((~ ((P @ N2))) => (?[M : nat]: ((ord_less_nat @ M @ N2) & (~ ((P @ M))))))) => (P @ N))))). % infinite_descent
thf(fact_101_nat__less__induct, axiom,
    ((![P : nat > $o, N : nat]: ((![N2 : nat]: ((![M : nat]: ((ord_less_nat @ M @ N2) => (P @ M))) => (P @ N2))) => (P @ N))))). % nat_less_induct
thf(fact_102_less__irrefl__nat, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_irrefl_nat
thf(fact_103_less__not__refl3, axiom,
    ((![S : nat, T : nat]: ((ord_less_nat @ S @ T) => (~ ((S = T))))))). % less_not_refl3
thf(fact_104_less__not__refl2, axiom,
    ((![N : nat, M2 : nat]: ((ord_less_nat @ N @ M2) => (~ ((M2 = N))))))). % less_not_refl2
thf(fact_105_less__not__refl, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_not_refl
thf(fact_106_nat__neq__iff, axiom,
    ((![M2 : nat, N : nat]: ((~ ((M2 = N))) = (((ord_less_nat @ M2 @ N)) | ((ord_less_nat @ N @ M2))))))). % nat_neq_iff
thf(fact_107_abs__typeE, axiom,
    ((![E : nat > type, T : dB, T2 : type]: ((typing @ E @ (abs @ T) @ T2) => (~ ((![U2 : type, V : type]: (~ ((typing @ (shift_type @ E @ zero_zero_nat @ U2) @ T @ V)))))))))). % abs_typeE
thf(fact_108_Lambda, axiom,
    ((![R : dB]: ((it @ R) => (it @ (abs @ R)))))). % Lambda
thf(fact_109_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_110_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_111_of__nat__eq__iff, axiom,
    ((![M2 : nat, N : nat]: (((semiri2019852685at_int @ M2) = (semiri2019852685at_int @ N)) = (M2 = N))))). % of_nat_eq_iff
thf(fact_112_of__nat__0, axiom,
    (((semiri1382578993at_nat @ zero_zero_nat) = zero_zero_nat))). % of_nat_0
thf(fact_113_of__nat__0, axiom,
    (((semiri2019852685at_int @ zero_zero_nat) = zero_zero_int))). % of_nat_0
thf(fact_114_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_115_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_116_of__nat__eq__0__iff, axiom,
    ((![M2 : nat]: (((semiri1382578993at_nat @ M2) = zero_zero_nat) = (M2 = zero_zero_nat))))). % of_nat_eq_0_iff
thf(fact_117_of__nat__eq__0__iff, axiom,
    ((![M2 : nat]: (((semiri2019852685at_int @ M2) = zero_zero_int) = (M2 = zero_zero_nat))))). % of_nat_eq_0_iff
thf(fact_118_of__nat__less__iff, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_nat @ (semiri1382578993at_nat @ M2) @ (semiri1382578993at_nat @ N)) = (ord_less_nat @ M2 @ N))))). % of_nat_less_iff
thf(fact_119_of__nat__less__iff, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_int @ (semiri2019852685at_int @ M2) @ (semiri2019852685at_int @ N)) = (ord_less_nat @ M2 @ N))))). % of_nat_less_iff
thf(fact_120_of__nat__less__imp__less, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_nat @ (semiri1382578993at_nat @ M2) @ (semiri1382578993at_nat @ N)) => (ord_less_nat @ M2 @ N))))). % of_nat_less_imp_less
thf(fact_121_of__nat__less__imp__less, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_int @ (semiri2019852685at_int @ M2) @ (semiri2019852685at_int @ N)) => (ord_less_nat @ M2 @ N))))). % of_nat_less_imp_less
thf(fact_122_less__imp__of__nat__less, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_nat @ M2 @ N) => (ord_less_nat @ (semiri1382578993at_nat @ M2) @ (semiri1382578993at_nat @ N)))))). % less_imp_of_nat_less
thf(fact_123_less__imp__of__nat__less, axiom,
    ((![M2 : nat, N : nat]: ((ord_less_nat @ M2 @ N) => (ord_less_int @ (semiri2019852685at_int @ M2) @ (semiri2019852685at_int @ N)))))). % less_imp_of_nat_less
thf(fact_124_of__nat__less__0__iff, axiom,
    ((![M2 : nat]: (~ ((ord_less_nat @ (semiri1382578993at_nat @ M2) @ zero_zero_nat)))))). % of_nat_less_0_iff
thf(fact_125_of__nat__less__0__iff, axiom,
    ((![M2 : nat]: (~ ((ord_less_int @ (semiri2019852685at_int @ M2) @ zero_zero_int)))))). % of_nat_less_0_iff
thf(fact_126_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_127_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_128_less__int__code_I1_J, axiom,
    ((~ ((ord_less_int @ zero_zero_int @ zero_zero_int))))). % less_int_code(1)
thf(fact_129_nat__int__comparison_I2_J, axiom,
    ((ord_less_nat = (^[A5 : nat]: (^[B : nat]: (ord_less_int @ (semiri2019852685at_int @ A5) @ (semiri2019852685at_int @ B))))))). % nat_int_comparison(2)
thf(fact_130_int__ops_I1_J, axiom,
    (((semiri2019852685at_int @ zero_zero_nat) = zero_zero_int))). % int_ops(1)
thf(fact_131_verit__comp__simplify1_I1_J, axiom,
    ((![A2 : nat]: (~ ((ord_less_nat @ A2 @ A2)))))). % verit_comp_simplify1(1)
thf(fact_132_verit__comp__simplify1_I1_J, axiom,
    ((![A2 : int]: (~ ((ord_less_int @ A2 @ A2)))))). % verit_comp_simplify1(1)
thf(fact_133_neg__int__cases, axiom,
    ((![K : int]: ((ord_less_int @ K @ zero_zero_int) => (~ ((![N2 : nat]: ((K = (uminus_uminus_int @ (semiri2019852685at_int @ N2))) => (~ ((ord_less_nat @ zero_zero_nat @ N2))))))))))). % neg_int_cases
thf(fact_134_zero__less__nat__eq, axiom,
    ((![Z : int]: ((ord_less_nat @ zero_zero_nat @ (nat2 @ Z)) = (ord_less_int @ zero_zero_int @ Z))))). % zero_less_nat_eq
thf(fact_135_add_Oinverse__inverse, axiom,
    ((![A2 : int]: ((uminus_uminus_int @ (uminus_uminus_int @ A2)) = A2)))). % add.inverse_inverse
thf(fact_136_neg__equal__iff__equal, axiom,
    ((![A2 : int, B2 : int]: (((uminus_uminus_int @ A2) = (uminus_uminus_int @ B2)) = (A2 = B2))))). % neg_equal_iff_equal
thf(fact_137_add_Oinverse__neutral, axiom,
    (((uminus_uminus_int @ zero_zero_int) = zero_zero_int))). % add.inverse_neutral
thf(fact_138_neg__0__equal__iff__equal, axiom,
    ((![A2 : int]: ((zero_zero_int = (uminus_uminus_int @ A2)) = (zero_zero_int = A2))))). % neg_0_equal_iff_equal

% Conjectures (1)
thf(conj_0, conjecture,
    ((listsp_dB @ it @ (cons_dB @ (lift @ ta2 @ zero_zero_nat) @ nil_dB)))).
