% 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_116__5210316_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:37:34.626

% Could-be-implicit typings (6)
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__LambdaType__Otype, type,
    type : $tType).
thf(ty_n_t__Lambda__OdB, type,
    dB : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).

% Explicit typings (39)
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_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_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_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_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_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_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_Nat_Osize__class_Osize_001t__Lambda__OdB, type,
    size_size_dB : dB > nat).
thf(sy_c_member_001t__Lambda__OdB, type,
    member_dB : dB > set_dB > $o).
thf(sy_v_T_H1____, type,
    t_1 : type).
thf(sy_v_T_H____, type,
    t : type).
thf(sy_v_T____, type,
    t2 : type).
thf(sy_v_e1____, type,
    e1 : nat > type).
thf(sy_v_e____, type,
    e : nat > type).
thf(sy_v_i1____, type,
    i1 : nat).
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,
    t3 : 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 @ t3))). % \<open>IT t\<close>
thf(fact_1_Var_Oprems_I2_J, axiom,
    ((it @ u1))). % Var.prems(2)
thf(fact_2_local_ONil, axiom,
    ((rs = nil_dB))). % local.Nil
thf(fact_3_uIT, axiom,
    ((it @ u))). % uIT
thf(fact_4_True, axiom,
    ((n = i))). % True
thf(fact_5_Var__IT, axiom,
    ((![N : nat]: (it @ (var @ N))))). % Var_IT
thf(fact_6_app__Var__IT, axiom,
    ((![T : dB, I : nat]: ((it @ T) => (it @ (app @ T @ (var @ I))))))). % app_Var_IT
thf(fact_7_subst__Var__IT, axiom,
    ((![R : dB, I : nat, J : nat]: ((it @ R) => (it @ (subst @ R @ (var @ I) @ J)))))). % subst_Var_IT
thf(fact_8_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_9_subst__eq, axiom,
    ((![K : nat, U : dB]: ((subst @ (var @ K) @ U @ K) = U)))). % subst_eq
thf(fact_10_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_11_dB_Oinject_I1_J, axiom,
    ((![X1 : nat, Y1 : nat]: (((var @ X1) = (var @ Y1)) = (X1 = Y1))))). % dB.inject(1)
thf(fact_12_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_13_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_14_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_15_dB_Odistinct_I1_J, axiom,
    ((![X1 : nat, X21 : dB, X22 : dB]: (~ (((var @ X1) = (app @ X21 @ X22))))))). % dB.distinct(1)
thf(fact_16_nT, axiom,
    ((typing @ (shift_type @ e @ i @ t2) @ (foldl_dB_dB @ app @ (var @ n) @ rs) @ t))). % nT
thf(fact_17_IT_OVar, axiom,
    ((![Rs : list_dB, N : nat]: ((listsp_dB @ it @ Rs) => (it @ (foldl_dB_dB @ app @ (var @ N) @ Rs)))))). % IT.Var
thf(fact_18_uT, axiom,
    ((typing @ e @ u @ t2))). % uT
thf(fact_19_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_20_dB_Oinject_I3_J, axiom,
    ((![X3 : dB, Y3 : dB]: (((abs @ X3) = (abs @ Y3)) = (X3 = Y3))))). % dB.inject(3)
thf(fact_21_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_22_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_23_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_24_dB_Odistinct_I5_J, axiom,
    ((![X21 : dB, X22 : dB, X3 : dB]: (~ (((app @ X21 @ X22) = (abs @ X3))))))). % dB.distinct(5)
thf(fact_25_dB_Odistinct_I3_J, axiom,
    ((![X1 : nat, X3 : dB]: (~ (((var @ X1) = (abs @ X3))))))). % dB.distinct(3)
thf(fact_26_Lambda, axiom,
    ((![R : dB]: ((it @ R) => (it @ (abs @ R)))))). % Lambda
thf(fact_27_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_28_dB_Oinduct, axiom,
    ((![P : dB > $o, DB : dB]: ((![X : nat]: (P @ (var @ X))) => ((![X1a : dB, X2 : dB]: ((P @ X1a) => ((P @ X2) => (P @ (app @ X1a @ X2))))) => ((![X : dB]: ((P @ X) => (P @ (abs @ X)))) => (P @ DB))))))). % dB.induct
thf(fact_29_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_30_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_31_MI2, axiom,
    ((![T1 : type, T2 : type, T : dB, E : nat > type, I : nat, T3 : type, U : dB]: ((t2 = (fun @ T1 @ T2)) => ((it @ T) => ((typing @ (shift_type @ E @ I @ T2) @ T @ T3) => ((it @ U) => ((typing @ E @ U @ T2) => (it @ (subst @ T @ U @ I)))))))))). % MI2
thf(fact_32_MI1, axiom,
    ((![T1 : type, T2 : type, T : dB, E : nat > type, I : nat, T3 : type, U : dB]: ((t2 = (fun @ T1 @ T2)) => ((it @ T) => ((typing @ (shift_type @ E @ I @ T1) @ T @ T3) => ((it @ U) => ((typing @ E @ U @ T1) => (it @ (subst @ T @ U @ I)))))))))). % MI1
thf(fact_33_listsp__simps_I1_J, axiom,
    ((![A : dB > $o]: (listsp_dB @ A @ nil_dB)))). % listsp_simps(1)
thf(fact_34_var__app__type__eq, axiom,
    ((![E : nat > type, I : nat, Ts : list_dB, T3 : type, U3 : type]: ((typing @ E @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ T3) => ((typing @ E @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ U3) => (T3 = U3)))))). % var_app_type_eq
thf(fact_35_subst__lemma, axiom,
    ((![E : nat > type, T : dB, T3 : type, E2 : nat > type, U : dB, U3 : type, I : nat]: ((typing @ E @ T @ T3) => ((typing @ E2 @ U @ U3) => ((E = (shift_type @ E2 @ I @ U3)) => (typing @ E2 @ (subst @ T @ U @ I) @ T3))))))). % subst_lemma
thf(fact_36_shift__eq, axiom,
    ((![I : nat, J : nat, E : nat > type, T3 : type]: ((I = J) => ((shift_type @ E @ I @ T3 @ J) = T3))))). % shift_eq
thf(fact_37_IT_Osimps, axiom,
    ((it = (^[A2 : dB]: (((?[Rs2 : list_dB]: (?[N3 : nat]: (((A2 = (foldl_dB_dB @ app @ (var @ N3) @ Rs2))) & ((listsp_dB @ it @ Rs2)))))) | ((((?[R2 : dB]: (((A2 = (abs @ R2))) & ((it @ R2))))) | ((?[R2 : dB]: (?[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.simps
thf(fact_38_IT_Ocases, axiom,
    ((![A3 : dB]: ((it @ A3) => ((![Rs3 : list_dB]: ((?[N2 : nat]: (A3 = (foldl_dB_dB @ app @ (var @ N2) @ Rs3))) => (~ ((listsp_dB @ it @ Rs3))))) => ((![R3 : dB]: ((A3 = (abs @ R3)) => (~ ((it @ R3))))) => (~ ((![R3 : dB, 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.cases
thf(fact_39_typing_OVar, axiom,
    ((![Env : nat > type, X4 : nat, T3 : type]: (((Env @ X4) = T3) => (typing @ Env @ (var @ X4) @ T3))))). % typing.Var
thf(fact_40_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_41_type__induct, axiom,
    ((![P : type > $o, T3 : type]: ((![T4 : type]: ((![T12 : type, T22 : type]: ((T4 = (fun @ T12 @ T22)) => (P @ T12))) => ((![T12 : type, T22 : type]: ((T4 = (fun @ T12 @ T22)) => (P @ T22))) => (P @ T4)))) => (P @ T3))))). % type_induct
thf(fact_42_Abs, axiom,
    ((![Env : nat > type, T3 : type, T : dB, U3 : type]: ((typing @ (shift_type @ Env @ zero_zero_nat @ T3) @ T @ U3) => (typing @ Env @ (abs @ T) @ (fun @ T3 @ U3)))))). % Abs
thf(fact_43_typing__elims_I3_J, axiom,
    ((![E : nat > type, T : dB, T3 : type]: ((typing @ E @ (abs @ T) @ T3) => (~ ((![T4 : type, U4 : type]: ((T3 = (fun @ T4 @ U4)) => (~ ((typing @ (shift_type @ E @ zero_zero_nat @ T4) @ T @ U4))))))))))). % typing_elims(3)
thf(fact_44_App, axiom,
    ((![Env : nat > type, S : dB, T3 : type, U3 : type, T : dB]: ((typing @ Env @ S @ (fun @ T3 @ U3)) => ((typing @ Env @ T @ T3) => (typing @ Env @ (app @ S @ T) @ U3)))))). % App
thf(fact_45_typing__elims_I2_J, axiom,
    ((![E : nat > type, T : dB, U : dB, T3 : type]: ((typing @ E @ (app @ T @ U) @ T3) => (~ ((![T4 : type]: ((typing @ E @ T @ (fun @ T4 @ T3)) => (~ ((typing @ E @ U @ T4))))))))))). % typing_elims(2)
thf(fact_46_typing_Oinducts, axiom,
    ((![X1 : nat > type, X23 : dB, X3 : type, P : (nat > type) > dB > type > $o]: ((typing @ X1 @ X23 @ X3) => ((![Env2 : nat > type, X : nat, T4 : type]: (((Env2 @ X) = T4) => (P @ Env2 @ (var @ X) @ T4))) => ((![Env2 : nat > type, T4 : type, T5 : dB, U4 : type]: ((typing @ (shift_type @ Env2 @ zero_zero_nat @ T4) @ T5 @ U4) => ((P @ (shift_type @ Env2 @ zero_zero_nat @ T4) @ T5 @ U4) => (P @ Env2 @ (abs @ T5) @ (fun @ T4 @ U4))))) => ((![Env2 : nat > type, S3 : dB, T4 : type, U4 : type, T5 : dB]: ((typing @ Env2 @ S3 @ (fun @ T4 @ U4)) => ((P @ Env2 @ S3 @ (fun @ T4 @ U4)) => ((typing @ Env2 @ T5 @ T4) => ((P @ Env2 @ T5 @ T4) => (P @ Env2 @ (app @ S3 @ T5) @ U4)))))) => (P @ X1 @ X23 @ X3)))))))). % typing.inducts
thf(fact_47_typing_Osimps, axiom,
    ((typing = (^[A1 : nat > type]: (^[A22 : dB]: (^[A32 : type]: (((?[Env3 : nat > type]: (?[X5 : nat]: (?[T6 : type]: (((A1 = Env3)) & ((((A22 = (var @ X5))) & ((((A32 = T6)) & (((Env3 @ X5) = T6))))))))))) | ((((?[Env3 : nat > type]: (?[T6 : type]: (?[T7 : dB]: (?[U5 : type]: (((A1 = Env3)) & ((((A22 = (abs @ T7))) & ((((A32 = (fun @ T6 @ U5))) & ((typing @ (shift_type @ Env3 @ zero_zero_nat @ T6) @ T7 @ U5)))))))))))) | ((?[Env3 : nat > type]: (?[S2 : dB]: (?[T6 : type]: (?[U5 : type]: (?[T7 : dB]: (((A1 = Env3)) & ((((A22 = (app @ S2 @ T7))) & ((((A32 = U5)) & ((((typing @ Env3 @ S2 @ (fun @ T6 @ U5))) & ((typing @ Env3 @ T7 @ T6)))))))))))))))))))))))). % typing.simps
thf(fact_48_typing_Ocases, axiom,
    ((![A12 : nat > type, A23 : dB, A33 : type]: ((typing @ A12 @ A23 @ A33) => ((![X : nat]: ((A23 = (var @ X)) => (~ (((A12 @ X) = A33))))) => ((![T4 : type, T5 : dB]: ((A23 = (abs @ T5)) => (![U4 : type]: ((A33 = (fun @ T4 @ U4)) => (~ ((typing @ (shift_type @ A12 @ zero_zero_nat @ T4) @ T5 @ U4))))))) => (~ ((![S3 : dB, T4 : type, U4 : type, T5 : dB]: ((A23 = (app @ S3 @ T5)) => ((A33 = U4) => ((typing @ A12 @ S3 @ (fun @ T4 @ U4)) => (~ ((typing @ A12 @ T5 @ T4))))))))))))))). % typing.cases
thf(fact_49_abs__typeE, axiom,
    ((![E : nat > type, T : dB, T3 : type]: ((typing @ E @ (abs @ T) @ T3) => (~ ((![U4 : type, V : type]: (~ ((typing @ (shift_type @ E @ zero_zero_nat @ U4) @ T @ V)))))))))). % abs_typeE
thf(fact_50_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_51_foldl__Nil, axiom,
    ((![F : dB > dB > dB, A3 : dB]: ((foldl_dB_dB @ F @ A3 @ nil_dB) = A3)))). % foldl_Nil
thf(fact_52_listsp_ONil, axiom,
    ((![A : dB > $o]: (listsp_dB @ A @ nil_dB)))). % listsp.Nil
thf(fact_53_typing__elims_I1_J, axiom,
    ((![E : nat > type, I : nat, T3 : type]: ((typing @ E @ (var @ I) @ T3) => ((E @ I) = T3))))). % typing_elims(1)
thf(fact_54_Var_Oprems_I1_J, axiom,
    ((typing @ (shift_type @ e1 @ i1 @ t2) @ (foldl_dB_dB @ app @ (var @ n) @ rs) @ t_1))). % Var.prems(1)
thf(fact_55_dB_Osize__gen_I1_J, axiom,
    ((![X1 : nat]: ((size_dB @ (var @ X1)) = zero_zero_nat)))). % dB.size_gen(1)
thf(fact_56_list__ex1__simps_I1_J, axiom,
    ((![P : dB > $o]: (~ ((list_ex1_dB @ P @ nil_dB)))))). % list_ex1_simps(1)
thf(fact_57_Var_Oprems_I3_J, axiom,
    ((typing @ e1 @ u1 @ t2))). % Var.prems(3)
thf(fact_58_substn__subst__0, axiom,
    ((![T : dB, S : dB]: ((substn @ T @ S @ zero_zero_nat) = (subst @ T @ S @ zero_zero_nat))))). % substn_subst_0
thf(fact_59_listsp__conj__eq, axiom,
    ((![A : dB > $o, B : dB > $o]: ((listsp_dB @ (^[X5 : dB]: (((A @ X5)) & ((B @ X5))))) = (^[X5 : list_dB]: (((listsp_dB @ A @ X5)) & ((listsp_dB @ B @ X5)))))))). % listsp_conj_eq
thf(fact_60_substn_Osimps_I2_J, axiom,
    ((![T : dB, U : dB, S : dB, K : nat]: ((substn @ (app @ T @ U) @ S @ K) = (app @ (substn @ T @ S @ K) @ (substn @ U @ S @ K)))))). % substn.simps(2)
thf(fact_61_zero__reorient, axiom,
    ((![X4 : nat]: ((zero_zero_nat = X4) = (X4 = zero_zero_nat))))). % zero_reorient
thf(fact_62_IT_Oinducts, axiom,
    ((![X4 : dB, P : dB > $o]: ((it @ X4) => ((![Rs3 : list_dB, N2 : nat]: ((listsp_dB @ (^[X5 : dB]: (((it @ X5)) & ((P @ X5)))) @ Rs3) => (P @ (foldl_dB_dB @ app @ (var @ N2) @ Rs3)))) => ((![R3 : dB]: ((it @ R3) => ((P @ R3) => (P @ (abs @ R3))))) => ((![R3 : dB, S3 : dB, Ss3 : list_dB]: ((it @ (foldl_dB_dB @ app @ (subst @ R3 @ S3 @ zero_zero_nat) @ Ss3)) => ((P @ (foldl_dB_dB @ app @ (subst @ R3 @ S3 @ zero_zero_nat) @ Ss3)) => ((it @ S3) => ((P @ S3) => (P @ (foldl_dB_dB @ app @ (app @ (abs @ R3) @ S3) @ Ss3))))))) => (P @ X4)))))))). % IT.inducts
thf(fact_63_substn__subst__n, axiom,
    ((substn = (^[T7 : dB]: (^[S2 : dB]: (^[N3 : nat]: (subst @ T7 @ (liftn @ N3 @ S2 @ zero_zero_nat) @ N3))))))). % substn_subst_n
thf(fact_64_Apps__dB__induct, axiom,
    ((![P : dB > $o, T : dB]: ((![N2 : nat, Ts2 : list_dB]: ((![X6 : dB]: ((member_dB @ X6 @ (set_dB2 @ Ts2)) => (P @ X6))) => (P @ (foldl_dB_dB @ app @ (var @ N2) @ Ts2)))) => ((![U2 : dB]: ((P @ U2) => (![Ts2 : list_dB]: ((![X6 : dB]: ((member_dB @ X6 @ (set_dB2 @ Ts2)) => (P @ X6))) => (P @ (foldl_dB_dB @ app @ (abs @ U2) @ Ts2)))))) => (P @ T)))))). % Apps_dB_induct
thf(fact_65_beta_Oinducts, axiom,
    ((![X1 : dB, X23 : dB, P : dB > dB > $o]: ((beta @ X1 @ X23) => ((![S3 : dB, T5 : dB]: (P @ (app @ (abs @ S3) @ T5) @ (subst @ S3 @ T5 @ zero_zero_nat))) => ((![S3 : dB, T5 : dB, U2 : dB]: ((beta @ S3 @ T5) => ((P @ S3 @ T5) => (P @ (app @ S3 @ U2) @ (app @ T5 @ U2))))) => ((![S3 : dB, T5 : dB, U2 : dB]: ((beta @ S3 @ T5) => ((P @ S3 @ T5) => (P @ (app @ U2 @ S3) @ (app @ U2 @ T5))))) => ((![S3 : dB, T5 : dB]: ((beta @ S3 @ T5) => ((P @ S3 @ T5) => (P @ (abs @ S3) @ (abs @ T5))))) => (P @ X1 @ X23))))))))). % beta.inducts
thf(fact_66_beta_Osimps, axiom,
    ((beta = (^[A1 : dB]: (^[A22 : dB]: (((?[S2 : dB]: (?[T7 : dB]: (((A1 = (app @ (abs @ S2) @ T7))) & ((A22 = (subst @ S2 @ T7 @ zero_zero_nat))))))) | ((((?[S2 : dB]: (?[T7 : dB]: (?[U6 : dB]: (((A1 = (app @ S2 @ U6))) & ((((A22 = (app @ T7 @ U6))) & ((beta @ S2 @ T7))))))))) | ((((?[S2 : dB]: (?[T7 : dB]: (?[U6 : dB]: (((A1 = (app @ U6 @ S2))) & ((((A22 = (app @ U6 @ T7))) & ((beta @ S2 @ T7))))))))) | ((?[S2 : dB]: (?[T7 : dB]: (((A1 = (abs @ S2))) & ((((A22 = (abs @ T7))) & ((beta @ S2 @ T7)))))))))))))))))). % beta.simps
thf(fact_67_in__listspI, axiom,
    ((![Xs : list_dB, A : dB > $o]: ((![X : dB]: ((member_dB @ X @ (set_dB2 @ Xs)) => (A @ X))) => (listsp_dB @ A @ Xs))))). % in_listspI
thf(fact_68_liftn__0, axiom,
    ((![T : dB, K : nat]: ((liftn @ zero_zero_nat @ T @ K) = T)))). % liftn_0
thf(fact_69_appL, axiom,
    ((![S : dB, T : dB, U : dB]: ((beta @ S @ T) => (beta @ (app @ S @ U) @ (app @ T @ U)))))). % appL
thf(fact_70_appR, axiom,
    ((![S : dB, T : dB, U : dB]: ((beta @ S @ T) => (beta @ (app @ U @ S) @ (app @ U @ T)))))). % appR
thf(fact_71_foldl__cong, axiom,
    ((![A3 : dB, B2 : dB, L : list_dB, K : list_dB, F : dB > dB > dB, G : dB > dB > dB]: ((A3 = B2) => ((L = K) => ((![A4 : dB, X : dB]: ((member_dB @ X @ (set_dB2 @ L)) => ((F @ A4 @ X) = (G @ A4 @ X)))) => ((foldl_dB_dB @ F @ A3 @ L) = (foldl_dB_dB @ G @ B2 @ K)))))))). % foldl_cong
thf(fact_72_subject__reduction, axiom,
    ((![E : nat > type, T : dB, T3 : type, T8 : dB]: ((typing @ E @ T @ T3) => ((beta @ T @ T8) => (typing @ E @ T8 @ T3)))))). % subject_reduction
thf(fact_73_abs, axiom,
    ((![S : dB, T : dB]: ((beta @ S @ T) => (beta @ (abs @ S) @ (abs @ T)))))). % abs
thf(fact_74_beta__cases_I2_J, axiom,
    ((![R : dB, S : dB]: ((beta @ (abs @ R) @ S) => (~ ((![T5 : dB]: ((S = (abs @ T5)) => (~ ((beta @ R @ T5))))))))))). % beta_cases(2)
thf(fact_75_beta__cases_I1_J, axiom,
    ((![I : nat, T : dB]: (~ ((beta @ (var @ I) @ T)))))). % beta_cases(1)
thf(fact_76_in__listsp__conv__set, axiom,
    ((listsp_dB = (^[A5 : dB > $o]: (^[Xs2 : list_dB]: (![X5 : dB]: (((member_dB @ X5 @ (set_dB2 @ Xs2))) => ((A5 @ X5))))))))). % in_listsp_conv_set
thf(fact_77_in__listspD, axiom,
    ((![A : dB > $o, Xs : list_dB]: ((listsp_dB @ A @ Xs) => (![X6 : dB]: ((member_dB @ X6 @ (set_dB2 @ Xs)) => (A @ X6))))))). % in_listspD
thf(fact_78_subst__preserves__beta, axiom,
    ((![R : dB, S : dB, T : dB, I : nat]: ((beta @ R @ S) => (beta @ (subst @ R @ T @ I) @ (subst @ S @ T @ I)))))). % subst_preserves_beta
thf(fact_79_list__ex1__iff, axiom,
    ((list_ex1_dB = (^[P2 : dB > $o]: (^[Xs2 : list_dB]: (?[X5 : dB]: (((((member_dB @ X5 @ (set_dB2 @ Xs2))) & ((P2 @ X5)))) & ((![Y2 : dB]: (((((member_dB @ Y2 @ (set_dB2 @ Xs2))) & ((P2 @ Y2)))) => ((Y2 = X5)))))))))))). % list_ex1_iff
thf(fact_80_liftn_Osimps_I2_J, axiom,
    ((![N : nat, S : dB, T : dB, K : nat]: ((liftn @ N @ (app @ S @ T) @ K) = (app @ (liftn @ N @ S @ K) @ (liftn @ N @ T @ K)))))). % liftn.simps(2)
thf(fact_81_beta__cases_I3_J, axiom,
    ((![S : dB, T : dB, U : dB]: ((beta @ (app @ S @ T) @ U) => ((![S3 : dB]: ((S = (abs @ S3)) => (~ ((U = (subst @ S3 @ T @ zero_zero_nat)))))) => ((![T5 : dB]: ((U = (app @ T5 @ T)) => (~ ((beta @ S @ T5))))) => (~ ((![T5 : dB]: ((U = (app @ S @ T5)) => (~ ((beta @ T @ T5))))))))))))). % beta_cases(3)
thf(fact_82_beta, axiom,
    ((![S : dB, T : dB]: (beta @ (app @ (abs @ S) @ T) @ (subst @ S @ T @ zero_zero_nat))))). % beta
thf(fact_83_beta_Ocases, axiom,
    ((![A12 : dB, A23 : dB]: ((beta @ A12 @ A23) => ((![S3 : dB, T5 : dB]: ((A12 = (app @ (abs @ S3) @ T5)) => (~ ((A23 = (subst @ S3 @ T5 @ zero_zero_nat)))))) => ((![S3 : dB, T5 : dB, U2 : dB]: ((A12 = (app @ S3 @ U2)) => ((A23 = (app @ T5 @ U2)) => (~ ((beta @ S3 @ T5)))))) => ((![S3 : dB, T5 : dB, U2 : dB]: ((A12 = (app @ U2 @ S3)) => ((A23 = (app @ U2 @ T5)) => (~ ((beta @ S3 @ T5)))))) => (~ ((![S3 : dB]: ((A12 = (abs @ S3)) => (![T5 : dB]: ((A23 = (abs @ T5)) => (~ ((beta @ S3 @ T5)))))))))))))))). % beta.cases
thf(fact_84_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_85_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_86_lem, axiom,
    ((![P : dB > $o, T : dB, N : nat]: ((![N2 : nat, Ts2 : list_dB]: ((![X6 : dB]: ((member_dB @ X6 @ (set_dB2 @ Ts2)) => (P @ X6))) => (P @ (foldl_dB_dB @ app @ (var @ N2) @ Ts2)))) => ((![U2 : dB]: ((P @ U2) => (![Ts2 : list_dB]: ((![X6 : dB]: ((member_dB @ X6 @ (set_dB2 @ Ts2)) => (P @ X6))) => (P @ (foldl_dB_dB @ app @ (abs @ U2) @ Ts2)))))) => (((size_size_dB @ T) = N) => (P @ T))))))). % lem
thf(fact_87_count__notin, axiom,
    ((![X4 : dB, Xs : list_dB]: ((~ ((member_dB @ X4 @ (set_dB2 @ Xs)))) => ((count_list_dB @ Xs @ X4) = zero_zero_nat))))). % count_notin
thf(fact_88_dB_Osize_I4_J, axiom,
    ((![X1 : nat]: ((size_size_dB @ (var @ X1)) = zero_zero_nat)))). % dB.size(4)
thf(fact_89_count__list_Osimps_I1_J, axiom,
    ((![Y : dB]: ((count_list_dB @ nil_dB @ Y) = zero_zero_nat)))). % count_list.simps(1)
thf(fact_90_head__Var__reduction, axiom,
    ((![N : nat, Rs : list_dB, V2 : dB]: ((beta @ (foldl_dB_dB @ app @ (var @ N) @ Rs) @ V2) => (?[Ss3 : list_dB]: ((step1_dB @ beta @ Rs @ Ss3) & (V2 = (foldl_dB_dB @ app @ (var @ N) @ Ss3)))))))). % head_Var_reduction
thf(fact_91_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_92_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 @ (^[T7 : dB]: (subst @ T7 @ U @ I)) @ Ts)))))). % subst_map
thf(fact_93_map__ident, axiom,
    (((map_dB_dB @ (^[X5 : dB]: X5)) = (^[Xs2 : list_dB]: Xs2)))). % map_ident
thf(fact_94_Nil__is__map__conv, axiom,
    ((![F : dB > dB, Xs : list_dB]: ((nil_dB = (map_dB_dB @ F @ Xs)) = (Xs = nil_dB))))). % Nil_is_map_conv
thf(fact_95_map__is__Nil__conv, axiom,
    ((![F : dB > dB, Xs : list_dB]: (((map_dB_dB @ F @ Xs) = nil_dB) = (Xs = nil_dB))))). % map_is_Nil_conv
thf(fact_96_list_Omap__disc__iff, axiom,
    ((![F : dB > dB, A3 : list_dB]: (((map_dB_dB @ F @ A3) = nil_dB) = (A3 = nil_dB))))). % list.map_disc_iff
thf(fact_97_map__eq__conv, axiom,
    ((![F : dB > dB, Xs : list_dB, G : dB > dB]: (((map_dB_dB @ F @ Xs) = (map_dB_dB @ G @ Xs)) = (![X5 : dB]: (((member_dB @ X5 @ (set_dB2 @ Xs))) => (((F @ X5) = (G @ X5))))))))). % map_eq_conv
thf(fact_98_map__ext, axiom,
    ((![Xs : list_dB, F : dB > dB, G : dB > dB]: ((![X : dB]: ((member_dB @ X @ (set_dB2 @ Xs)) => ((F @ X) = (G @ X)))) => ((map_dB_dB @ F @ Xs) = (map_dB_dB @ G @ Xs)))))). % map_ext
thf(fact_99_map__idI, axiom,
    ((![Xs : list_dB, F : dB > dB]: ((![X : dB]: ((member_dB @ X @ (set_dB2 @ Xs)) => ((F @ X) = X))) => ((map_dB_dB @ F @ Xs) = Xs))))). % map_idI
thf(fact_100_map__cong, axiom,
    ((![Xs : list_dB, Ys : list_dB, F : dB > dB, G : dB > dB]: ((Xs = Ys) => ((![X : dB]: ((member_dB @ X @ (set_dB2 @ Ys)) => ((F @ X) = (G @ X)))) => ((map_dB_dB @ F @ Xs) = (map_dB_dB @ G @ Ys))))))). % map_cong
thf(fact_101_ex__map__conv, axiom,
    ((![Ys : list_dB, F : dB > dB]: ((?[Xs2 : list_dB]: (Ys = (map_dB_dB @ F @ Xs2))) = (![X5 : dB]: (((member_dB @ X5 @ (set_dB2 @ Ys))) => ((?[Y2 : dB]: (X5 = (F @ Y2)))))))))). % ex_map_conv
thf(fact_102_list_Omap__cong, axiom,
    ((![X4 : list_dB, Ya : list_dB, F : dB > dB, G : dB > dB]: ((X4 = Ya) => ((![Z : dB]: ((member_dB @ Z @ (set_dB2 @ Ya)) => ((F @ Z) = (G @ Z)))) => ((map_dB_dB @ F @ X4) = (map_dB_dB @ G @ Ya))))))). % list.map_cong
thf(fact_103_list_Omap__cong0, axiom,
    ((![X4 : list_dB, F : dB > dB, G : dB > dB]: ((![Z : dB]: ((member_dB @ Z @ (set_dB2 @ X4)) => ((F @ Z) = (G @ Z)))) => ((map_dB_dB @ F @ X4) = (map_dB_dB @ G @ X4)))))). % list.map_cong0
thf(fact_104_list_Oinj__map__strong, axiom,
    ((![X4 : list_dB, Xa : list_dB, F : dB > dB, Fa : dB > dB]: ((![Z : dB, Za : dB]: ((member_dB @ Z @ (set_dB2 @ X4)) => ((member_dB @ Za @ (set_dB2 @ Xa)) => (((F @ Z) = (Fa @ Za)) => (Z = Za))))) => (((map_dB_dB @ F @ X4) = (map_dB_dB @ Fa @ Xa)) => (X4 = Xa)))))). % list.inj_map_strong
thf(fact_105_foldl__map, axiom,
    ((![G : dB > dB > dB, A3 : dB, F : dB > dB, Xs : list_dB]: ((foldl_dB_dB @ G @ A3 @ (map_dB_dB @ F @ Xs)) = (foldl_dB_dB @ (^[A2 : dB]: (^[X5 : dB]: (G @ A2 @ (F @ X5)))) @ A3 @ Xs))))). % foldl_map
thf(fact_106_list_Omap__ident, axiom,
    ((![T : list_dB]: ((map_dB_dB @ (^[X5 : dB]: X5) @ T) = T)))). % list.map_ident
thf(fact_107_list_Osimps_I8_J, axiom,
    ((![F : dB > dB]: ((map_dB_dB @ F @ nil_dB) = nil_dB)))). % list.simps(8)
thf(fact_108_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)))))) => (~ ((![T5 : dB]: ((R = (abs @ T5)) => (![U2 : dB, Us : list_dB]: ((Rs = (cons_dB @ U2 @ Us)) => (~ ((S = (foldl_dB_dB @ app @ (subst @ T5 @ U2 @ zero_zero_nat) @ Us)))))))))))))))). % apps_betasE
thf(fact_109_ex__step1I, axiom,
    ((![X4 : dB, Xs : list_dB, R : dB > dB > $o, Y : dB]: ((member_dB @ X4 @ (set_dB2 @ Xs)) => ((R @ Y @ X4) => (?[Ys2 : list_dB]: ((step1_dB @ R @ Ys2 @ Xs) & (member_dB @ Y @ (set_dB2 @ Ys2))))))))). % ex_step1I
thf(fact_110_not__Nil__step1, axiom,
    ((![R : dB > dB > $o, Xs : list_dB]: (~ ((step1_dB @ R @ nil_dB @ Xs)))))). % not_Nil_step1
thf(fact_111_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_112_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_113_Cons__eq__map__D, axiom,
    ((![X4 : dB, Xs : list_dB, F : dB > dB, Ys : list_dB]: (((cons_dB @ X4 @ Xs) = (map_dB_dB @ F @ Ys)) => (?[Z : dB, Zs : list_dB]: ((Ys = (cons_dB @ Z @ Zs)) & ((X4 = (F @ Z)) & (Xs = (map_dB_dB @ F @ Zs))))))))). % Cons_eq_map_D
thf(fact_114_map__eq__Cons__D, axiom,
    ((![F : dB > dB, Xs : list_dB, Y : dB, Ys : list_dB]: (((map_dB_dB @ F @ Xs) = (cons_dB @ Y @ Ys)) => (?[Z : dB, Zs : list_dB]: ((Xs = (cons_dB @ Z @ Zs)) & (((F @ Z) = Y) & ((map_dB_dB @ F @ Zs) = Ys)))))))). % map_eq_Cons_D
thf(fact_115_Cons__eq__map__conv, axiom,
    ((![X4 : dB, Xs : list_dB, F : dB > dB, Ys : list_dB]: (((cons_dB @ X4 @ Xs) = (map_dB_dB @ F @ Ys)) = (?[Z2 : dB]: (?[Zs2 : list_dB]: (((Ys = (cons_dB @ Z2 @ Zs2))) & ((((X4 = (F @ Z2))) & ((Xs = (map_dB_dB @ F @ Zs2)))))))))))). % Cons_eq_map_conv
thf(fact_116_map__eq__Cons__conv, axiom,
    ((![F : dB > dB, Xs : list_dB, Y : dB, Ys : list_dB]: (((map_dB_dB @ F @ Xs) = (cons_dB @ Y @ Ys)) = (?[Z2 : dB]: (?[Zs2 : list_dB]: (((Xs = (cons_dB @ Z2 @ Zs2))) & (((((F @ Z2) = Y)) & (((map_dB_dB @ F @ Zs2) = Ys))))))))))). % map_eq_Cons_conv
thf(fact_117_foldl__Cons, axiom,
    ((![F : dB > dB > dB, A3 : dB, X4 : dB, Xs : list_dB]: ((foldl_dB_dB @ F @ A3 @ (cons_dB @ X4 @ Xs)) = (foldl_dB_dB @ F @ (F @ A3 @ X4) @ Xs))))). % foldl_Cons
thf(fact_118_listsp_OCons, axiom,
    ((![A : dB > $o, A3 : dB, L : list_dB]: ((A @ A3) => ((listsp_dB @ A @ L) => (listsp_dB @ A @ (cons_dB @ A3 @ L))))))). % listsp.Cons
thf(fact_119_listspE, axiom,
    ((![A : dB > $o, X4 : dB, L : list_dB]: ((listsp_dB @ A @ (cons_dB @ X4 @ L)) => (~ (((A @ X4) => (~ ((listsp_dB @ A @ L)))))))))). % listspE
thf(fact_120_listsp__simps_I2_J, axiom,
    ((![A : dB > $o, X4 : dB, Xs : list_dB]: ((listsp_dB @ A @ (cons_dB @ X4 @ Xs)) = (((A @ X4)) & ((listsp_dB @ A @ Xs))))))). % listsp_simps(2)
thf(fact_121_transpose_Ocases, axiom,
    ((![X4 : list_list_dB]: ((~ ((X4 = nil_list_dB))) => ((![Xss : list_list_dB]: (~ ((X4 = (cons_list_dB @ nil_dB @ Xss))))) => (~ ((![X : dB, Xs3 : list_dB, Xss : list_list_dB]: (~ ((X4 = (cons_list_dB @ (cons_dB @ X @ Xs3) @ Xss)))))))))))). % transpose.cases
thf(fact_122_not__Cons__self2, axiom,
    ((![X4 : dB, Xs : list_dB]: (~ (((cons_dB @ X4 @ Xs) = Xs)))))). % not_Cons_self2
thf(fact_123_list_Odistinct_I1_J, axiom,
    ((![X21 : dB, X22 : list_dB]: (~ ((nil_dB = (cons_dB @ X21 @ X22))))))). % list.distinct(1)
thf(fact_124_list_OdiscI, axiom,
    ((![List : list_dB, X21 : dB, X22 : list_dB]: ((List = (cons_dB @ X21 @ X22)) => (~ ((List = nil_dB))))))). % list.discI
thf(fact_125_list_Oexhaust, axiom,
    ((![Y : list_dB]: ((~ ((Y = nil_dB))) => (~ ((![X212 : dB, X222 : list_dB]: (~ ((Y = (cons_dB @ X212 @ X222))))))))))). % list.exhaust
thf(fact_126_list_Oinducts, axiom,
    ((![P : list_dB > $o, List : list_dB]: ((P @ nil_dB) => ((![X12 : dB, X2 : list_dB]: ((P @ X2) => (P @ (cons_dB @ X12 @ X2)))) => (P @ List)))))). % list.inducts
thf(fact_127_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_128_list__induct2_H, axiom,
    ((![P : list_dB > list_dB > $o, Xs : list_dB, Ys : list_dB]: ((P @ nil_dB @ nil_dB) => ((![X : dB, Xs3 : list_dB]: (P @ (cons_dB @ X @ Xs3) @ nil_dB)) => ((![Y4 : dB, Ys2 : list_dB]: (P @ nil_dB @ (cons_dB @ Y4 @ Ys2))) => ((![X : dB, Xs3 : list_dB, Y4 : dB, Ys2 : list_dB]: ((P @ Xs3 @ Ys2) => (P @ (cons_dB @ X @ Xs3) @ (cons_dB @ Y4 @ Ys2)))) => (P @ Xs @ Ys)))))))). % list_induct2'
thf(fact_129_splice_Oinduct, axiom,
    ((![P : list_dB > list_dB > $o, A0 : list_dB, A12 : list_dB]: ((![X_1 : list_dB]: (P @ nil_dB @ X_1)) => ((![X : dB, Xs3 : list_dB, Ys2 : list_dB]: ((P @ Ys2 @ Xs3) => (P @ (cons_dB @ X @ Xs3) @ Ys2))) => (P @ A0 @ A12)))))). % splice.induct
thf(fact_130_induct__list012, axiom,
    ((![P : list_dB > $o, Xs : list_dB]: ((P @ nil_dB) => ((![X : dB]: (P @ (cons_dB @ X @ nil_dB))) => ((![X : dB, Y4 : dB, Zs : list_dB]: ((P @ Zs) => ((P @ (cons_dB @ Y4 @ Zs)) => (P @ (cons_dB @ X @ (cons_dB @ Y4 @ Zs)))))) => (P @ Xs))))))). % induct_list012
thf(fact_131_shuffles_Oinduct, axiom,
    ((![P : list_dB > list_dB > $o, A0 : list_dB, A12 : list_dB]: ((![X_1 : list_dB]: (P @ nil_dB @ X_1)) => ((![Xs3 : list_dB]: (P @ Xs3 @ nil_dB)) => ((![X : dB, Xs3 : list_dB, Y4 : dB, Ys2 : list_dB]: ((P @ Xs3 @ (cons_dB @ Y4 @ Ys2)) => ((P @ (cons_dB @ X @ Xs3) @ Ys2) => (P @ (cons_dB @ X @ Xs3) @ (cons_dB @ Y4 @ Ys2))))) => (P @ A0 @ A12))))))). % shuffles.induct
thf(fact_132_remdups__adj_Ocases, axiom,
    ((![X4 : list_dB]: ((~ ((X4 = nil_dB))) => ((![X : dB]: (~ ((X4 = (cons_dB @ X @ nil_dB))))) => (~ ((![X : dB, Y4 : dB, Xs3 : list_dB]: (~ ((X4 = (cons_dB @ X @ (cons_dB @ Y4 @ Xs3))))))))))))). % remdups_adj.cases
thf(fact_133_sorted__wrt_Oinduct, axiom,
    ((![P : (dB > dB > $o) > list_dB > $o, A0 : dB > dB > $o, A12 : list_dB]: ((![P3 : dB > dB > $o]: (P @ P3 @ nil_dB)) => ((![P3 : dB > dB > $o, X : dB, Ys2 : list_dB]: ((P @ P3 @ Ys2) => (P @ P3 @ (cons_dB @ X @ Ys2)))) => (P @ A0 @ A12)))))). % sorted_wrt.induct
thf(fact_134_remdups__adj_Oinduct, axiom,
    ((![P : list_dB > $o, A0 : list_dB]: ((P @ nil_dB) => ((![X : dB]: (P @ (cons_dB @ X @ nil_dB))) => ((![X : dB, Y4 : dB, Xs3 : list_dB]: (((X = Y4) => (P @ (cons_dB @ X @ Xs3))) => (((~ ((X = Y4))) => (P @ (cons_dB @ Y4 @ Xs3))) => (P @ (cons_dB @ X @ (cons_dB @ Y4 @ Xs3)))))) => (P @ A0))))))). % remdups_adj.induct
thf(fact_135_successively_Oinduct, axiom,
    ((![P : (dB > dB > $o) > list_dB > $o, A0 : dB > dB > $o, A12 : list_dB]: ((![P3 : dB > dB > $o]: (P @ P3 @ nil_dB)) => ((![P3 : dB > dB > $o, X : dB]: (P @ P3 @ (cons_dB @ X @ nil_dB))) => ((![P3 : dB > dB > $o, X : dB, Y4 : dB, Xs3 : list_dB]: ((P @ P3 @ (cons_dB @ Y4 @ Xs3)) => (P @ P3 @ (cons_dB @ X @ (cons_dB @ Y4 @ Xs3))))) => (P @ A0 @ A12))))))). % successively.induct
thf(fact_136_list__nonempty__induct, axiom,
    ((![Xs : list_dB, P : list_dB > $o]: ((~ ((Xs = nil_dB))) => ((![X : dB]: (P @ (cons_dB @ X @ nil_dB))) => ((![X : dB, Xs3 : list_dB]: ((~ ((Xs3 = nil_dB))) => ((P @ Xs3) => (P @ (cons_dB @ X @ Xs3))))) => (P @ Xs))))))). % list_nonempty_induct
thf(fact_137_map__tailrec__rev_Oinduct, axiom,
    ((![P : (dB > dB) > list_dB > list_dB > $o, A0 : dB > dB, A12 : list_dB, A23 : 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 @ A12 @ A23)))))). % map_tailrec_rev.induct

% Conjectures (1)
thf(conj_0, conjecture,
    ((it @ (subst @ (foldl_dB_dB @ app @ (var @ n) @ rs) @ u @ i)))).
