% 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_56__5209798_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:37:06.899

% Could-be-implicit typings (5)
thf(ty_n_t__List__Olist_It__List__Olist_It__Lambda__OdB_J_J, type,
    list_list_dB : $tType).
thf(ty_n_t__List__Olist_It__Lambda__OdB_J, type,
    list_dB : $tType).
thf(ty_n_t__Set__Oset_It__Lambda__OdB_J, type,
    set_dB : $tType).
thf(ty_n_t__Lambda__OdB, type,
    dB : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).

% Explicit typings (31)
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_InductTermi_OIT, type,
    it : dB > $o).
thf(sy_c_Lambda_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_Oappend_001t__Lambda__OdB, type,
    append_dB : list_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_Oinsert_001t__Lambda__OdB, type,
    insert_dB : dB > list_dB > list_dB).
thf(sy_c_List_Olist_OCons_001t__Lambda__OdB, type,
    cons_dB : dB > list_dB > list_dB).
thf(sy_c_List_Olist_OCons_001t__List__Olist_It__Lambda__OdB_J, type,
    cons_list_dB : list_dB > list_list_dB > list_list_dB).
thf(sy_c_List_Olist_ONil_001t__Lambda__OdB, type,
    nil_dB : list_dB).
thf(sy_c_List_Olist_ONil_001t__List__Olist_It__Lambda__OdB_J, type,
    nil_list_dB : list_list_dB).
thf(sy_c_List_Olist_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_Set_Othe__elem_001t__Lambda__OdB, type,
    the_elem_dB : set_dB > dB).
thf(sy_c_member_001t__Lambda__OdB, type,
    member_dB : dB > set_dB > $o).
thf(sy_v_ia, type,
    ia : nat).
thf(sy_v_ja, type,
    ja : nat).
thf(sy_v_ra, type,
    ra : dB).
thf(sy_v_s, type,
    s : dB).
thf(sy_v_ss, type,
    ss : list_dB).

% Relevant facts (138)
thf(fact_0_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_1_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_2_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_3_subst__eq, axiom,
    ((![K : nat, U : dB]: ((subst @ (var @ K) @ U @ K) = U)))). % subst_eq
thf(fact_4_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_5_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_6_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_7_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_8_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_9_dB_Oexhaust, axiom,
    ((![Y : dB]: ((![X1 : nat]: (~ ((Y = (var @ X1))))) => ((![X21 : dB, X22 : dB]: (~ ((Y = (app @ X21 @ X22))))) => (~ ((![X3 : dB]: (~ ((Y = (abs @ X3)))))))))))). % dB.exhaust
thf(fact_10_dB_Oinject_I1_J, axiom,
    ((![X12 : nat, Y1 : nat]: (((var @ X12) = (var @ Y1)) = (X12 = Y1))))). % dB.inject(1)
thf(fact_11_dB_Oinject_I3_J, axiom,
    ((![X32 : dB, Y3 : dB]: (((abs @ X32) = (abs @ Y3)) = (X32 = Y3))))). % dB.inject(3)
thf(fact_12_dB_Oinject_I2_J, axiom,
    ((![X212 : dB, X222 : dB, Y21 : dB, Y22 : dB]: (((app @ X212 @ X222) = (app @ Y21 @ Y22)) = (((X212 = Y21)) & ((X222 = Y22))))))). % dB.inject(2)
thf(fact_13_dB_Odistinct_I5_J, axiom,
    ((![X212 : dB, X222 : dB, X32 : dB]: (~ (((app @ X212 @ X222) = (abs @ X32))))))). % dB.distinct(5)
thf(fact_14_dB_Odistinct_I1_J, axiom,
    ((![X12 : nat, X212 : dB, X222 : dB]: (~ (((var @ X12) = (app @ X212 @ X222))))))). % dB.distinct(1)
thf(fact_15_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_16_dB_Odistinct_I3_J, axiom,
    ((![X12 : nat, X32 : dB]: (~ (((var @ X12) = (abs @ X32))))))). % dB.distinct(3)
thf(fact_17_Lambda, axiom,
    ((![R : dB]: ((it @ R) => (it @ (abs @ R)))))). % Lambda
thf(fact_18_IT_Osimps, axiom,
    ((it = (^[A : dB]: (((?[Rs2 : list_dB]: (?[N3 : nat]: (((A = (foldl_dB_dB @ app @ (var @ N3) @ Rs2))) & ((listsp_dB @ it @ Rs2)))))) | ((((?[R2 : dB]: (((A = (abs @ R2))) & ((it @ R2))))) | ((?[R2 : dB]: (?[S2 : dB]: (?[Ss2 : list_dB]: (((A = (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_19_IT_Ocases, axiom,
    ((![A2 : dB]: ((it @ A2) => ((![Rs3 : list_dB]: ((?[N2 : nat]: (A2 = (foldl_dB_dB @ app @ (var @ N2) @ Rs3))) => (~ ((listsp_dB @ it @ Rs3))))) => ((![R3 : dB]: ((A2 = (abs @ R3)) => (~ ((it @ R3))))) => (~ ((![R3 : dB, S3 : dB, Ss3 : list_dB]: ((A2 = (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_20_dB_Osize__gen_I1_J, axiom,
    ((![X12 : nat]: ((size_dB @ (var @ X12)) = zero_zero_nat)))). % dB.size_gen(1)
thf(fact_21_IT_OVar, axiom,
    ((![Rs : list_dB, N : nat]: ((listsp_dB @ it @ Rs) => (it @ (foldl_dB_dB @ app @ (var @ N) @ Rs)))))). % IT.Var
thf(fact_22_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_23_Apps__dB__induct, axiom,
    ((![P : dB > $o, T : dB]: ((![N2 : nat, Ts2 : list_dB]: ((![X4 : dB]: ((member_dB @ X4 @ (set_dB2 @ Ts2)) => (P @ X4))) => (P @ (foldl_dB_dB @ app @ (var @ N2) @ Ts2)))) => ((![U2 : dB]: ((P @ U2) => (![Ts2 : list_dB]: ((![X4 : dB]: ((member_dB @ X4 @ (set_dB2 @ Ts2)) => (P @ X4))) => (P @ (foldl_dB_dB @ app @ (abs @ U2) @ Ts2)))))) => (P @ T)))))). % Apps_dB_induct
thf(fact_24_beta_Oinducts, axiom,
    ((![X12 : dB, X23 : dB, P : dB > dB > $o]: ((beta @ X12 @ X23) => ((![S3 : dB, T2 : dB]: (P @ (app @ (abs @ S3) @ T2) @ (subst @ S3 @ T2 @ zero_zero_nat))) => ((![S3 : dB, T2 : dB, U2 : dB]: ((beta @ S3 @ T2) => ((P @ S3 @ T2) => (P @ (app @ S3 @ U2) @ (app @ T2 @ U2))))) => ((![S3 : dB, T2 : dB, U2 : dB]: ((beta @ S3 @ T2) => ((P @ S3 @ T2) => (P @ (app @ U2 @ S3) @ (app @ U2 @ T2))))) => ((![S3 : dB, T2 : dB]: ((beta @ S3 @ T2) => ((P @ S3 @ T2) => (P @ (abs @ S3) @ (abs @ T2))))) => (P @ X12 @ X23))))))))). % beta.inducts
thf(fact_25_beta_Osimps, axiom,
    ((beta = (^[A1 : dB]: (^[A22 : dB]: (((?[S2 : dB]: (?[T3 : dB]: (((A1 = (app @ (abs @ S2) @ T3))) & ((A22 = (subst @ S2 @ T3 @ zero_zero_nat))))))) | ((((?[S2 : dB]: (?[T3 : dB]: (?[U3 : dB]: (((A1 = (app @ S2 @ U3))) & ((((A22 = (app @ T3 @ U3))) & ((beta @ S2 @ T3))))))))) | ((((?[S2 : dB]: (?[T3 : dB]: (?[U3 : dB]: (((A1 = (app @ U3 @ S2))) & ((((A22 = (app @ U3 @ T3))) & ((beta @ S2 @ T3))))))))) | ((?[S2 : dB]: (?[T3 : dB]: (((A1 = (abs @ S2))) & ((((A22 = (abs @ T3))) & ((beta @ S2 @ T3)))))))))))))))))). % beta.simps
thf(fact_26_beta_Ocases, axiom,
    ((![A12 : dB, A23 : dB]: ((beta @ A12 @ A23) => ((![S3 : dB, T2 : dB]: ((A12 = (app @ (abs @ S3) @ T2)) => (~ ((A23 = (subst @ S3 @ T2 @ zero_zero_nat)))))) => ((![S3 : dB, T2 : dB, U2 : dB]: ((A12 = (app @ S3 @ U2)) => ((A23 = (app @ T2 @ U2)) => (~ ((beta @ S3 @ T2)))))) => ((![S3 : dB, T2 : dB, U2 : dB]: ((A12 = (app @ U2 @ S3)) => ((A23 = (app @ U2 @ T2)) => (~ ((beta @ S3 @ T2)))))) => (~ ((![S3 : dB]: ((A12 = (abs @ S3)) => (![T2 : dB]: ((A23 = (abs @ T2)) => (~ ((beta @ S3 @ T2)))))))))))))))). % beta.cases
thf(fact_27_appL, axiom,
    ((![S : dB, T : dB, U : dB]: ((beta @ S @ T) => (beta @ (app @ S @ U) @ (app @ T @ U)))))). % appL
thf(fact_28_appR, axiom,
    ((![S : dB, T : dB, U : dB]: ((beta @ S @ T) => (beta @ (app @ U @ S) @ (app @ U @ T)))))). % appR
thf(fact_29_beta__cases_I2_J, axiom,
    ((![R : dB, S : dB]: ((beta @ (abs @ R) @ S) => (~ ((![T2 : dB]: ((S = (abs @ T2)) => (~ ((beta @ R @ T2))))))))))). % beta_cases(2)
thf(fact_30_abs, axiom,
    ((![S : dB, T : dB]: ((beta @ S @ T) => (beta @ (abs @ S) @ (abs @ T)))))). % abs
thf(fact_31_beta__cases_I1_J, axiom,
    ((![I : nat, T : dB]: (~ ((beta @ (var @ I) @ T)))))). % beta_cases(1)
thf(fact_32_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_33_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_34_zero__reorient, axiom,
    ((![X5 : nat]: ((zero_zero_nat = X5) = (X5 = zero_zero_nat))))). % zero_reorient
thf(fact_35_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)))))) => ((![T2 : dB]: ((U = (app @ T2 @ T)) => (~ ((beta @ S @ T2))))) => (~ ((![T2 : dB]: ((U = (app @ S @ T2)) => (~ ((beta @ T @ T2))))))))))))). % beta_cases(3)
thf(fact_36_beta, axiom,
    ((![S : dB, T : dB]: (beta @ (app @ (abs @ S) @ T) @ (subst @ S @ T @ zero_zero_nat))))). % beta
thf(fact_37_in__listspI, axiom,
    ((![Xs : list_dB, A3 : dB > $o]: ((![X : dB]: ((member_dB @ X @ (set_dB2 @ Xs)) => (A3 @ X))) => (listsp_dB @ A3 @ Xs))))). % in_listspI
thf(fact_38_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_39_lem, axiom,
    ((![P : dB > $o, T : dB, N : nat]: ((![N2 : nat, Ts2 : list_dB]: ((![X4 : dB]: ((member_dB @ X4 @ (set_dB2 @ Ts2)) => (P @ X4))) => (P @ (foldl_dB_dB @ app @ (var @ N2) @ Ts2)))) => ((![U2 : dB]: ((P @ U2) => (![Ts2 : list_dB]: ((![X4 : dB]: ((member_dB @ X4 @ (set_dB2 @ Ts2)) => (P @ X4))) => (P @ (foldl_dB_dB @ app @ (abs @ U2) @ Ts2)))))) => (((size_size_dB @ T) = N) => (P @ T))))))). % lem
thf(fact_40_substn__subst__n, axiom,
    ((substn = (^[T3 : dB]: (^[S2 : dB]: (^[N3 : nat]: (subst @ T3 @ (liftn @ N3 @ S2 @ zero_zero_nat) @ N3))))))). % substn_subst_n
thf(fact_41_in__listsp__conv__set, axiom,
    ((listsp_dB = (^[A4 : dB > $o]: (^[Xs2 : list_dB]: (![X6 : dB]: (((member_dB @ X6 @ (set_dB2 @ Xs2))) => ((A4 @ X6))))))))). % in_listsp_conv_set
thf(fact_42_in__listspD, axiom,
    ((![A3 : dB > $o, Xs : list_dB]: ((listsp_dB @ A3 @ Xs) => (![X4 : dB]: ((member_dB @ X4 @ (set_dB2 @ Xs)) => (A3 @ X4))))))). % in_listspD
thf(fact_43_foldl__cong, axiom,
    ((![A2 : dB, B : dB, L : list_dB, K : list_dB, F : dB > dB > dB, G : dB > dB > dB]: ((A2 = B) => ((L = K) => ((![A5 : dB, X : dB]: ((member_dB @ X @ (set_dB2 @ L)) => ((F @ A5 @ X) = (G @ A5 @ X)))) => ((foldl_dB_dB @ F @ A2 @ L) = (foldl_dB_dB @ G @ B @ K)))))))). % foldl_cong
thf(fact_44_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_45_listsp__simps_I1_J, axiom,
    ((![A3 : dB > $o]: (listsp_dB @ A3 @ nil_dB)))). % listsp_simps(1)
thf(fact_46_liftn__0, axiom,
    ((![T : dB, K : nat]: ((liftn @ zero_zero_nat @ T @ K) = T)))). % liftn_0
thf(fact_47_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_48_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_49_foldl__Nil, axiom,
    ((![F : dB > dB > dB, A2 : dB]: ((foldl_dB_dB @ F @ A2 @ nil_dB) = A2)))). % foldl_Nil
thf(fact_50_listsp_ONil, axiom,
    ((![A3 : dB > $o]: (listsp_dB @ A3 @ nil_dB)))). % listsp.Nil
thf(fact_51_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_52_dB_Osize_I4_J, axiom,
    ((![X12 : nat]: ((size_size_dB @ (var @ X12)) = zero_zero_nat)))). % dB.size(4)
thf(fact_53_count__notin, axiom,
    ((![X5 : dB, Xs : list_dB]: ((~ ((member_dB @ X5 @ (set_dB2 @ Xs)))) => ((count_list_dB @ Xs @ X5) = zero_zero_nat))))). % count_notin
thf(fact_54_head__Var__reduction, axiom,
    ((![N : nat, Rs : list_dB, V : dB]: ((beta @ (foldl_dB_dB @ app @ (var @ N) @ Rs) @ V) => (?[Ss3 : list_dB]: ((step1_dB @ beta @ Rs @ Ss3) & (V = (foldl_dB_dB @ app @ (var @ N) @ Ss3)))))))). % head_Var_reduction
thf(fact_55_list__ex1__simps_I1_J, axiom,
    ((![P : dB > $o]: (~ ((list_ex1_dB @ P @ nil_dB)))))). % list_ex1_simps(1)
thf(fact_56_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_57_IT_Oinducts, axiom,
    ((![X5 : dB, P : dB > $o]: ((it @ X5) => ((![Rs3 : list_dB, N2 : nat]: ((listsp_dB @ (^[X6 : dB]: (((it @ X6)) & ((P @ X6)))) @ 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 @ X5)))))))). % IT.inducts
thf(fact_58_count__list_Osimps_I1_J, axiom,
    ((![Y : dB]: ((count_list_dB @ nil_dB @ Y) = zero_zero_nat)))). % count_list.simps(1)
thf(fact_59_listsp__conj__eq, axiom,
    ((![A3 : dB > $o, B2 : dB > $o]: ((listsp_dB @ (^[X6 : dB]: (((A3 @ X6)) & ((B2 @ X6))))) = (^[X6 : list_dB]: (((listsp_dB @ A3 @ X6)) & ((listsp_dB @ B2 @ X6)))))))). % listsp_conj_eq
thf(fact_60_list__ex1__iff, axiom,
    ((list_ex1_dB = (^[P2 : dB > $o]: (^[Xs2 : list_dB]: (?[X6 : dB]: (((((member_dB @ X6 @ (set_dB2 @ Xs2))) & ((P2 @ X6)))) & ((![Y2 : dB]: (((((member_dB @ Y2 @ (set_dB2 @ Xs2))) & ((P2 @ Y2)))) => ((Y2 = X6)))))))))))). % list_ex1_iff
thf(fact_61_can__select__set__list__ex1, axiom,
    ((![P : dB > $o, A3 : list_dB]: ((can_select_dB @ P @ (set_dB2 @ A3)) = (list_ex1_dB @ P @ A3))))). % can_select_set_list_ex1
thf(fact_62_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)))))) => (~ ((![T2 : dB]: ((R = (abs @ T2)) => (![U2 : dB, Us : list_dB]: ((Rs = (cons_dB @ U2 @ Us)) => (~ ((S = (foldl_dB_dB @ app @ (subst @ T2 @ U2 @ zero_zero_nat) @ Us)))))))))))))))). % apps_betasE
thf(fact_63_ex__step1I, axiom,
    ((![X5 : dB, Xs : list_dB, R : dB > dB > $o, Y : dB]: ((member_dB @ X5 @ (set_dB2 @ Xs)) => ((R @ Y @ X5) => (?[Ys : list_dB]: ((step1_dB @ R @ Ys @ Xs) & (member_dB @ Y @ (set_dB2 @ Ys))))))))). % ex_step1I
thf(fact_64_not__Nil__step1, axiom,
    ((![R : dB > dB > $o, Xs : list_dB]: (~ ((step1_dB @ R @ nil_dB @ Xs)))))). % not_Nil_step1
thf(fact_65_not__step1__Nil, axiom,
    ((![R : dB > dB > $o, Xs : list_dB]: (~ ((step1_dB @ R @ Xs @ nil_dB)))))). % not_step1_Nil
thf(fact_66_list_Oinject, axiom,
    ((![X212 : dB, X222 : list_dB, Y21 : dB, Y22 : list_dB]: (((cons_dB @ X212 @ X222) = (cons_dB @ Y21 @ Y22)) = (((X212 = Y21)) & ((X222 = Y22))))))). % list.inject
thf(fact_67_Cons__step1__Cons, axiom,
    ((![R : dB > dB > $o, Y : dB, Ys2 : list_dB, X5 : dB, Xs : list_dB]: ((step1_dB @ R @ (cons_dB @ Y @ Ys2) @ (cons_dB @ X5 @ Xs)) = (((((R @ Y @ X5)) & ((Xs = Ys2)))) | ((((X5 = Y)) & ((step1_dB @ R @ Ys2 @ Xs))))))))). % Cons_step1_Cons
thf(fact_68_transpose_Ocases, axiom,
    ((![X5 : list_list_dB]: ((~ ((X5 = nil_list_dB))) => ((![Xss : list_list_dB]: (~ ((X5 = (cons_list_dB @ nil_dB @ Xss))))) => (~ ((![X : dB, Xs3 : list_dB, Xss : list_list_dB]: (~ ((X5 = (cons_list_dB @ (cons_dB @ X @ Xs3) @ Xss)))))))))))). % transpose.cases
thf(fact_69_Cons__step1E, axiom,
    ((![R : dB > dB > $o, Ys2 : list_dB, X5 : dB, Xs : list_dB]: ((step1_dB @ R @ Ys2 @ (cons_dB @ X5 @ Xs)) => ((![Y4 : dB]: ((Ys2 = (cons_dB @ Y4 @ Xs)) => (~ ((R @ Y4 @ X5))))) => (~ ((![Zs : list_dB]: ((Ys2 = (cons_dB @ X5 @ Zs)) => (~ ((step1_dB @ R @ Zs @ Xs)))))))))))). % Cons_step1E
thf(fact_70_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, A5 : dB, As : list_dB, Bs : list_dB]: ((P @ F2 @ As @ (cons_dB @ (F2 @ A5) @ Bs)) => (P @ F2 @ (cons_dB @ A5 @ As) @ Bs))) => (P @ A0 @ A12 @ A23)))))). % map_tailrec_rev.induct
thf(fact_71_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_72_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_73_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_74_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, Ys : list_dB]: ((P @ P3 @ Ys) => (P @ P3 @ (cons_dB @ X @ Ys)))) => (P @ A0 @ A12)))))). % sorted_wrt.induct
thf(fact_75_remdups__adj_Ocases, axiom,
    ((![X5 : list_dB]: ((~ ((X5 = nil_dB))) => ((![X : dB]: (~ ((X5 = (cons_dB @ X @ nil_dB))))) => (~ ((![X : dB, Y4 : dB, Xs3 : list_dB]: (~ ((X5 = (cons_dB @ X @ (cons_dB @ Y4 @ Xs3))))))))))))). % remdups_adj.cases
thf(fact_76_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, Ys : list_dB]: ((P @ Xs3 @ (cons_dB @ Y4 @ Ys)) => ((P @ (cons_dB @ X @ Xs3) @ Ys) => (P @ (cons_dB @ X @ Xs3) @ (cons_dB @ Y4 @ Ys))))) => (P @ A0 @ A12))))))). % shuffles.induct
thf(fact_77_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_78_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, Ys : list_dB]: ((P @ Ys @ Xs3) => (P @ (cons_dB @ X @ Xs3) @ Ys))) => (P @ A0 @ A12)))))). % splice.induct
thf(fact_79_list__induct2_H, axiom,
    ((![P : list_dB > list_dB > $o, Xs : list_dB, Ys2 : list_dB]: ((P @ nil_dB @ nil_dB) => ((![X : dB, Xs3 : list_dB]: (P @ (cons_dB @ X @ Xs3) @ nil_dB)) => ((![Y4 : dB, Ys : list_dB]: (P @ nil_dB @ (cons_dB @ Y4 @ Ys))) => ((![X : dB, Xs3 : list_dB, Y4 : dB, Ys : list_dB]: ((P @ Xs3 @ Ys) => (P @ (cons_dB @ X @ Xs3) @ (cons_dB @ Y4 @ Ys)))) => (P @ Xs @ Ys2)))))))). % list_induct2'
thf(fact_80_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_81_list_Oinducts, axiom,
    ((![P : list_dB > $o, List : list_dB]: ((P @ nil_dB) => ((![X1 : dB, X2 : list_dB]: ((P @ X2) => (P @ (cons_dB @ X1 @ X2)))) => (P @ List)))))). % list.inducts
thf(fact_82_list_Oexhaust, axiom,
    ((![Y : list_dB]: ((~ ((Y = nil_dB))) => (~ ((![X21 : dB, X22 : list_dB]: (~ ((Y = (cons_dB @ X21 @ X22))))))))))). % list.exhaust
thf(fact_83_list_OdiscI, axiom,
    ((![List : list_dB, X212 : dB, X222 : list_dB]: ((List = (cons_dB @ X212 @ X222)) => (~ ((List = nil_dB))))))). % list.discI
thf(fact_84_list_Odistinct_I1_J, axiom,
    ((![X212 : dB, X222 : list_dB]: (~ ((nil_dB = (cons_dB @ X212 @ X222))))))). % list.distinct(1)
thf(fact_85_list_Oset__intros_I2_J, axiom,
    ((![Y : dB, X222 : list_dB, X212 : dB]: ((member_dB @ Y @ (set_dB2 @ X222)) => (member_dB @ Y @ (set_dB2 @ (cons_dB @ X212 @ X222))))))). % list.set_intros(2)
thf(fact_86_list_Oset__intros_I1_J, axiom,
    ((![X212 : dB, X222 : list_dB]: (member_dB @ X212 @ (set_dB2 @ (cons_dB @ X212 @ X222)))))). % list.set_intros(1)
thf(fact_87_set__ConsD, axiom,
    ((![Y : dB, X5 : dB, Xs : list_dB]: ((member_dB @ Y @ (set_dB2 @ (cons_dB @ X5 @ Xs))) => ((Y = X5) | (member_dB @ Y @ (set_dB2 @ Xs))))))). % set_ConsD
thf(fact_88_list_Oset__cases, axiom,
    ((![E : dB, A2 : list_dB]: ((member_dB @ E @ (set_dB2 @ A2)) => ((![Z2 : list_dB]: (~ ((A2 = (cons_dB @ E @ Z2))))) => (~ ((![Z1 : dB, Z2 : list_dB]: ((A2 = (cons_dB @ Z1 @ Z2)) => (~ ((member_dB @ E @ (set_dB2 @ Z2))))))))))))). % list.set_cases
thf(fact_89_not__Cons__self2, axiom,
    ((![X5 : dB, Xs : list_dB]: (~ (((cons_dB @ X5 @ Xs) = Xs)))))). % not_Cons_self2
thf(fact_90_foldl__Cons, axiom,
    ((![F : dB > dB > dB, A2 : dB, X5 : dB, Xs : list_dB]: ((foldl_dB_dB @ F @ A2 @ (cons_dB @ X5 @ Xs)) = (foldl_dB_dB @ F @ (F @ A2 @ X5) @ Xs))))). % foldl_Cons
thf(fact_91_listsp__simps_I2_J, axiom,
    ((![A3 : dB > $o, X5 : dB, Xs : list_dB]: ((listsp_dB @ A3 @ (cons_dB @ X5 @ Xs)) = (((A3 @ X5)) & ((listsp_dB @ A3 @ Xs))))))). % listsp_simps(2)
thf(fact_92_listspE, axiom,
    ((![A3 : dB > $o, X5 : dB, L : list_dB]: ((listsp_dB @ A3 @ (cons_dB @ X5 @ L)) => (~ (((A3 @ X5) => (~ ((listsp_dB @ A3 @ L)))))))))). % listspE
thf(fact_93_listsp_OCons, axiom,
    ((![A3 : dB > $o, A2 : dB, L : list_dB]: ((A3 @ A2) => ((listsp_dB @ A3 @ L) => (listsp_dB @ A3 @ (cons_dB @ A2 @ L))))))). % listsp.Cons
thf(fact_94_listsp_Oinducts, axiom,
    ((![A3 : dB > $o, X5 : list_dB, P : list_dB > $o]: ((listsp_dB @ A3 @ X5) => ((P @ nil_dB) => ((![A5 : dB, L2 : list_dB]: ((A3 @ A5) => ((listsp_dB @ A3 @ L2) => ((P @ L2) => (P @ (cons_dB @ A5 @ L2)))))) => (P @ X5))))))). % listsp.inducts
thf(fact_95_listsp_Osimps, axiom,
    ((listsp_dB = (^[A4 : dB > $o]: (^[A : list_dB]: (((A = nil_dB)) | ((?[B3 : dB]: (?[L3 : list_dB]: (((A = (cons_dB @ B3 @ L3))) & ((((A4 @ B3)) & ((listsp_dB @ A4 @ L3)))))))))))))). % listsp.simps
thf(fact_96_listsp_Ocases, axiom,
    ((![A3 : dB > $o, A2 : list_dB]: ((listsp_dB @ A3 @ A2) => ((~ ((A2 = nil_dB))) => (~ ((![A5 : dB, L2 : list_dB]: ((A2 = (cons_dB @ A5 @ L2)) => ((A3 @ A5) => (~ ((listsp_dB @ A3 @ L2))))))))))))). % listsp.cases
thf(fact_97_the__elem__set, axiom,
    ((![X5 : dB]: ((the_elem_dB @ (set_dB2 @ (cons_dB @ X5 @ nil_dB))) = X5)))). % the_elem_set
thf(fact_98_app__last, axiom,
    ((![T : dB, Ts : list_dB, U : dB]: ((app @ (foldl_dB_dB @ app @ T @ Ts) @ U) = (foldl_dB_dB @ app @ T @ (append_dB @ Ts @ (cons_dB @ U @ nil_dB))))))). % app_last
thf(fact_99_App__eq__foldl__conv, axiom,
    ((![R : dB, S : dB, T : dB, Ts : list_dB]: (((app @ R @ S) = (foldl_dB_dB @ app @ T @ Ts)) = (((((Ts = nil_dB)) => (((app @ R @ S) = T)))) & ((((~ ((Ts = nil_dB)))) => ((?[Ss2 : list_dB]: (((Ts = (append_dB @ Ss2 @ (cons_dB @ S @ nil_dB)))) & ((R = (foldl_dB_dB @ app @ T @ Ss2))))))))))))). % App_eq_foldl_conv
thf(fact_100_not__in__set__insert, axiom,
    ((![X5 : dB, Xs : list_dB]: ((~ ((member_dB @ X5 @ (set_dB2 @ Xs)))) => ((insert_dB @ X5 @ Xs) = (cons_dB @ X5 @ Xs)))))). % not_in_set_insert
thf(fact_101_append_Oassoc, axiom,
    ((![A2 : list_dB, B : list_dB, C : list_dB]: ((append_dB @ (append_dB @ A2 @ B) @ C) = (append_dB @ A2 @ (append_dB @ B @ C)))))). % append.assoc
thf(fact_102_append__assoc, axiom,
    ((![Xs : list_dB, Ys2 : list_dB, Zs2 : list_dB]: ((append_dB @ (append_dB @ Xs @ Ys2) @ Zs2) = (append_dB @ Xs @ (append_dB @ Ys2 @ Zs2)))))). % append_assoc
thf(fact_103_append__same__eq, axiom,
    ((![Ys2 : list_dB, Xs : list_dB, Zs2 : list_dB]: (((append_dB @ Ys2 @ Xs) = (append_dB @ Zs2 @ Xs)) = (Ys2 = Zs2))))). % append_same_eq
thf(fact_104_same__append__eq, axiom,
    ((![Xs : list_dB, Ys2 : list_dB, Zs2 : list_dB]: (((append_dB @ Xs @ Ys2) = (append_dB @ Xs @ Zs2)) = (Ys2 = Zs2))))). % same_append_eq
thf(fact_105_append_Oright__neutral, axiom,
    ((![A2 : list_dB]: ((append_dB @ A2 @ nil_dB) = A2)))). % append.right_neutral
thf(fact_106_append__is__Nil__conv, axiom,
    ((![Xs : list_dB, Ys2 : list_dB]: (((append_dB @ Xs @ Ys2) = nil_dB) = (((Xs = nil_dB)) & ((Ys2 = nil_dB))))))). % append_is_Nil_conv
thf(fact_107_Nil__is__append__conv, axiom,
    ((![Xs : list_dB, Ys2 : list_dB]: ((nil_dB = (append_dB @ Xs @ Ys2)) = (((Xs = nil_dB)) & ((Ys2 = nil_dB))))))). % Nil_is_append_conv
thf(fact_108_self__append__conv2, axiom,
    ((![Ys2 : list_dB, Xs : list_dB]: ((Ys2 = (append_dB @ Xs @ Ys2)) = (Xs = nil_dB))))). % self_append_conv2
thf(fact_109_append__self__conv2, axiom,
    ((![Xs : list_dB, Ys2 : list_dB]: (((append_dB @ Xs @ Ys2) = Ys2) = (Xs = nil_dB))))). % append_self_conv2
thf(fact_110_self__append__conv, axiom,
    ((![Xs : list_dB, Ys2 : list_dB]: ((Xs = (append_dB @ Xs @ Ys2)) = (Ys2 = nil_dB))))). % self_append_conv
thf(fact_111_append__self__conv, axiom,
    ((![Xs : list_dB, Ys2 : list_dB]: (((append_dB @ Xs @ Ys2) = Xs) = (Ys2 = nil_dB))))). % append_self_conv
thf(fact_112_append__Nil2, axiom,
    ((![Xs : list_dB]: ((append_dB @ Xs @ nil_dB) = Xs)))). % append_Nil2
thf(fact_113_foldl__append, axiom,
    ((![F : dB > dB > dB, A2 : dB, Xs : list_dB, Ys2 : list_dB]: ((foldl_dB_dB @ F @ A2 @ (append_dB @ Xs @ Ys2)) = (foldl_dB_dB @ F @ (foldl_dB_dB @ F @ A2 @ Xs) @ Ys2))))). % foldl_append
thf(fact_114_append__in__listsp__conv, axiom,
    ((![A3 : dB > $o, Xs : list_dB, Ys2 : list_dB]: ((listsp_dB @ A3 @ (append_dB @ Xs @ Ys2)) = (((listsp_dB @ A3 @ Xs)) & ((listsp_dB @ A3 @ Ys2))))))). % append_in_listsp_conv
thf(fact_115_in__set__insert, axiom,
    ((![X5 : dB, Xs : list_dB]: ((member_dB @ X5 @ (set_dB2 @ Xs)) => ((insert_dB @ X5 @ Xs) = Xs))))). % in_set_insert
thf(fact_116_append1__eq__conv, axiom,
    ((![Xs : list_dB, X5 : dB, Ys2 : list_dB, Y : dB]: (((append_dB @ Xs @ (cons_dB @ X5 @ nil_dB)) = (append_dB @ Ys2 @ (cons_dB @ Y @ nil_dB))) = (((Xs = Ys2)) & ((X5 = Y))))))). % append1_eq_conv
thf(fact_117_insert__Nil, axiom,
    ((![X5 : dB]: ((insert_dB @ X5 @ nil_dB) = (cons_dB @ X5 @ nil_dB))))). % insert_Nil
thf(fact_118_append__step1I, axiom,
    ((![R : dB > dB > $o, Ys2 : list_dB, Xs : list_dB, Vs : list_dB, Us2 : list_dB]: ((((step1_dB @ R @ Ys2 @ Xs) & (Vs = Us2)) | ((Ys2 = Xs) & (step1_dB @ R @ Vs @ Us2))) => (step1_dB @ R @ (append_dB @ Ys2 @ Vs) @ (append_dB @ Xs @ Us2)))))). % append_step1I
thf(fact_119_append__Cons, axiom,
    ((![X5 : dB, Xs : list_dB, Ys2 : list_dB]: ((append_dB @ (cons_dB @ X5 @ Xs) @ Ys2) = (cons_dB @ X5 @ (append_dB @ Xs @ Ys2)))))). % append_Cons
thf(fact_120_Cons__eq__appendI, axiom,
    ((![X5 : dB, Xs1 : list_dB, Ys2 : list_dB, Xs : list_dB, Zs2 : list_dB]: (((cons_dB @ X5 @ Xs1) = Ys2) => ((Xs = (append_dB @ Xs1 @ Zs2)) => ((cons_dB @ X5 @ Xs) = (append_dB @ Ys2 @ Zs2))))))). % Cons_eq_appendI
thf(fact_121_append__eq__appendI, axiom,
    ((![Xs : list_dB, Xs1 : list_dB, Zs2 : list_dB, Ys2 : list_dB, Us2 : list_dB]: (((append_dB @ Xs @ Xs1) = Zs2) => ((Ys2 = (append_dB @ Xs1 @ Us2)) => ((append_dB @ Xs @ Ys2) = (append_dB @ Zs2 @ Us2))))))). % append_eq_appendI
thf(fact_122_append__eq__append__conv2, axiom,
    ((![Xs : list_dB, Ys2 : list_dB, Zs2 : list_dB, Ts : list_dB]: (((append_dB @ Xs @ Ys2) = (append_dB @ Zs2 @ Ts)) = (?[Us3 : list_dB]: (((((Xs = (append_dB @ Zs2 @ Us3))) & (((append_dB @ Us3 @ Ys2) = Ts)))) | (((((append_dB @ Xs @ Us3) = Zs2)) & ((Ys2 = (append_dB @ Us3 @ Ts))))))))))). % append_eq_append_conv2
thf(fact_123_eq__Nil__appendI, axiom,
    ((![Xs : list_dB, Ys2 : list_dB]: ((Xs = Ys2) => (Xs = (append_dB @ nil_dB @ Ys2)))))). % eq_Nil_appendI
thf(fact_124_append__Nil, axiom,
    ((![Ys2 : list_dB]: ((append_dB @ nil_dB @ Ys2) = Ys2)))). % append_Nil
thf(fact_125_append_Oleft__neutral, axiom,
    ((![A2 : list_dB]: ((append_dB @ nil_dB @ A2) = A2)))). % append.left_neutral
thf(fact_126_rev__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 @ (append_dB @ Xs3 @ (cons_dB @ X @ nil_dB)))))) => (P @ Xs))))))). % rev_nonempty_induct
thf(fact_127_append__eq__Cons__conv, axiom,
    ((![Ys2 : list_dB, Zs2 : list_dB, X5 : dB, Xs : list_dB]: (((append_dB @ Ys2 @ Zs2) = (cons_dB @ X5 @ Xs)) = (((((Ys2 = nil_dB)) & ((Zs2 = (cons_dB @ X5 @ Xs))))) | ((?[Ys4 : list_dB]: (((Ys2 = (cons_dB @ X5 @ Ys4))) & (((append_dB @ Ys4 @ Zs2) = Xs)))))))))). % append_eq_Cons_conv
thf(fact_128_Cons__eq__append__conv, axiom,
    ((![X5 : dB, Xs : list_dB, Ys2 : list_dB, Zs2 : list_dB]: (((cons_dB @ X5 @ Xs) = (append_dB @ Ys2 @ Zs2)) = (((((Ys2 = nil_dB)) & (((cons_dB @ X5 @ Xs) = Zs2)))) | ((?[Ys4 : list_dB]: ((((cons_dB @ X5 @ Ys4) = Ys2)) & ((Xs = (append_dB @ Ys4 @ Zs2))))))))))). % Cons_eq_append_conv
thf(fact_129_rev__exhaust, axiom,
    ((![Xs : list_dB]: ((~ ((Xs = nil_dB))) => (~ ((![Ys : list_dB, Y4 : dB]: (~ ((Xs = (append_dB @ Ys @ (cons_dB @ Y4 @ nil_dB)))))))))))). % rev_exhaust
thf(fact_130_rev__induct, axiom,
    ((![P : list_dB > $o, Xs : list_dB]: ((P @ nil_dB) => ((![X : dB, Xs3 : list_dB]: ((P @ Xs3) => (P @ (append_dB @ Xs3 @ (cons_dB @ X @ nil_dB))))) => (P @ Xs)))))). % rev_induct
thf(fact_131_split__list__first__prop__iff, axiom,
    ((![Xs : list_dB, P : dB > $o]: ((?[X6 : dB]: (((member_dB @ X6 @ (set_dB2 @ Xs))) & ((P @ X6)))) = (?[Ys3 : list_dB]: (?[X6 : dB]: (((?[Zs3 : list_dB]: (Xs = (append_dB @ Ys3 @ (cons_dB @ X6 @ Zs3))))) & ((((P @ X6)) & ((![Y2 : dB]: (((member_dB @ Y2 @ (set_dB2 @ Ys3))) => ((~ ((P @ Y2)))))))))))))))). % split_list_first_prop_iff
thf(fact_132_split__list__last__prop__iff, axiom,
    ((![Xs : list_dB, P : dB > $o]: ((?[X6 : dB]: (((member_dB @ X6 @ (set_dB2 @ Xs))) & ((P @ X6)))) = (?[Ys3 : list_dB]: (?[X6 : dB]: (?[Zs3 : list_dB]: (((Xs = (append_dB @ Ys3 @ (cons_dB @ X6 @ Zs3)))) & ((((P @ X6)) & ((![Y2 : dB]: (((member_dB @ Y2 @ (set_dB2 @ Zs3))) => ((~ ((P @ Y2))))))))))))))))). % split_list_last_prop_iff
thf(fact_133_in__set__conv__decomp__first, axiom,
    ((![X5 : dB, Xs : list_dB]: ((member_dB @ X5 @ (set_dB2 @ Xs)) = (?[Ys3 : list_dB]: (?[Zs3 : list_dB]: (((Xs = (append_dB @ Ys3 @ (cons_dB @ X5 @ Zs3)))) & ((~ ((member_dB @ X5 @ (set_dB2 @ Ys3)))))))))))). % in_set_conv_decomp_first
thf(fact_134_in__set__conv__decomp__last, axiom,
    ((![X5 : dB, Xs : list_dB]: ((member_dB @ X5 @ (set_dB2 @ Xs)) = (?[Ys3 : list_dB]: (?[Zs3 : list_dB]: (((Xs = (append_dB @ Ys3 @ (cons_dB @ X5 @ Zs3)))) & ((~ ((member_dB @ X5 @ (set_dB2 @ Zs3)))))))))))). % in_set_conv_decomp_last
thf(fact_135_split__list__first__propE, axiom,
    ((![Xs : list_dB, P : dB > $o]: ((?[X4 : dB]: ((member_dB @ X4 @ (set_dB2 @ Xs)) & (P @ X4))) => (~ ((![Ys : list_dB, X : dB]: ((?[Zs : list_dB]: (Xs = (append_dB @ Ys @ (cons_dB @ X @ Zs)))) => ((P @ X) => (~ ((![Xa : dB]: ((member_dB @ Xa @ (set_dB2 @ Ys)) => (~ ((P @ Xa)))))))))))))))). % split_list_first_propE
thf(fact_136_split__list__last__propE, axiom,
    ((![Xs : list_dB, P : dB > $o]: ((?[X4 : dB]: ((member_dB @ X4 @ (set_dB2 @ Xs)) & (P @ X4))) => (~ ((![Ys : list_dB, X : dB, Zs : list_dB]: ((Xs = (append_dB @ Ys @ (cons_dB @ X @ Zs))) => ((P @ X) => (~ ((![Xa : dB]: ((member_dB @ Xa @ (set_dB2 @ Zs)) => (~ ((P @ Xa)))))))))))))))). % split_list_last_propE
thf(fact_137_split__list__first__prop, axiom,
    ((![Xs : list_dB, P : dB > $o]: ((?[X4 : dB]: ((member_dB @ X4 @ (set_dB2 @ Xs)) & (P @ X4))) => (?[Ys : list_dB, X : dB]: ((?[Zs : list_dB]: (Xs = (append_dB @ Ys @ (cons_dB @ X @ Zs)))) & ((P @ X) & (![Xa : dB]: ((member_dB @ Xa @ (set_dB2 @ Ys)) => (~ ((P @ Xa)))))))))))). % split_list_first_prop

% Conjectures (5)
thf(conj_0, hypothesis,
    ((it @ (foldl_dB_dB @ app @ (subst @ ra @ s @ zero_zero_nat) @ ss)))).
thf(conj_1, hypothesis,
    ((![I2 : nat, J : nat]: (it @ (subst @ (foldl_dB_dB @ app @ (subst @ ra @ s @ zero_zero_nat) @ ss) @ (var @ I2) @ J))))).
thf(conj_2, hypothesis,
    ((it @ s))).
thf(conj_3, hypothesis,
    ((![I2 : nat, J : nat]: (it @ (subst @ s @ (var @ I2) @ J))))).
thf(conj_4, conjecture,
    ((it @ (subst @ (foldl_dB_dB @ app @ (app @ (abs @ ra) @ s) @ ss) @ (var @ ia) @ ja)))).
