% 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_226__5211820_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:38:29.541

% 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__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 (35)
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_Oappend_001t__Lambda__OdB, type,
    append_dB : list_dB > list_dB > list_dB).
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_Olistsp_001t__Lambda__OdB, type,
    listsp_dB : (dB > $o) > list_dB > $o).
thf(sy_c_List_On__lists_001t__Lambda__OdB, type,
    n_lists_dB : nat > list_dB > list_list_dB).
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_a____, type,
    a : dB).
thf(sy_v_as____, type,
    as : list_dB).
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_r____, type,
    r : dB).
thf(sy_v_t____, type,
    t3 : dB).
thf(sy_v_u1____, type,
    u1 : dB).
thf(sy_v_u____, type,
    u : dB).

% Relevant facts (142)
thf(fact_0_Beta_Ohyps_I3_J, axiom,
    ((it @ a))). % Beta.hyps(3)
thf(fact_1_Beta_Ohyps_I1_J, axiom,
    ((it @ (foldl_dB_dB @ app @ (subst @ r @ a @ zero_zero_nat) @ as)))). % Beta.hyps(1)
thf(fact_2__092_060open_062IT_At_092_060close_062, axiom,
    ((it @ t3))). % \<open>IT t\<close>
thf(fact_3_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_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_beta__cases_I3_J, axiom,
    ((![S : dB, T : dB, U : dB]: ((beta @ (app @ S @ T) @ U) => ((![S2 : dB]: ((S = (abs @ S2)) => (~ ((U = (subst @ S2 @ 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_6_beta, axiom,
    ((![S : dB, T : dB]: (beta @ (app @ (abs @ S) @ T) @ (subst @ S @ T @ zero_zero_nat))))). % beta
thf(fact_7_beta_Ocases, axiom,
    ((![A1 : dB, A2 : dB]: ((beta @ A1 @ A2) => ((![S2 : dB, T2 : dB]: ((A1 = (app @ (abs @ S2) @ T2)) => (~ ((A2 = (subst @ S2 @ T2 @ zero_zero_nat)))))) => ((![S2 : dB, T2 : dB, U2 : dB]: ((A1 = (app @ S2 @ U2)) => ((A2 = (app @ T2 @ U2)) => (~ ((beta @ S2 @ T2)))))) => ((![S2 : dB, T2 : dB, U2 : dB]: ((A1 = (app @ U2 @ S2)) => ((A2 = (app @ U2 @ T2)) => (~ ((beta @ S2 @ T2)))))) => (~ ((![S2 : dB]: ((A1 = (abs @ S2)) => (![T2 : dB]: ((A2 = (abs @ T2)) => (~ ((beta @ S2 @ T2)))))))))))))))). % beta.cases
thf(fact_8_beta_Osimps, axiom,
    ((beta = (^[A12 : dB]: (^[A22 : dB]: (((?[S3 : dB]: (?[T3 : dB]: (((A12 = (app @ (abs @ S3) @ T3))) & ((A22 = (subst @ S3 @ T3 @ zero_zero_nat))))))) | ((((?[S3 : dB]: (?[T3 : dB]: (?[U3 : dB]: (((A12 = (app @ S3 @ U3))) & ((((A22 = (app @ T3 @ U3))) & ((beta @ S3 @ T3))))))))) | ((((?[S3 : dB]: (?[T3 : dB]: (?[U3 : dB]: (((A12 = (app @ U3 @ S3))) & ((((A22 = (app @ U3 @ T3))) & ((beta @ S3 @ T3))))))))) | ((?[S3 : dB]: (?[T3 : dB]: (((A12 = (abs @ S3))) & ((((A22 = (abs @ T3))) & ((beta @ S3 @ T3)))))))))))))))))). % beta.simps
thf(fact_9_beta_Oinducts, axiom,
    ((![X1 : dB, X2 : dB, P : dB > dB > $o]: ((beta @ X1 @ X2) => ((![S2 : dB, T2 : dB]: (P @ (app @ (abs @ S2) @ T2) @ (subst @ S2 @ T2 @ zero_zero_nat))) => ((![S2 : dB, T2 : dB, U2 : dB]: ((beta @ S2 @ T2) => ((P @ S2 @ T2) => (P @ (app @ S2 @ U2) @ (app @ T2 @ U2))))) => ((![S2 : dB, T2 : dB, U2 : dB]: ((beta @ S2 @ T2) => ((P @ S2 @ T2) => (P @ (app @ U2 @ S2) @ (app @ U2 @ T2))))) => ((![S2 : dB, T2 : dB]: ((beta @ S2 @ T2) => ((P @ S2 @ T2) => (P @ (abs @ S2) @ (abs @ T2))))) => (P @ X1 @ X2))))))))). % beta.inducts
thf(fact_10_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_11_dB_Oinject_I3_J, axiom,
    ((![X3 : dB, Y3 : dB]: (((abs @ X3) = (abs @ Y3)) = (X3 = Y3))))). % dB.inject(3)
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_uIT, axiom,
    ((it @ u))). % uIT
thf(fact_14_Beta_Oprems_I2_J, axiom,
    ((it @ u1))). % Beta.prems(2)
thf(fact_15_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_16_appR, axiom,
    ((![S : dB, T : dB, U : dB]: ((beta @ S @ T) => (beta @ (app @ U @ S) @ (app @ U @ T)))))). % appR
thf(fact_17_appL, axiom,
    ((![S : dB, T : dB, U : dB]: ((beta @ S @ T) => (beta @ (app @ S @ U) @ (app @ T @ U)))))). % appL
thf(fact_18_dB_Odistinct_I5_J, axiom,
    ((![X21 : dB, X22 : dB, X3 : dB]: (~ (((app @ X21 @ X22) = (abs @ X3))))))). % dB.distinct(5)
thf(fact_19_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_20_abs, axiom,
    ((![S : dB, T : dB]: ((beta @ S @ T) => (beta @ (abs @ S) @ (abs @ T)))))). % abs
thf(fact_21_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_22_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_23_IT_OBeta, 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))))))). % IT.Beta
thf(fact_24_Lambda, axiom,
    ((![R : dB]: ((it @ R) => (it @ (abs @ R)))))). % Lambda
thf(fact_25_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_26_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_27_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_28_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_29_T, axiom,
    ((typing @ (shift_type @ e @ i @ t2) @ (foldl_dB_dB @ app @ (app @ (abs @ r) @ a) @ as) @ t))). % T
thf(fact_30_apps__betasE, axiom,
    ((![R : dB, Rs : list_dB, S : dB]: ((beta @ (foldl_dB_dB @ app @ R @ Rs) @ S) => ((![R2 : dB]: ((beta @ R @ R2) => (~ ((S = (foldl_dB_dB @ app @ R2 @ Rs)))))) => ((![Rs2 : list_dB]: ((step1_dB @ beta @ Rs @ Rs2) => (~ ((S = (foldl_dB_dB @ app @ R @ Rs2)))))) => (~ ((![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_31_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_32_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_33_uT, axiom,
    ((typing @ e @ u @ t2))). % uT
thf(fact_34_dB_Oinject_I1_J, axiom,
    ((![X1 : nat, Y1 : nat]: (((var @ X1) = (var @ Y1)) = (X1 = Y1))))). % dB.inject(1)
thf(fact_35_subst__eq, axiom,
    ((![K : nat, U : dB]: ((subst @ (var @ K) @ U @ K) = U)))). % subst_eq
thf(fact_36_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_37_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_38_dB_Odistinct_I1_J, axiom,
    ((![X1 : nat, X21 : dB, X22 : dB]: (~ (((var @ X1) = (app @ X21 @ X22))))))). % dB.distinct(1)
thf(fact_39_beta__cases_I1_J, axiom,
    ((![I : nat, T : dB]: (~ ((beta @ (var @ I) @ T)))))). % beta_cases(1)
thf(fact_40_dB_Odistinct_I3_J, axiom,
    ((![X1 : nat, X3 : dB]: (~ (((var @ X1) = (abs @ X3))))))). % dB.distinct(3)
thf(fact_41_Var__IT, axiom,
    ((![N : nat]: (it @ (var @ N))))). % Var_IT
thf(fact_42_head__Var__reduction, axiom,
    ((![N : nat, Rs : list_dB, V : dB]: ((beta @ (foldl_dB_dB @ app @ (var @ N) @ Rs) @ V) => (?[Ss2 : list_dB]: ((step1_dB @ beta @ Rs @ Ss2) & (V = (foldl_dB_dB @ app @ (var @ N) @ Ss2)))))))). % head_Var_reduction
thf(fact_43_dB_Oinduct, axiom,
    ((![P : dB > $o, DB : dB]: ((![X4 : nat]: (P @ (var @ X4))) => ((![X1a : dB, X23 : dB]: ((P @ X1a) => ((P @ X23) => (P @ (app @ X1a @ X23))))) => ((![X4 : dB]: ((P @ X4) => (P @ (abs @ X4)))) => (P @ DB))))))). % dB.induct
thf(fact_44_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_45_app__Var__IT, axiom,
    ((![T : dB, I : nat]: ((it @ T) => (it @ (app @ T @ (var @ I))))))). % app_Var_IT
thf(fact_46_subst__Var__IT, axiom,
    ((![R : dB, I : nat, J : nat]: ((it @ R) => (it @ (subst @ R @ (var @ I) @ J)))))). % subst_Var_IT
thf(fact_47_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_48_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_49_MI2, axiom,
    ((![T1 : type, T22 : type, T : dB, E : nat > type, I : nat, T4 : type, U : dB]: ((t2 = (fun @ T1 @ T22)) => ((it @ T) => ((typing @ (shift_type @ E @ I @ T22) @ T @ T4) => ((it @ U) => ((typing @ E @ U @ T22) => (it @ (subst @ T @ U @ I)))))))))). % MI2
thf(fact_50_MI1, axiom,
    ((![T1 : type, T22 : type, T : dB, E : nat > type, I : nat, T4 : type, U : dB]: ((t2 = (fun @ T1 @ T22)) => ((it @ T) => ((typing @ (shift_type @ E @ I @ T1) @ T @ T4) => ((it @ U) => ((typing @ E @ U @ T1) => (it @ (subst @ T @ U @ I)))))))))). % MI1
thf(fact_51_Cons__step1__Cons, axiom,
    ((![R : dB > dB > $o, Y : dB, Ys : list_dB, X : dB, Xs : list_dB]: ((step1_dB @ R @ (cons_dB @ Y @ Ys) @ (cons_dB @ X @ Xs)) = (((((R @ Y @ X)) & ((Xs = Ys)))) | ((((X = Y)) & ((step1_dB @ R @ Ys @ Xs))))))))). % Cons_step1_Cons
thf(fact_52_abs__typeE, axiom,
    ((![E : nat > type, T : dB, T4 : type]: ((typing @ E @ (abs @ T) @ T4) => (~ ((![U4 : type, V2 : type]: (~ ((typing @ (shift_type @ E @ zero_zero_nat @ U4) @ T @ V2)))))))))). % abs_typeE
thf(fact_53_var__app__type__eq, axiom,
    ((![E : nat > type, I : nat, Ts : list_dB, T4 : type, U5 : type]: ((typing @ E @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ T4) => ((typing @ E @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ U5) => (T4 = U5)))))). % var_app_type_eq
thf(fact_54_subst__lemma, axiom,
    ((![E : nat > type, T : dB, T4 : type, E2 : nat > type, U : dB, U5 : type, I : nat]: ((typing @ E @ T @ T4) => ((typing @ E2 @ U @ U5) => ((E = (shift_type @ E2 @ I @ U5)) => (typing @ E2 @ (subst @ T @ U @ I) @ T4))))))). % subst_lemma
thf(fact_55_shift__eq, axiom,
    ((![I : nat, J : nat, E : nat > type, T4 : type]: ((I = J) => ((shift_type @ E @ I @ T4 @ J) = T4))))). % shift_eq
thf(fact_56_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_57_type__induct, axiom,
    ((![P : type > $o, T4 : type]: ((![T5 : type]: ((![T12 : type, T23 : type]: ((T5 = (fun @ T12 @ T23)) => (P @ T12))) => ((![T12 : type, T23 : type]: ((T5 = (fun @ T12 @ T23)) => (P @ T23))) => (P @ T5)))) => (P @ T4))))). % type_induct
thf(fact_58_typing__elims_I2_J, axiom,
    ((![E : nat > type, T : dB, U : dB, T4 : type]: ((typing @ E @ (app @ T @ U) @ T4) => (~ ((![T5 : type]: ((typing @ E @ T @ (fun @ T5 @ T4)) => (~ ((typing @ E @ U @ T5))))))))))). % typing_elims(2)
thf(fact_59_App, axiom,
    ((![Env : nat > type, S : dB, T4 : type, U5 : type, T : dB]: ((typing @ Env @ S @ (fun @ T4 @ U5)) => ((typing @ Env @ T @ T4) => (typing @ Env @ (app @ S @ T) @ U5)))))). % App
thf(fact_60_typing__elims_I3_J, axiom,
    ((![E : nat > type, T : dB, T4 : type]: ((typing @ E @ (abs @ T) @ T4) => (~ ((![T5 : type, U4 : type]: ((T4 = (fun @ T5 @ U4)) => (~ ((typing @ (shift_type @ E @ zero_zero_nat @ T5) @ T @ U4))))))))))). % typing_elims(3)
thf(fact_61_Abs, axiom,
    ((![Env : nat > type, T4 : type, T : dB, U5 : type]: ((typing @ (shift_type @ Env @ zero_zero_nat @ T4) @ T @ U5) => (typing @ Env @ (abs @ T) @ (fun @ T4 @ U5)))))). % Abs
thf(fact_62_typing_Oinducts, axiom,
    ((![X1 : nat > type, X2 : dB, X3 : type, P : (nat > type) > dB > type > $o]: ((typing @ X1 @ X2 @ X3) => ((![Env2 : nat > type, X4 : nat, T5 : type]: (((Env2 @ X4) = T5) => (P @ Env2 @ (var @ X4) @ T5))) => ((![Env2 : nat > type, T5 : type, T2 : dB, U4 : type]: ((typing @ (shift_type @ Env2 @ zero_zero_nat @ T5) @ T2 @ U4) => ((P @ (shift_type @ Env2 @ zero_zero_nat @ T5) @ T2 @ U4) => (P @ Env2 @ (abs @ T2) @ (fun @ T5 @ U4))))) => ((![Env2 : nat > type, S2 : dB, T5 : type, U4 : type, T2 : dB]: ((typing @ Env2 @ S2 @ (fun @ T5 @ U4)) => ((P @ Env2 @ S2 @ (fun @ T5 @ U4)) => ((typing @ Env2 @ T2 @ T5) => ((P @ Env2 @ T2 @ T5) => (P @ Env2 @ (app @ S2 @ T2) @ U4)))))) => (P @ X1 @ X2 @ X3)))))))). % typing.inducts
thf(fact_63_typing_Osimps, axiom,
    ((typing = (^[A12 : nat > type]: (^[A22 : dB]: (^[A3 : type]: (((?[Env3 : nat > type]: (?[X5 : nat]: (?[T6 : type]: (((A12 = Env3)) & ((((A22 = (var @ X5))) & ((((A3 = T6)) & (((Env3 @ X5) = T6))))))))))) | ((((?[Env3 : nat > type]: (?[T6 : type]: (?[T3 : dB]: (?[U6 : type]: (((A12 = Env3)) & ((((A22 = (abs @ T3))) & ((((A3 = (fun @ T6 @ U6))) & ((typing @ (shift_type @ Env3 @ zero_zero_nat @ T6) @ T3 @ U6)))))))))))) | ((?[Env3 : nat > type]: (?[S3 : dB]: (?[T6 : type]: (?[U6 : type]: (?[T3 : dB]: (((A12 = Env3)) & ((((A22 = (app @ S3 @ T3))) & ((((A3 = U6)) & ((((typing @ Env3 @ S3 @ (fun @ T6 @ U6))) & ((typing @ Env3 @ T3 @ T6)))))))))))))))))))))))). % typing.simps
thf(fact_64_typing_Ocases, axiom,
    ((![A1 : nat > type, A2 : dB, A32 : type]: ((typing @ A1 @ A2 @ A32) => ((![X4 : nat]: ((A2 = (var @ X4)) => (~ (((A1 @ X4) = A32))))) => ((![T5 : type, T2 : dB]: ((A2 = (abs @ T2)) => (![U4 : type]: ((A32 = (fun @ T5 @ U4)) => (~ ((typing @ (shift_type @ A1 @ zero_zero_nat @ T5) @ T2 @ U4))))))) => (~ ((![S2 : dB, T5 : type, U4 : type, T2 : dB]: ((A2 = (app @ S2 @ T2)) => ((A32 = U4) => ((typing @ A1 @ S2 @ (fun @ T5 @ U4)) => (~ ((typing @ A1 @ T2 @ T5))))))))))))))). % typing.cases
thf(fact_65_Cons__step1E, axiom,
    ((![R : dB > dB > $o, Ys : list_dB, X : dB, Xs : list_dB]: ((step1_dB @ R @ Ys @ (cons_dB @ X @ Xs)) => ((![Y2 : dB]: ((Ys = (cons_dB @ Y2 @ Xs)) => (~ ((R @ Y2 @ X))))) => (~ ((![Zs : list_dB]: ((Ys = (cons_dB @ X @ Zs)) => (~ ((step1_dB @ R @ Zs @ Xs)))))))))))). % Cons_step1E
thf(fact_66_not__Nil__step1, axiom,
    ((![R : dB > dB > $o, Xs : list_dB]: (~ ((step1_dB @ R @ nil_dB @ Xs)))))). % not_Nil_step1
thf(fact_67_not__step1__Nil, axiom,
    ((![R : dB > dB > $o, Xs : list_dB]: (~ ((step1_dB @ R @ Xs @ nil_dB)))))). % not_step1_Nil
thf(fact_68_subject__reduction, axiom,
    ((![E : nat > type, T : dB, T4 : type, T7 : dB]: ((typing @ E @ T @ T4) => ((beta @ T @ T7) => (typing @ E @ T7 @ T4)))))). % subject_reduction
thf(fact_69_typing__elims_I1_J, axiom,
    ((![E : nat > type, I : nat, T4 : type]: ((typing @ E @ (var @ I) @ T4) => ((E @ I) = T4))))). % typing_elims(1)
thf(fact_70_typing_OVar, axiom,
    ((![Env : nat > type, X : nat, T4 : type]: (((Env @ X) = T4) => (typing @ Env @ (var @ X) @ T4))))). % typing.Var
thf(fact_71_SI1, axiom,
    ((![E : nat > type, I : nat, T8 : type, U : dB]: ((typing @ (shift_type @ E @ I @ t2) @ (foldl_dB_dB @ app @ (subst @ r @ a @ zero_zero_nat) @ as) @ T8) => ((it @ U) => ((typing @ E @ U @ t2) => (it @ (subst @ (foldl_dB_dB @ app @ (subst @ r @ a @ zero_zero_nat) @ as) @ U @ I)))))))). % SI1
thf(fact_72_substn__subst__n, axiom,
    ((substn = (^[T3 : dB]: (^[S3 : dB]: (^[N3 : nat]: (subst @ T3 @ (liftn @ N3 @ S3 @ zero_zero_nat) @ N3))))))). % substn_subst_n
thf(fact_73_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_74_IT_Osimps, axiom,
    ((it = (^[A : dB]: (((?[Rs3 : list_dB]: (?[N3 : nat]: (((A = (foldl_dB_dB @ app @ (var @ N3) @ Rs3))) & ((listsp_dB @ it @ Rs3)))))) | ((((?[R3 : dB]: (((A = (abs @ R3))) & ((it @ R3))))) | ((?[R3 : dB]: (?[S3 : dB]: (?[Ss3 : list_dB]: (((A = (foldl_dB_dB @ app @ (app @ (abs @ R3) @ S3) @ Ss3))) & ((((it @ (foldl_dB_dB @ app @ (subst @ R3 @ S3 @ zero_zero_nat) @ Ss3))) & ((it @ S3)))))))))))))))). % IT.simps
thf(fact_75_IT_Ocases, axiom,
    ((![A4 : dB]: ((it @ A4) => ((![Rs4 : list_dB]: ((?[N2 : nat]: (A4 = (foldl_dB_dB @ app @ (var @ N2) @ Rs4))) => (~ ((listsp_dB @ it @ Rs4))))) => ((![R4 : dB]: ((A4 = (abs @ R4)) => (~ ((it @ R4))))) => (~ ((![R4 : dB, S2 : dB, Ss2 : list_dB]: ((A4 = (foldl_dB_dB @ app @ (app @ (abs @ R4) @ S2) @ Ss2)) => ((it @ (foldl_dB_dB @ app @ (subst @ R4 @ S2 @ zero_zero_nat) @ Ss2)) => (~ ((it @ S2)))))))))))))). % IT.cases
thf(fact_76_Beta_Oprems_I3_J, axiom,
    ((typing @ e1 @ u1 @ t2))). % Beta.prems(3)
thf(fact_77_listsp__conj__eq, axiom,
    ((![A5 : dB > $o, B : dB > $o]: ((listsp_dB @ (^[X5 : dB]: (((A5 @ X5)) & ((B @ X5))))) = (^[X5 : list_dB]: (((listsp_dB @ A5 @ X5)) & ((listsp_dB @ B @ X5)))))))). % listsp_conj_eq
thf(fact_78_listsp__simps_I1_J, axiom,
    ((![A5 : dB > $o]: (listsp_dB @ A5 @ nil_dB)))). % listsp_simps(1)
thf(fact_79_SI2, axiom,
    ((![E : nat > type, I : nat, T8 : type, U : dB]: ((typing @ (shift_type @ E @ I @ t2) @ a @ T8) => ((it @ U) => ((typing @ E @ U @ t2) => (it @ (subst @ a @ U @ I)))))))). % SI2
thf(fact_80_liftn__0, axiom,
    ((![T : dB, K : nat]: ((liftn @ zero_zero_nat @ T @ K) = T)))). % liftn_0
thf(fact_81_Beta_Oprems_I1_J, axiom,
    ((typing @ (shift_type @ e1 @ i1 @ t2) @ (foldl_dB_dB @ app @ (app @ (abs @ r) @ a) @ as) @ t_1))). % Beta.prems(1)
thf(fact_82_listsp_OCons, axiom,
    ((![A5 : dB > $o, A4 : dB, L : list_dB]: ((A5 @ A4) => ((listsp_dB @ A5 @ L) => (listsp_dB @ A5 @ (cons_dB @ A4 @ L))))))). % listsp.Cons
thf(fact_83_listspE, axiom,
    ((![A5 : dB > $o, X : dB, L : list_dB]: ((listsp_dB @ A5 @ (cons_dB @ X @ L)) => (~ (((A5 @ X) => (~ ((listsp_dB @ A5 @ L)))))))))). % listspE
thf(fact_84_listsp__simps_I2_J, axiom,
    ((![A5 : dB > $o, X : dB, Xs : list_dB]: ((listsp_dB @ A5 @ (cons_dB @ X @ Xs)) = (((A5 @ X)) & ((listsp_dB @ A5 @ Xs))))))). % listsp_simps(2)
thf(fact_85_listsp_ONil, axiom,
    ((![A5 : dB > $o]: (listsp_dB @ A5 @ nil_dB)))). % listsp.Nil
thf(fact_86_listsp_Oinducts, axiom,
    ((![A5 : dB > $o, X : list_dB, P : list_dB > $o]: ((listsp_dB @ A5 @ X) => ((P @ nil_dB) => ((![A6 : dB, L2 : list_dB]: ((A5 @ A6) => ((listsp_dB @ A5 @ L2) => ((P @ L2) => (P @ (cons_dB @ A6 @ L2)))))) => (P @ X))))))). % listsp.inducts
thf(fact_87_listsp_Osimps, axiom,
    ((listsp_dB = (^[A7 : dB > $o]: (^[A : list_dB]: (((A = nil_dB)) | ((?[B2 : dB]: (?[L3 : list_dB]: (((A = (cons_dB @ B2 @ L3))) & ((((A7 @ B2)) & ((listsp_dB @ A7 @ L3)))))))))))))). % listsp.simps
thf(fact_88_listsp_Ocases, axiom,
    ((![A5 : dB > $o, A4 : list_dB]: ((listsp_dB @ A5 @ A4) => ((~ ((A4 = nil_dB))) => (~ ((![A6 : dB, L2 : list_dB]: ((A4 = (cons_dB @ A6 @ L2)) => ((A5 @ A6) => (~ ((listsp_dB @ A5 @ L2))))))))))))). % listsp.cases
thf(fact_89_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_90_IT_Oinducts, axiom,
    ((![X : dB, P : dB > $o]: ((it @ X) => ((![Rs4 : list_dB, N2 : nat]: ((listsp_dB @ (^[X5 : dB]: (((it @ X5)) & ((P @ X5)))) @ Rs4) => (P @ (foldl_dB_dB @ app @ (var @ N2) @ Rs4)))) => ((![R4 : dB]: ((it @ R4) => ((P @ R4) => (P @ (abs @ R4))))) => ((![R4 : dB, S2 : dB, Ss2 : list_dB]: ((it @ (foldl_dB_dB @ app @ (subst @ R4 @ S2 @ zero_zero_nat) @ Ss2)) => ((P @ (foldl_dB_dB @ app @ (subst @ R4 @ S2 @ zero_zero_nat) @ Ss2)) => ((it @ S2) => ((P @ S2) => (P @ (foldl_dB_dB @ app @ (app @ (abs @ R4) @ S2) @ Ss2))))))) => (P @ X)))))))). % IT.inducts
thf(fact_91_not__Cons__self2, axiom,
    ((![X : dB, Xs : list_dB]: (~ (((cons_dB @ X @ Xs) = Xs)))))). % not_Cons_self2
thf(fact_92_IT_OVar, axiom,
    ((![Rs : list_dB, N : nat]: ((listsp_dB @ it @ Rs) => (it @ (foldl_dB_dB @ app @ (var @ N) @ Rs)))))). % IT.Var
thf(fact_93_map__tailrec__rev_Oinduct, axiom,
    ((![P : (dB > dB) > list_dB > list_dB > $o, A0 : dB > dB, A1 : list_dB, A2 : list_dB]: ((![F : dB > dB, X_1 : list_dB]: (P @ F @ nil_dB @ X_1)) => ((![F : dB > dB, A6 : dB, As : list_dB, Bs : list_dB]: ((P @ F @ As @ (cons_dB @ (F @ A6) @ Bs)) => (P @ F @ (cons_dB @ A6 @ As) @ Bs))) => (P @ A0 @ A1 @ A2)))))). % map_tailrec_rev.induct
thf(fact_94_list__nonempty__induct, axiom,
    ((![Xs : list_dB, P : list_dB > $o]: ((~ ((Xs = nil_dB))) => ((![X4 : dB]: (P @ (cons_dB @ X4 @ nil_dB))) => ((![X4 : dB, Xs2 : list_dB]: ((~ ((Xs2 = nil_dB))) => ((P @ Xs2) => (P @ (cons_dB @ X4 @ Xs2))))) => (P @ Xs))))))). % list_nonempty_induct
thf(fact_95_successively_Oinduct, axiom,
    ((![P : (dB > dB > $o) > list_dB > $o, A0 : dB > dB > $o, A1 : list_dB]: ((![P2 : dB > dB > $o]: (P @ P2 @ nil_dB)) => ((![P2 : dB > dB > $o, X4 : dB]: (P @ P2 @ (cons_dB @ X4 @ nil_dB))) => ((![P2 : dB > dB > $o, X4 : dB, Y2 : dB, Xs2 : list_dB]: ((P @ P2 @ (cons_dB @ Y2 @ Xs2)) => (P @ P2 @ (cons_dB @ X4 @ (cons_dB @ Y2 @ Xs2))))) => (P @ A0 @ A1))))))). % successively.induct
thf(fact_96_remdups__adj_Oinduct, axiom,
    ((![P : list_dB > $o, A0 : list_dB]: ((P @ nil_dB) => ((![X4 : dB]: (P @ (cons_dB @ X4 @ nil_dB))) => ((![X4 : dB, Y2 : dB, Xs2 : list_dB]: (((X4 = Y2) => (P @ (cons_dB @ X4 @ Xs2))) => (((~ ((X4 = Y2))) => (P @ (cons_dB @ Y2 @ Xs2))) => (P @ (cons_dB @ X4 @ (cons_dB @ Y2 @ Xs2)))))) => (P @ A0))))))). % remdups_adj.induct
thf(fact_97_sorted__wrt_Oinduct, axiom,
    ((![P : (dB > dB > $o) > list_dB > $o, A0 : dB > dB > $o, A1 : list_dB]: ((![P2 : dB > dB > $o]: (P @ P2 @ nil_dB)) => ((![P2 : dB > dB > $o, X4 : dB, Ys2 : list_dB]: ((P @ P2 @ Ys2) => (P @ P2 @ (cons_dB @ X4 @ Ys2)))) => (P @ A0 @ A1)))))). % sorted_wrt.induct
thf(fact_98_remdups__adj_Ocases, axiom,
    ((![X : list_dB]: ((~ ((X = nil_dB))) => ((![X4 : dB]: (~ ((X = (cons_dB @ X4 @ nil_dB))))) => (~ ((![X4 : dB, Y2 : dB, Xs2 : list_dB]: (~ ((X = (cons_dB @ X4 @ (cons_dB @ Y2 @ Xs2))))))))))))). % remdups_adj.cases
thf(fact_99_transpose_Ocases, axiom,
    ((![X : list_list_dB]: ((~ ((X = nil_list_dB))) => ((![Xss : list_list_dB]: (~ ((X = (cons_list_dB @ nil_dB @ Xss))))) => (~ ((![X4 : dB, Xs2 : list_dB, Xss : list_list_dB]: (~ ((X = (cons_list_dB @ (cons_dB @ X4 @ Xs2) @ Xss)))))))))))). % transpose.cases
thf(fact_100_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)) => ((![X4 : dB, Xs2 : list_dB, Y2 : dB, Ys2 : list_dB]: ((P @ Xs2 @ (cons_dB @ Y2 @ Ys2)) => ((P @ (cons_dB @ X4 @ Xs2) @ Ys2) => (P @ (cons_dB @ X4 @ Xs2) @ (cons_dB @ Y2 @ Ys2))))) => (P @ A0 @ A1))))))). % shuffles.induct
thf(fact_101_induct__list012, axiom,
    ((![P : list_dB > $o, Xs : list_dB]: ((P @ nil_dB) => ((![X4 : dB]: (P @ (cons_dB @ X4 @ nil_dB))) => ((![X4 : dB, Y2 : dB, Zs : list_dB]: ((P @ Zs) => ((P @ (cons_dB @ Y2 @ Zs)) => (P @ (cons_dB @ X4 @ (cons_dB @ Y2 @ Zs)))))) => (P @ Xs))))))). % induct_list012
thf(fact_102_splice_Oinduct, axiom,
    ((![P : list_dB > list_dB > $o, A0 : list_dB, A1 : list_dB]: ((![X_1 : list_dB]: (P @ nil_dB @ X_1)) => ((![X4 : dB, Xs2 : list_dB, Ys2 : list_dB]: ((P @ Ys2 @ Xs2) => (P @ (cons_dB @ X4 @ Xs2) @ Ys2))) => (P @ A0 @ A1)))))). % splice.induct
thf(fact_103_list__induct2_H, axiom,
    ((![P : list_dB > list_dB > $o, Xs : list_dB, Ys : list_dB]: ((P @ nil_dB @ nil_dB) => ((![X4 : dB, Xs2 : list_dB]: (P @ (cons_dB @ X4 @ Xs2) @ nil_dB)) => ((![Y2 : dB, Ys2 : list_dB]: (P @ nil_dB @ (cons_dB @ Y2 @ Ys2))) => ((![X4 : dB, Xs2 : list_dB, Y2 : dB, Ys2 : list_dB]: ((P @ Xs2 @ Ys2) => (P @ (cons_dB @ X4 @ Xs2) @ (cons_dB @ Y2 @ Ys2)))) => (P @ Xs @ Ys)))))))). % list_induct2'
thf(fact_104_neq__Nil__conv, axiom,
    ((![Xs : list_dB]: ((~ ((Xs = nil_dB))) = (?[Y4 : dB]: (?[Ys3 : list_dB]: (Xs = (cons_dB @ Y4 @ Ys3)))))))). % neq_Nil_conv
thf(fact_105_list_Oinducts, axiom,
    ((![P : list_dB > $o, List : list_dB]: ((P @ nil_dB) => ((![X12 : dB, X23 : list_dB]: ((P @ X23) => (P @ (cons_dB @ X12 @ X23)))) => (P @ List)))))). % list.inducts
thf(fact_106_list_Oexhaust, axiom,
    ((![Y : list_dB]: ((~ ((Y = nil_dB))) => (~ ((![X212 : dB, X222 : list_dB]: (~ ((Y = (cons_dB @ X212 @ X222))))))))))). % list.exhaust
thf(fact_107_list_OdiscI, axiom,
    ((![List : list_dB, X21 : dB, X22 : list_dB]: ((List = (cons_dB @ X21 @ X22)) => (~ ((List = nil_dB))))))). % list.discI
thf(fact_108_list_Odistinct_I1_J, axiom,
    ((![X21 : dB, X22 : list_dB]: (~ ((nil_dB = (cons_dB @ X21 @ X22))))))). % list.distinct(1)
thf(fact_109_foldl__Cons, axiom,
    ((![F2 : dB > dB > dB, A4 : dB, X : dB, Xs : list_dB]: ((foldl_dB_dB @ F2 @ A4 @ (cons_dB @ X @ Xs)) = (foldl_dB_dB @ F2 @ (F2 @ A4 @ X) @ Xs))))). % foldl_Cons
thf(fact_110_foldl__Nil, axiom,
    ((![F2 : dB > dB > dB, A4 : dB]: ((foldl_dB_dB @ F2 @ A4 @ nil_dB) = A4)))). % foldl_Nil
thf(fact_111_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_112_dB_Osize__gen_I1_J, axiom,
    ((![X1 : nat]: ((size_dB @ (var @ X1)) = zero_zero_nat)))). % dB.size_gen(1)
thf(fact_113_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_114_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)))) => ((?[Ss3 : list_dB]: (((Ts = (append_dB @ Ss3 @ (cons_dB @ S @ nil_dB)))) & ((R = (foldl_dB_dB @ app @ T @ Ss3))))))))))))). % App_eq_foldl_conv
thf(fact_115_same__append__eq, axiom,
    ((![Xs : list_dB, Ys : list_dB, Zs2 : list_dB]: (((append_dB @ Xs @ Ys) = (append_dB @ Xs @ Zs2)) = (Ys = Zs2))))). % same_append_eq
thf(fact_116_append__same__eq, axiom,
    ((![Ys : list_dB, Xs : list_dB, Zs2 : list_dB]: (((append_dB @ Ys @ Xs) = (append_dB @ Zs2 @ Xs)) = (Ys = Zs2))))). % append_same_eq
thf(fact_117_append__assoc, axiom,
    ((![Xs : list_dB, Ys : list_dB, Zs2 : list_dB]: ((append_dB @ (append_dB @ Xs @ Ys) @ Zs2) = (append_dB @ Xs @ (append_dB @ Ys @ Zs2)))))). % append_assoc
thf(fact_118_append_Oassoc, axiom,
    ((![A4 : list_dB, B3 : list_dB, C : list_dB]: ((append_dB @ (append_dB @ A4 @ B3) @ C) = (append_dB @ A4 @ (append_dB @ B3 @ C)))))). % append.assoc
thf(fact_119_append__Nil2, axiom,
    ((![Xs : list_dB]: ((append_dB @ Xs @ nil_dB) = Xs)))). % append_Nil2
thf(fact_120_append__self__conv, axiom,
    ((![Xs : list_dB, Ys : list_dB]: (((append_dB @ Xs @ Ys) = Xs) = (Ys = nil_dB))))). % append_self_conv
thf(fact_121_self__append__conv, axiom,
    ((![Xs : list_dB, Ys : list_dB]: ((Xs = (append_dB @ Xs @ Ys)) = (Ys = nil_dB))))). % self_append_conv
thf(fact_122_append__self__conv2, axiom,
    ((![Xs : list_dB, Ys : list_dB]: (((append_dB @ Xs @ Ys) = Ys) = (Xs = nil_dB))))). % append_self_conv2
thf(fact_123_self__append__conv2, axiom,
    ((![Ys : list_dB, Xs : list_dB]: ((Ys = (append_dB @ Xs @ Ys)) = (Xs = nil_dB))))). % self_append_conv2
thf(fact_124_Nil__is__append__conv, axiom,
    ((![Xs : list_dB, Ys : list_dB]: ((nil_dB = (append_dB @ Xs @ Ys)) = (((Xs = nil_dB)) & ((Ys = nil_dB))))))). % Nil_is_append_conv
thf(fact_125_append__is__Nil__conv, axiom,
    ((![Xs : list_dB, Ys : list_dB]: (((append_dB @ Xs @ Ys) = nil_dB) = (((Xs = nil_dB)) & ((Ys = nil_dB))))))). % append_is_Nil_conv
thf(fact_126_append_Oright__neutral, axiom,
    ((![A4 : list_dB]: ((append_dB @ A4 @ nil_dB) = A4)))). % append.right_neutral
thf(fact_127_foldl__append, axiom,
    ((![F2 : dB > dB > dB, A4 : dB, Xs : list_dB, Ys : list_dB]: ((foldl_dB_dB @ F2 @ A4 @ (append_dB @ Xs @ Ys)) = (foldl_dB_dB @ F2 @ (foldl_dB_dB @ F2 @ A4 @ Xs) @ Ys))))). % foldl_append
thf(fact_128_append__in__listsp__conv, axiom,
    ((![A5 : dB > $o, Xs : list_dB, Ys : list_dB]: ((listsp_dB @ A5 @ (append_dB @ Xs @ Ys)) = (((listsp_dB @ A5 @ Xs)) & ((listsp_dB @ A5 @ Ys))))))). % append_in_listsp_conv
thf(fact_129_append1__eq__conv, axiom,
    ((![Xs : list_dB, X : dB, Ys : list_dB, Y : dB]: (((append_dB @ Xs @ (cons_dB @ X @ nil_dB)) = (append_dB @ Ys @ (cons_dB @ Y @ nil_dB))) = (((Xs = Ys)) & ((X = Y))))))). % append1_eq_conv
thf(fact_130_append__Cons, axiom,
    ((![X : dB, Xs : list_dB, Ys : list_dB]: ((append_dB @ (cons_dB @ X @ Xs) @ Ys) = (cons_dB @ X @ (append_dB @ Xs @ Ys)))))). % append_Cons
thf(fact_131_Cons__eq__appendI, axiom,
    ((![X : dB, Xs1 : list_dB, Ys : list_dB, Xs : list_dB, Zs2 : list_dB]: (((cons_dB @ X @ Xs1) = Ys) => ((Xs = (append_dB @ Xs1 @ Zs2)) => ((cons_dB @ X @ Xs) = (append_dB @ Ys @ Zs2))))))). % Cons_eq_appendI
thf(fact_132_append__eq__append__conv2, axiom,
    ((![Xs : list_dB, Ys : list_dB, Zs2 : list_dB, Ts : list_dB]: (((append_dB @ Xs @ Ys) = (append_dB @ Zs2 @ Ts)) = (?[Us2 : list_dB]: (((((Xs = (append_dB @ Zs2 @ Us2))) & (((append_dB @ Us2 @ Ys) = Ts)))) | (((((append_dB @ Xs @ Us2) = Zs2)) & ((Ys = (append_dB @ Us2 @ Ts))))))))))). % append_eq_append_conv2
thf(fact_133_append__eq__appendI, axiom,
    ((![Xs : list_dB, Xs1 : list_dB, Zs2 : list_dB, Ys : list_dB, Us3 : list_dB]: (((append_dB @ Xs @ Xs1) = Zs2) => ((Ys = (append_dB @ Xs1 @ Us3)) => ((append_dB @ Xs @ Ys) = (append_dB @ Zs2 @ Us3))))))). % append_eq_appendI
thf(fact_134_eq__Nil__appendI, axiom,
    ((![Xs : list_dB, Ys : list_dB]: ((Xs = Ys) => (Xs = (append_dB @ nil_dB @ Ys)))))). % eq_Nil_appendI
thf(fact_135_append__Nil, axiom,
    ((![Ys : list_dB]: ((append_dB @ nil_dB @ Ys) = Ys)))). % append_Nil
thf(fact_136_append_Oleft__neutral, axiom,
    ((![A4 : list_dB]: ((append_dB @ nil_dB @ A4) = A4)))). % append.left_neutral
thf(fact_137_append__step1I, axiom,
    ((![R : dB > dB > $o, Ys : list_dB, Xs : list_dB, Vs : list_dB, Us3 : list_dB]: ((((step1_dB @ R @ Ys @ Xs) & (Vs = Us3)) | ((Ys = Xs) & (step1_dB @ R @ Vs @ Us3))) => (step1_dB @ R @ (append_dB @ Ys @ Vs) @ (append_dB @ Xs @ Us3)))))). % append_step1I
thf(fact_138_rev__exhaust2, axiom,
    ((![Xs : list_dB]: ((~ ((Xs = nil_dB))) => (~ ((![Ys2 : list_dB, Y2 : dB]: (~ ((Xs = (append_dB @ Ys2 @ (cons_dB @ Y2 @ nil_dB)))))))))))). % rev_exhaust2
thf(fact_139_rev__induct, axiom,
    ((![P : list_dB > $o, Xs : list_dB]: ((P @ nil_dB) => ((![X4 : dB, Xs2 : list_dB]: ((P @ Xs2) => (P @ (append_dB @ Xs2 @ (cons_dB @ X4 @ nil_dB))))) => (P @ Xs)))))). % rev_induct
thf(fact_140_rev__exhaust, axiom,
    ((![Xs : list_dB]: ((~ ((Xs = nil_dB))) => (~ ((![Ys2 : list_dB, Y2 : dB]: (~ ((Xs = (append_dB @ Ys2 @ (cons_dB @ Y2 @ nil_dB)))))))))))). % rev_exhaust
thf(fact_141_Cons__eq__append__conv, axiom,
    ((![X : dB, Xs : list_dB, Ys : list_dB, Zs2 : list_dB]: (((cons_dB @ X @ Xs) = (append_dB @ Ys @ Zs2)) = (((((Ys = nil_dB)) & (((cons_dB @ X @ Xs) = Zs2)))) | ((?[Ys4 : list_dB]: ((((cons_dB @ X @ Ys4) = Ys)) & ((Xs = (append_dB @ Ys4 @ Zs2))))))))))). % Cons_eq_append_conv

% Conjectures (1)
thf(conj_0, conjecture,
    ((beta @ (foldl_dB_dB @ app @ (app @ (abs @ r) @ a) @ as) @ (foldl_dB_dB @ app @ (subst @ r @ a @ zero_zero_nat) @ as)))).
