% 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_278__5212398_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:38:51.940

% Could-be-implicit typings (5)
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 (28)
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_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_Olift, type,
    lift : 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_Ofoldl_001t__Lambda__OdB_001t__Lambda__OdB, type,
    foldl_dB_dB : (dB > dB > dB) > dB > list_dB > dB).
thf(sy_c_List_Olist_ONil_001t__Lambda__OdB, type,
    nil_dB : list_dB).
thf(sy_c_List_Olist_Oset_001t__Lambda__OdB, type,
    set_dB2 : list_dB > set_dB).
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_Orderings_Oord__class_Oless__eq_001_062_It__Lambda__OdB_M_062_It__Lambda__OdB_M_Eo_J_J, type,
    ord_less_eq_dB_dB_o : (dB > dB > $o) > (dB > dB > $o) > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Lambda__OdB_M_Eo_J, type,
    ord_less_eq_dB_o : (dB > $o) > (dB > $o) > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__List__Olist_It__Lambda__OdB_J_M_Eo_J, type,
    ord_le2037264764t_dB_o : (list_dB > $o) > (list_dB > $o) > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Set__Oset_It__Lambda__OdB_J, type,
    ord_less_eq_set_dB : set_dB > set_dB > $o).
thf(sy_c_Relation_Oconversep_001t__Lambda__OdB_001t__Lambda__OdB, type,
    conversep_dB_dB : (dB > dB > $o) > dB > dB > $o).
thf(sy_c_Relation_Oconversep_001t__List__Olist_It__Lambda__OdB_J_001t__List__Olist_It__Lambda__OdB_J, type,
    conver499110749ist_dB : (list_dB > list_dB > $o) > list_dB > list_dB > $o).
thf(sy_c_Transitive__Closure_Ortranclp_001t__Lambda__OdB, type,
    transi72828568clp_dB : (dB > dB > $o) > dB > dB > $o).
thf(sy_c_Wellfounded_Oaccp_001t__Lambda__OdB, type,
    accp_dB : (dB > dB > $o) > dB > $o).
thf(sy_c_Wellfounded_Oaccp_001t__List__Olist_It__Lambda__OdB_J, type,
    accp_list_dB : (list_dB > list_dB > $o) > list_dB > $o).
thf(sy_c_Wellfounded_OwfP_001t__Lambda__OdB, type,
    wfP_dB : (dB > dB > $o) > $o).
thf(sy_c_member_001t__Lambda__OdB, type,
    member_dB : dB > set_dB > $o).
thf(sy_v_T, type,
    t : type).
thf(sy_v_e, type,
    e : nat > type).
thf(sy_v_t, type,
    t2 : dB).

% Relevant facts (99)
thf(fact_0_conversep__eq, axiom,
    (((conversep_dB_dB @ (^[Y : dB]: (^[Z : dB]: (Y = Z)))) = (^[Y : dB]: (^[Z : dB]: (Y = Z)))))). % conversep_eq
thf(fact_1_conversep__iff, axiom,
    ((conversep_dB_dB = (^[R : dB > dB > $o]: (^[A : dB]: (^[B : dB]: (R @ B @ A))))))). % conversep_iff
thf(fact_2_conversep__inject, axiom,
    ((![R2 : dB > dB > $o, S : dB > dB > $o]: (((conversep_dB_dB @ R2) = (conversep_dB_dB @ S)) = (R2 = S))))). % conversep_inject
thf(fact_3_conversep__conversep, axiom,
    ((![R2 : dB > dB > $o]: ((conversep_dB_dB @ (conversep_dB_dB @ R2)) = R2)))). % conversep_conversep
thf(fact_4_subject__reduction, axiom,
    ((![E : nat > type, T : dB, T2 : type, T3 : dB]: ((typing @ E @ T @ T2) => ((beta @ T @ T3) => (typing @ E @ T3 @ T2)))))). % subject_reduction
thf(fact_5_conversepD, axiom,
    ((![R2 : dB > dB > $o, B2 : dB, A2 : dB]: ((conversep_dB_dB @ R2 @ B2 @ A2) => (R2 @ A2 @ B2))))). % conversepD
thf(fact_6_conversep_Ocases, axiom,
    ((![R2 : dB > dB > $o, A1 : dB, A22 : dB]: ((conversep_dB_dB @ R2 @ A1 @ A22) => (R2 @ A22 @ A1))))). % conversep.cases
thf(fact_7_conversep_Osimps, axiom,
    ((conversep_dB_dB = (^[R : dB > dB > $o]: (^[A12 : dB]: (^[A23 : dB]: (?[A : dB]: (?[B : dB]: (((A12 = B)) & ((((A23 = A)) & ((R @ A @ B))))))))))))). % conversep.simps
thf(fact_8_conversep_Ointros, axiom,
    ((![R2 : dB > dB > $o, A2 : dB, B2 : dB]: ((R2 @ A2 @ B2) => (conversep_dB_dB @ R2 @ B2 @ A2))))). % conversep.intros
thf(fact_9_conversep_Oinducts, axiom,
    ((![R2 : dB > dB > $o, X1 : dB, X2 : dB, P : dB > dB > $o]: ((conversep_dB_dB @ R2 @ X1 @ X2) => ((![A3 : dB, B3 : dB]: ((R2 @ A3 @ B3) => (P @ B3 @ A3))) => (P @ X1 @ X2)))))). % conversep.inducts
thf(fact_10_accp_Ocases, axiom,
    ((![R2 : dB > dB > $o, A2 : dB]: ((accp_dB @ R2 @ A2) => (![Y2 : dB]: ((R2 @ Y2 @ A2) => (accp_dB @ R2 @ Y2))))))). % accp.cases
thf(fact_11_accp_Osimps, axiom,
    ((accp_dB = (^[R : dB > dB > $o]: (^[A : dB]: (?[X : dB]: (((A = X)) & ((![Y3 : dB]: (((R @ Y3 @ X)) => ((accp_dB @ R @ Y3)))))))))))). % accp.simps
thf(fact_12_accp__induct__rule, axiom,
    ((![R2 : dB > dB > $o, A2 : dB, P : dB > $o]: ((accp_dB @ R2 @ A2) => ((![X3 : dB]: ((accp_dB @ R2 @ X3) => ((![Y2 : dB]: ((R2 @ Y2 @ X3) => (P @ Y2))) => (P @ X3)))) => (P @ A2)))))). % accp_induct_rule
thf(fact_13_not__accp__down, axiom,
    ((![R3 : dB > dB > $o, X4 : dB]: ((~ ((accp_dB @ R3 @ X4))) => (~ ((![Z2 : dB]: ((R3 @ Z2 @ X4) => (accp_dB @ R3 @ Z2))))))))). % not_accp_down
thf(fact_14_accp__downward, axiom,
    ((![R2 : dB > dB > $o, B2 : dB, A2 : dB]: ((accp_dB @ R2 @ B2) => ((R2 @ A2 @ B2) => (accp_dB @ R2 @ A2)))))). % accp_downward
thf(fact_15_accp_Oinducts, axiom,
    ((![R2 : dB > dB > $o, X4 : dB, P : dB > $o]: ((accp_dB @ R2 @ X4) => ((![X3 : dB]: ((![Y2 : dB]: ((R2 @ Y2 @ X3) => (accp_dB @ R2 @ Y2))) => ((![Y2 : dB]: ((R2 @ Y2 @ X3) => (P @ Y2))) => (P @ X3)))) => (P @ X4)))))). % accp.inducts
thf(fact_16_accp__induct, axiom,
    ((![R2 : dB > dB > $o, A2 : dB, P : dB > $o]: ((accp_dB @ R2 @ A2) => ((![X3 : dB]: ((accp_dB @ R2 @ X3) => ((![Y2 : dB]: ((R2 @ Y2 @ X3) => (P @ Y2))) => (P @ X3)))) => (P @ A2)))))). % accp_induct
thf(fact_17_accp_Ointros, axiom,
    ((![R2 : dB > dB > $o, X4 : dB]: ((![Y4 : dB]: ((R2 @ Y4 @ X4) => (accp_dB @ R2 @ Y4))) => (accp_dB @ R2 @ X4))))). % accp.intros
thf(fact_18_termi__implies__IT, axiom,
    ((![R2 : dB]: ((accp_dB @ (conversep_dB_dB @ beta) @ R2) => (it @ R2))))). % termi_implies_IT
thf(fact_19_IT__implies__termi, axiom,
    ((![T : dB]: ((it @ T) => (accp_dB @ (conversep_dB_dB @ beta) @ T))))). % IT_implies_termi
thf(fact_20_subject__reduction_H, axiom,
    ((![T : dB, T3 : dB, E : nat > type, T2 : type]: ((transi72828568clp_dB @ beta @ T @ T3) => ((typing @ E @ T @ T2) => (typing @ E @ T3 @ T2)))))). % subject_reduction'
thf(fact_21_wfP__accp__iff, axiom,
    ((wfP_dB = (^[R : dB > dB > $o]: (![X5 : dB]: (accp_dB @ R @ X5)))))). % wfP_accp_iff
thf(fact_22_accp__wfPI, axiom,
    ((![R2 : dB > dB > $o]: ((![X_1 : dB]: (accp_dB @ R2 @ X_1)) => (wfP_dB @ R2))))). % accp_wfPI
thf(fact_23_accp__wfPD, axiom,
    ((![R2 : dB > dB > $o, X4 : dB]: ((wfP_dB @ R2) => (accp_dB @ R2 @ X4))))). % accp_wfPD
thf(fact_24_type__implies__IT, axiom,
    ((![E : nat > type, T : dB, T2 : type]: ((typing @ E @ T @ T2) => (it @ T))))). % type_implies_IT
thf(fact_25_conversep__mono, axiom,
    ((![R2 : dB > dB > $o, S : dB > dB > $o]: ((ord_less_eq_dB_dB_o @ (conversep_dB_dB @ R2) @ (conversep_dB_dB @ S)) = (ord_less_eq_dB_dB_o @ R2 @ S))))). % conversep_mono
thf(fact_26_typing_OVar, axiom,
    ((![Env : nat > type, X4 : nat, T2 : type]: (((Env @ X4) = T2) => (typing @ Env @ (var @ X4) @ T2))))). % typing.Var
thf(fact_27_Var__IT, axiom,
    ((![N : nat]: (it @ (var @ N))))). % Var_IT
thf(fact_28_accp__downwards, axiom,
    ((![R2 : dB > dB > $o, A2 : dB, B2 : dB]: ((accp_dB @ R2 @ A2) => ((transi72828568clp_dB @ R2 @ B2 @ A2) => (accp_dB @ R2 @ B2)))))). % accp_downwards
thf(fact_29_accp__downwards__aux, axiom,
    ((![R2 : dB > dB > $o, B2 : dB, A2 : dB]: ((transi72828568clp_dB @ R2 @ B2 @ A2) => ((accp_dB @ R2 @ A2) => (accp_dB @ R2 @ B2)))))). % accp_downwards_aux
thf(fact_30_conversep__le__swap, axiom,
    ((![R2 : dB > dB > $o, S : dB > dB > $o]: ((ord_less_eq_dB_dB_o @ R2 @ (conversep_dB_dB @ S)) = (ord_less_eq_dB_dB_o @ (conversep_dB_dB @ R2) @ S))))). % conversep_le_swap
thf(fact_31_typing__elims_I1_J, axiom,
    ((![E : nat > type, I : nat, T2 : type]: ((typing @ E @ (var @ I) @ T2) => ((E @ I) = T2))))). % typing_elims(1)
thf(fact_32_dB_Oinject_I1_J, axiom,
    ((![X1 : nat, Y1 : nat]: (((var @ X1) = (var @ Y1)) = (X1 = Y1))))). % dB.inject(1)
thf(fact_33_r__into__rtranclp, axiom,
    ((![R2 : dB > dB > $o, X4 : dB, Y5 : dB]: ((R2 @ X4 @ Y5) => (transi72828568clp_dB @ R2 @ X4 @ Y5))))). % r_into_rtranclp
thf(fact_34_rtranclp__idemp, axiom,
    ((![R2 : dB > dB > $o]: ((transi72828568clp_dB @ (transi72828568clp_dB @ R2)) = (transi72828568clp_dB @ R2))))). % rtranclp_idemp
thf(fact_35_beta__cases_I1_J, axiom,
    ((![I : nat, T : dB]: (~ ((beta @ (var @ I) @ T)))))). % beta_cases(1)
thf(fact_36_leq__conversepI, axiom,
    ((![R3 : dB > dB > $o]: ((R3 = (^[Y : dB]: (^[Z : dB]: (Y = Z)))) => (ord_less_eq_dB_dB_o @ R3 @ (conversep_dB_dB @ R3)))))). % leq_conversepI
thf(fact_37_rtranclp__subset, axiom,
    ((![R3 : dB > dB > $o, S2 : dB > dB > $o]: ((ord_less_eq_dB_dB_o @ R3 @ S2) => ((ord_less_eq_dB_dB_o @ S2 @ (transi72828568clp_dB @ R3)) => ((transi72828568clp_dB @ S2) = (transi72828568clp_dB @ R3))))))). % rtranclp_subset
thf(fact_38_accp__subset__induct, axiom,
    ((![D : dB > $o, R3 : dB > dB > $o, X4 : dB, P : dB > $o]: ((ord_less_eq_dB_o @ D @ (accp_dB @ R3)) => ((![X3 : dB, Z2 : dB]: ((D @ X3) => ((R3 @ Z2 @ X3) => (D @ Z2)))) => ((D @ X4) => ((![X3 : dB]: ((D @ X3) => ((![Z3 : dB]: ((R3 @ Z3 @ X3) => (P @ Z3))) => (P @ X3)))) => (P @ X4)))))))). % accp_subset_induct
thf(fact_39_accp__subset, axiom,
    ((![R1 : dB > dB > $o, R22 : dB > dB > $o]: ((ord_less_eq_dB_dB_o @ R1 @ R22) => (ord_less_eq_dB_o @ (accp_dB @ R22) @ (accp_dB @ R1)))))). % accp_subset
thf(fact_40_mono__rtranclp, axiom,
    ((![X4 : dB > dB > $o, Y5 : dB > dB > $o, A2 : dB, B2 : dB]: ((![A3 : dB, B3 : dB]: ((X4 @ A3 @ B3) => (Y5 @ A3 @ B3))) => ((transi72828568clp_dB @ X4 @ A2 @ B2) => (transi72828568clp_dB @ Y5 @ A2 @ B2)))))). % mono_rtranclp
thf(fact_41_rtranclp_Ocases, axiom,
    ((![R2 : dB > dB > $o, A1 : dB, A22 : dB]: ((transi72828568clp_dB @ R2 @ A1 @ A22) => ((~ ((A22 = A1))) => (~ ((![B3 : dB]: ((transi72828568clp_dB @ R2 @ A1 @ B3) => (~ ((R2 @ B3 @ A22)))))))))))). % rtranclp.cases
thf(fact_42_rtranclp_Osimps, axiom,
    ((transi72828568clp_dB = (^[R : dB > dB > $o]: (^[A12 : dB]: (^[A23 : dB]: (((?[A : dB]: (((A12 = A)) & ((A23 = A))))) | ((?[A : dB]: (?[B : dB]: (?[C : dB]: (((A12 = A)) & ((((A23 = C)) & ((((transi72828568clp_dB @ R @ A @ B)) & ((R @ B @ C)))))))))))))))))). % rtranclp.simps
thf(fact_43_rtranclp__trans, axiom,
    ((![R2 : dB > dB > $o, X4 : dB, Y5 : dB, Z4 : dB]: ((transi72828568clp_dB @ R2 @ X4 @ Y5) => ((transi72828568clp_dB @ R2 @ Y5 @ Z4) => (transi72828568clp_dB @ R2 @ X4 @ Z4)))))). % rtranclp_trans
thf(fact_44_rtranclp__induct, axiom,
    ((![R2 : dB > dB > $o, A2 : dB, B2 : dB, P : dB > $o]: ((transi72828568clp_dB @ R2 @ A2 @ B2) => ((P @ A2) => ((![Y4 : dB, Z2 : dB]: ((transi72828568clp_dB @ R2 @ A2 @ Y4) => ((R2 @ Y4 @ Z2) => ((P @ Y4) => (P @ Z2))))) => (P @ B2))))))). % rtranclp_induct
thf(fact_45_rtranclp_Oinducts, axiom,
    ((![R2 : dB > dB > $o, X1 : dB, X2 : dB, P : dB > dB > $o]: ((transi72828568clp_dB @ R2 @ X1 @ X2) => ((![A3 : dB]: (P @ A3 @ A3)) => ((![A3 : dB, B3 : dB, C2 : dB]: ((transi72828568clp_dB @ R2 @ A3 @ B3) => ((P @ A3 @ B3) => ((R2 @ B3 @ C2) => (P @ A3 @ C2))))) => (P @ X1 @ X2))))))). % rtranclp.inducts
thf(fact_46_converse__rtranclpE, axiom,
    ((![R2 : dB > dB > $o, X4 : dB, Z4 : dB]: ((transi72828568clp_dB @ R2 @ X4 @ Z4) => ((~ ((X4 = Z4))) => (~ ((![Y4 : dB]: ((R2 @ X4 @ Y4) => (~ ((transi72828568clp_dB @ R2 @ Y4 @ Z4)))))))))))). % converse_rtranclpE
thf(fact_47_rtranclp_Ortrancl__refl, axiom,
    ((![R2 : dB > dB > $o, A2 : dB]: (transi72828568clp_dB @ R2 @ A2 @ A2)))). % rtranclp.rtrancl_refl
thf(fact_48_converse__rtranclp__induct, axiom,
    ((![R2 : dB > dB > $o, A2 : dB, B2 : dB, P : dB > $o]: ((transi72828568clp_dB @ R2 @ A2 @ B2) => ((P @ B2) => ((![Y4 : dB, Z2 : dB]: ((R2 @ Y4 @ Z2) => ((transi72828568clp_dB @ R2 @ Z2 @ B2) => ((P @ Z2) => (P @ Y4))))) => (P @ A2))))))). % converse_rtranclp_induct
thf(fact_49_rtranclp_Ortrancl__into__rtrancl, axiom,
    ((![R2 : dB > dB > $o, A2 : dB, B2 : dB, C3 : dB]: ((transi72828568clp_dB @ R2 @ A2 @ B2) => ((R2 @ B2 @ C3) => (transi72828568clp_dB @ R2 @ A2 @ C3)))))). % rtranclp.rtrancl_into_rtrancl
thf(fact_50_converse__rtranclp__into__rtranclp, axiom,
    ((![R2 : dB > dB > $o, A2 : dB, B2 : dB, C3 : dB]: ((R2 @ A2 @ B2) => ((transi72828568clp_dB @ R2 @ B2 @ C3) => (transi72828568clp_dB @ R2 @ A2 @ C3)))))). % converse_rtranclp_into_rtranclp
thf(fact_51_rtranclp__conversep, axiom,
    ((![R2 : dB > dB > $o]: ((transi72828568clp_dB @ (conversep_dB_dB @ R2)) = (conversep_dB_dB @ (transi72828568clp_dB @ R2)))))). % rtranclp_conversep
thf(fact_52_rtranclp__converseI, axiom,
    ((![R2 : dB > dB > $o, Y5 : dB, X4 : dB]: ((transi72828568clp_dB @ R2 @ Y5 @ X4) => (transi72828568clp_dB @ (conversep_dB_dB @ R2) @ X4 @ Y5))))). % rtranclp_converseI
thf(fact_53_rtranclp__converseD, axiom,
    ((![R2 : dB > dB > $o, X4 : dB, Y5 : dB]: ((transi72828568clp_dB @ (conversep_dB_dB @ R2) @ X4 @ Y5) => (transi72828568clp_dB @ R2 @ Y5 @ X4))))). % rtranclp_converseD
thf(fact_54_rtranclp__mono, axiom,
    ((![R2 : dB > dB > $o, S : dB > dB > $o]: ((ord_less_eq_dB_dB_o @ R2 @ S) => (ord_less_eq_dB_dB_o @ (transi72828568clp_dB @ R2) @ (transi72828568clp_dB @ S)))))). % rtranclp_mono
thf(fact_55_rtrancl__beta__Abs, axiom,
    ((![S : dB, S3 : dB]: ((transi72828568clp_dB @ beta @ S @ S3) => (transi72828568clp_dB @ beta @ (abs @ S) @ (abs @ S3)))))). % rtrancl_beta_Abs
thf(fact_56_dB_Oinject_I3_J, axiom,
    ((![X32 : dB, Y32 : dB]: (((abs @ X32) = (abs @ Y32)) = (X32 = Y32))))). % dB.inject(3)
thf(fact_57_abs, axiom,
    ((![S : dB, T : dB]: ((beta @ S @ T) => (beta @ (abs @ S) @ (abs @ T)))))). % abs
thf(fact_58_beta__cases_I2_J, axiom,
    ((![R2 : dB, S : dB]: ((beta @ (abs @ R2) @ S) => (~ ((![T4 : dB]: ((S = (abs @ T4)) => (~ ((beta @ R2 @ T4))))))))))). % beta_cases(2)
thf(fact_59_dB_Odistinct_I3_J, axiom,
    ((![X1 : nat, X32 : dB]: (~ (((var @ X1) = (abs @ X32))))))). % dB.distinct(3)
thf(fact_60_Lambda, axiom,
    ((![R2 : dB]: ((it @ R2) => (it @ (abs @ R2)))))). % Lambda
thf(fact_61_lift__preserves__beta_H, axiom,
    ((![R2 : dB, S : dB, I : nat]: ((transi72828568clp_dB @ beta @ R2 @ S) => (transi72828568clp_dB @ beta @ (lift @ R2 @ I) @ (lift @ S @ I)))))). % lift_preserves_beta'
thf(fact_62_rtrancl__beta__App, axiom,
    ((![S : dB, S3 : dB, T : dB, T3 : dB]: ((transi72828568clp_dB @ beta @ S @ S3) => ((transi72828568clp_dB @ beta @ T @ T3) => (transi72828568clp_dB @ beta @ (app @ S @ T) @ (app @ S3 @ T3))))))). % rtrancl_beta_App
thf(fact_63_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_64_lift__IT, axiom,
    ((![T : dB, I : nat]: ((it @ T) => (it @ (lift @ T @ I)))))). % lift_IT
thf(fact_65_dB_Odistinct_I5_J, axiom,
    ((![X21 : dB, X22 : dB, X32 : dB]: (~ (((app @ X21 @ X22) = (abs @ X32))))))). % dB.distinct(5)
thf(fact_66_appR, axiom,
    ((![S : dB, T : dB, U : dB]: ((beta @ S @ T) => (beta @ (app @ U @ S) @ (app @ U @ T)))))). % appR
thf(fact_67_appL, axiom,
    ((![S : dB, T : dB, U : dB]: ((beta @ S @ T) => (beta @ (app @ S @ U) @ (app @ T @ U)))))). % appL
thf(fact_68_dB_Odistinct_I1_J, axiom,
    ((![X1 : nat, X21 : dB, X22 : dB]: (~ (((var @ X1) = (app @ X21 @ X22))))))). % dB.distinct(1)
thf(fact_69_lift__preserves__beta, axiom,
    ((![R2 : dB, S : dB, I : nat]: ((beta @ R2 @ S) => (beta @ (lift @ R2 @ I) @ (lift @ S @ I)))))). % lift_preserves_beta
thf(fact_70_lift_Osimps_I2_J, axiom,
    ((![S : dB, T : dB, K : nat]: ((lift @ (app @ S @ T) @ K) = (app @ (lift @ S @ K) @ (lift @ T @ K)))))). % lift.simps(2)
thf(fact_71_dB_Oinduct, axiom,
    ((![P : dB > $o, DB : dB]: ((![X3 : nat]: (P @ (var @ X3))) => ((![X1a : dB, X23 : dB]: ((P @ X1a) => ((P @ X23) => (P @ (app @ X1a @ X23))))) => ((![X3 : dB]: ((P @ X3) => (P @ (abs @ X3)))) => (P @ DB))))))). % dB.induct
thf(fact_72_dB_Oexhaust, axiom,
    ((![Y5 : dB]: ((![X12 : nat]: (~ ((Y5 = (var @ X12))))) => ((![X212 : dB, X222 : dB]: (~ ((Y5 = (app @ X212 @ X222))))) => (~ ((![X33 : dB]: (~ ((Y5 = (abs @ X33)))))))))))). % dB.exhaust
thf(fact_73_app__Var__IT, axiom,
    ((![T : dB, I : nat]: ((it @ T) => (it @ (app @ T @ (var @ I))))))). % app_Var_IT
thf(fact_74_rtrancl__beta__AppL, axiom,
    ((![S : dB, S3 : dB, T : dB]: ((transi72828568clp_dB @ beta @ S @ S3) => (transi72828568clp_dB @ beta @ (app @ S @ T) @ (app @ S3 @ T)))))). % rtrancl_beta_AppL
thf(fact_75_rtrancl__beta__AppR, axiom,
    ((![T : dB, T3 : dB, S : dB]: ((transi72828568clp_dB @ beta @ T @ T3) => (transi72828568clp_dB @ beta @ (app @ S @ T) @ (app @ S @ T3)))))). % rtrancl_beta_AppR
thf(fact_76_lift__type, axiom,
    ((![E : nat > type, T : dB, T2 : type, I : nat, U2 : type]: ((typing @ E @ T @ T2) => (typing @ (shift_type @ E @ I @ U2) @ (lift @ T @ I) @ T2))))). % lift_type
thf(fact_77_var__app__type__eq, axiom,
    ((![E : nat > type, I : nat, Ts : list_dB, T2 : type, U2 : type]: ((typing @ E @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ T2) => ((typing @ E @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ U2) => (T2 = U2)))))). % var_app_type_eq
thf(fact_78_shift__eq, axiom,
    ((![I : nat, J : nat, E : nat > type, T2 : type]: ((I = J) => ((shift_type @ E @ I @ T2 @ J) = T2))))). % shift_eq
thf(fact_79_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_80_Abs__apps__eq__Abs__apps__conv, axiom,
    ((![R2 : dB, Rs : list_dB, S : dB, Ss : list_dB]: (((foldl_dB_dB @ app @ (abs @ R2) @ Rs) = (foldl_dB_dB @ app @ (abs @ S) @ Ss)) = (((R2 = S)) & ((Rs = Ss))))))). % Abs_apps_eq_Abs_apps_conv
thf(fact_81_apps__eq__tail__conv, axiom,
    ((![R2 : dB, Ts : list_dB, S : dB]: (((foldl_dB_dB @ app @ R2 @ Ts) = (foldl_dB_dB @ app @ S @ Ts)) = (R2 = S))))). % apps_eq_tail_conv
thf(fact_82_Var__apps__neq__Abs__apps, axiom,
    ((![N : nat, Ts : list_dB, R2 : dB, Ss : list_dB]: (~ (((foldl_dB_dB @ app @ (var @ N) @ Ts) = (foldl_dB_dB @ app @ (abs @ R2) @ Ss))))))). % Var_apps_neq_Abs_apps
thf(fact_83_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_84_ex__head__tail, axiom,
    ((![T : dB]: (?[Ts2 : list_dB, H : dB]: ((T = (foldl_dB_dB @ app @ H @ Ts2)) & ((?[N2 : nat]: (H = (var @ N2))) | (?[U3 : dB]: (H = (abs @ U3))))))))). % ex_head_tail
thf(fact_85_apps__preserves__beta2, axiom,
    ((![R2 : dB, S : dB, Ss : list_dB]: ((transi72828568clp_dB @ beta @ R2 @ S) => (transi72828568clp_dB @ beta @ (foldl_dB_dB @ app @ R2 @ Ss) @ (foldl_dB_dB @ app @ S @ Ss)))))). % apps_preserves_beta2
thf(fact_86_apps__preserves__beta, axiom,
    ((![R2 : dB, S : dB, Ss : list_dB]: ((beta @ R2 @ S) => (beta @ (foldl_dB_dB @ app @ R2 @ Ss) @ (foldl_dB_dB @ app @ S @ Ss)))))). % apps_preserves_beta
thf(fact_87_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_88_IT_OVar, axiom,
    ((![Rs : list_dB, N : nat]: ((listsp_dB @ it @ Rs) => (it @ (foldl_dB_dB @ app @ (var @ N) @ Rs)))))). % IT.Var
thf(fact_89_apps__preserves__betas, axiom,
    ((![Rs : list_dB, Ss : list_dB, R2 : dB]: ((step1_dB @ beta @ Rs @ Ss) => (beta @ (foldl_dB_dB @ app @ R2 @ Rs) @ (foldl_dB_dB @ app @ R2 @ Ss)))))). % apps_preserves_betas
thf(fact_90_step1__converse, axiom,
    ((![R2 : dB > dB > $o]: ((step1_dB @ (conversep_dB_dB @ R2)) = (conver499110749ist_dB @ (step1_dB @ R2)))))). % step1_converse
thf(fact_91_in__step1__converse, axiom,
    ((![R2 : dB > dB > $o, X4 : list_dB, Y5 : list_dB]: ((step1_dB @ (conversep_dB_dB @ R2) @ X4 @ Y5) = (conver499110749ist_dB @ (step1_dB @ R2) @ X4 @ Y5))))). % in_step1_converse
thf(fact_92_ListOrder_Olists__accD, axiom,
    ((![R2 : dB > dB > $o, Xs : list_dB]: ((listsp_dB @ (accp_dB @ R2) @ Xs) => (accp_list_dB @ (step1_dB @ R2) @ Xs))))). % ListOrder.lists_accD
thf(fact_93_ListOrder_Olists__accI, axiom,
    ((![R2 : dB > dB > $o, Xs : list_dB]: ((accp_list_dB @ (step1_dB @ R2) @ Xs) => (listsp_dB @ (accp_dB @ R2) @ Xs))))). % ListOrder.lists_accI
thf(fact_94_listsp__mono, axiom,
    ((![A4 : dB > $o, B4 : dB > $o]: ((ord_less_eq_dB_o @ A4 @ B4) => (ord_le2037264764t_dB_o @ (listsp_dB @ A4) @ (listsp_dB @ B4)))))). % listsp_mono
thf(fact_95_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)))) => ((![U3 : dB]: ((P @ U3) => (![Ts2 : list_dB]: ((![X6 : dB]: ((member_dB @ X6 @ (set_dB2 @ Ts2)) => (P @ X6))) => (P @ (foldl_dB_dB @ app @ (abs @ U3) @ Ts2)))))) => (P @ T)))))). % Apps_dB_induct
thf(fact_96_subset__code_I1_J, axiom,
    ((![Xs : list_dB, B4 : set_dB]: ((ord_less_eq_set_dB @ (set_dB2 @ Xs) @ B4) = (![X : dB]: (((member_dB @ X @ (set_dB2 @ Xs))) => ((member_dB @ X @ B4)))))))). % subset_code(1)
thf(fact_97_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)))) => ((![U3 : dB]: ((P @ U3) => (![Ts2 : list_dB]: ((![X6 : dB]: ((member_dB @ X6 @ (set_dB2 @ Ts2)) => (P @ X6))) => (P @ (foldl_dB_dB @ app @ (abs @ U3) @ Ts2)))))) => (((size_size_dB @ T) = N) => (P @ T))))))). % lem
thf(fact_98_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

% Conjectures (2)
thf(conj_0, hypothesis,
    ((typing @ e @ t2 @ t))).
thf(conj_1, conjecture,
    ((accp_dB @ (conversep_dB_dB @ beta) @ t2))).
