% 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_19__5209578_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:36:52.853

% Could-be-implicit typings (5)
thf(ty_n_t__List__Olist_It__Lambda__OdB_J, type,
    list_dB : $tType).
thf(ty_n_t__List__Olist_It__Nat__Onat_J, type,
    list_nat : $tType).
thf(ty_n_t__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 (30)
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat, type,
    one_one_nat : nat).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat, type,
    plus_plus_nat : nat > nat > nat).
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_Otype_Osize__type, type,
    size_type : type > nat).
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_Olift, type,
    lift : dB > nat > dB).
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_Ofoldl_001t__Lambda__OdB_001t__Lambda__OdB, type,
    foldl_dB_dB : (dB > dB > dB) > dB > list_dB > dB).
thf(sy_c_List_Ofoldl_001t__Nat__Onat_001t__Nat__Onat, type,
    foldl_nat_nat : (nat > nat > nat) > nat > list_nat > nat).
thf(sy_c_List_Olist_OCons_001t__Lambda__OdB, type,
    cons_dB : dB > list_dB > list_dB).
thf(sy_c_List_Olist_OCons_001t__Nat__Onat, type,
    cons_nat : nat > list_nat > list_nat).
thf(sy_c_List_Olist_Omap_001t__Lambda__OdB_001t__Nat__Onat, type,
    map_dB_nat : (dB > nat) > list_dB > list_nat).
thf(sy_c_List_Olistsp_001t__Lambda__OdB, type,
    listsp_dB : (dB > $o) > list_dB > $o).
thf(sy_c_Nat_OSuc, type,
    suc : nat > nat).
thf(sy_c_Nat_Osize__class_Osize_001t__LambdaType__Otype, type,
    size_size_type : type > nat).
thf(sy_c_Nat_Osize__class_Osize_001t__Lambda__OdB, type,
    size_size_dB : dB > nat).
thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Lambda__OdB_J, type,
    size_size_list_dB : list_dB > nat).
thf(sy_v_i, type,
    i : nat).
thf(sy_v_t, type,
    t : dB).

% Relevant facts (142)
thf(fact_0_subst__lift, axiom,
    ((![T : dB, K : nat, S : dB]: ((subst @ (lift @ T @ K) @ S @ K) = T)))). % subst_lift
thf(fact_1_Lambda, axiom,
    ((![R : dB]: ((it @ R) => (it @ (abs @ R)))))). % Lambda
thf(fact_2_lift__preserves__beta, axiom,
    ((![R : dB, S : dB, I : nat]: ((beta @ R @ S) => (beta @ (lift @ R @ I) @ (lift @ S @ I)))))). % lift_preserves_beta
thf(fact_3_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_4_liftn__lift, axiom,
    ((![N : nat, T : dB, K : nat]: ((liftn @ (suc @ N) @ T @ K) = (lift @ (liftn @ N @ T @ K) @ K))))). % liftn_lift
thf(fact_5_lift__type, axiom,
    ((![E : nat > type, T : dB, T2 : type, I : nat, U : type]: ((typing @ E @ T @ T2) => (typing @ (shift_type @ E @ I @ U) @ (lift @ T @ I) @ T2))))). % lift_type
thf(fact_6_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_7_dB_Oinject_I3_J, axiom,
    ((![X3 : dB, Y3 : dB]: (((abs @ X3) = (abs @ Y3)) = (X3 = Y3))))). % dB.inject(3)
thf(fact_8_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_9_beta__cases_I2_J, axiom,
    ((![R : dB, S : dB]: ((beta @ (abs @ R) @ S) => (~ ((![T3 : dB]: ((S = (abs @ T3)) => (~ ((beta @ R @ T3))))))))))). % beta_cases(2)
thf(fact_10_dB_Odistinct_I5_J, axiom,
    ((![X21 : dB, X22 : dB, X3 : dB]: (~ (((app @ X21 @ X22) = (abs @ X3))))))). % dB.distinct(5)
thf(fact_11_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_12_abs, axiom,
    ((![S : dB, T : dB]: ((beta @ S @ T) => (beta @ (abs @ S) @ (abs @ T)))))). % abs
thf(fact_13_appL, axiom,
    ((![S : dB, T : dB, U2 : dB]: ((beta @ S @ T) => (beta @ (app @ S @ U2) @ (app @ T @ U2)))))). % appL
thf(fact_14_appR, axiom,
    ((![S : dB, T : dB, U2 : dB]: ((beta @ S @ T) => (beta @ (app @ U2 @ S) @ (app @ U2 @ T)))))). % appR
thf(fact_15_subst__App, axiom,
    ((![T : dB, U2 : dB, S : dB, K : nat]: ((subst @ (app @ T @ U2) @ S @ K) = (app @ (subst @ T @ S @ K) @ (subst @ U2 @ S @ K)))))). % subst_App
thf(fact_16_subst__lemma, axiom,
    ((![E : nat > type, T : dB, T2 : type, E2 : nat > type, U2 : dB, U : type, I : nat]: ((typing @ E @ T @ T2) => ((typing @ E2 @ U2 @ U) => ((E = (shift_type @ E2 @ I @ U)) => (typing @ E2 @ (subst @ T @ U2 @ I) @ T2))))))). % subst_lemma
thf(fact_17_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_18_subject__reduction, axiom,
    ((![E : nat > type, T : dB, T2 : type, T4 : dB]: ((typing @ E @ T @ T2) => ((beta @ T @ T4) => (typing @ E @ T4 @ T2)))))). % subject_reduction
thf(fact_19_nat_Oinject, axiom,
    ((![X2 : nat, Y2 : nat]: (((suc @ X2) = (suc @ Y2)) = (X2 = Y2))))). % nat.inject
thf(fact_20_old_Onat_Oinject, axiom,
    ((![Nat : nat, Nat2 : nat]: (((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))). % old.nat.inject
thf(fact_21_beta__cases_I3_J, axiom,
    ((![S : dB, T : dB, U2 : dB]: ((beta @ (app @ S @ T) @ U2) => ((![S2 : dB]: ((S = (abs @ S2)) => (~ ((U2 = (subst @ S2 @ T @ zero_zero_nat)))))) => ((![T3 : dB]: ((U2 = (app @ T3 @ T)) => (~ ((beta @ S @ T3))))) => (~ ((![T3 : dB]: ((U2 = (app @ S @ T3)) => (~ ((beta @ T @ T3))))))))))))). % beta_cases(3)
thf(fact_22_beta, axiom,
    ((![S : dB, T : dB]: (beta @ (app @ (abs @ S) @ T) @ (subst @ S @ T @ zero_zero_nat))))). % beta
thf(fact_23_beta_Ocases, axiom,
    ((![A1 : dB, A2 : dB]: ((beta @ A1 @ A2) => ((![S2 : dB, T3 : dB]: ((A1 = (app @ (abs @ S2) @ T3)) => (~ ((A2 = (subst @ S2 @ T3 @ zero_zero_nat)))))) => ((![S2 : dB, T3 : dB, U3 : dB]: ((A1 = (app @ S2 @ U3)) => ((A2 = (app @ T3 @ U3)) => (~ ((beta @ S2 @ T3)))))) => ((![S2 : dB, T3 : dB, U3 : dB]: ((A1 = (app @ U3 @ S2)) => ((A2 = (app @ U3 @ T3)) => (~ ((beta @ S2 @ T3)))))) => (~ ((![S2 : dB]: ((A1 = (abs @ S2)) => (![T3 : dB]: ((A2 = (abs @ T3)) => (~ ((beta @ S2 @ T3)))))))))))))))). % beta.cases
thf(fact_24_beta_Osimps, axiom,
    ((beta = (^[A12 : dB]: (^[A22 : dB]: (((?[S3 : dB]: (?[T5 : dB]: (((A12 = (app @ (abs @ S3) @ T5))) & ((A22 = (subst @ S3 @ T5 @ zero_zero_nat))))))) | ((((?[S3 : dB]: (?[T5 : dB]: (?[U4 : dB]: (((A12 = (app @ S3 @ U4))) & ((((A22 = (app @ T5 @ U4))) & ((beta @ S3 @ T5))))))))) | ((((?[S3 : dB]: (?[T5 : dB]: (?[U4 : dB]: (((A12 = (app @ U4 @ S3))) & ((((A22 = (app @ U4 @ T5))) & ((beta @ S3 @ T5))))))))) | ((?[S3 : dB]: (?[T5 : dB]: (((A12 = (abs @ S3))) & ((((A22 = (abs @ T5))) & ((beta @ S3 @ T5)))))))))))))))))). % beta.simps
thf(fact_25_beta_Oinducts, axiom,
    ((![X1 : dB, X2 : dB, P : dB > dB > $o]: ((beta @ X1 @ X2) => ((![S2 : dB, T3 : dB]: (P @ (app @ (abs @ S2) @ T3) @ (subst @ S2 @ T3 @ zero_zero_nat))) => ((![S2 : dB, T3 : dB, U3 : dB]: ((beta @ S2 @ T3) => ((P @ S2 @ T3) => (P @ (app @ S2 @ U3) @ (app @ T3 @ U3))))) => ((![S2 : dB, T3 : dB, U3 : dB]: ((beta @ S2 @ T3) => ((P @ S2 @ T3) => (P @ (app @ U3 @ S2) @ (app @ U3 @ T3))))) => ((![S2 : dB, T3 : dB]: ((beta @ S2 @ T3) => ((P @ S2 @ T3) => (P @ (abs @ S2) @ (abs @ T3))))) => (P @ X1 @ X2))))))))). % beta.inducts
thf(fact_26_abs__typeE, axiom,
    ((![E : nat > type, T : dB, T2 : type]: ((typing @ E @ (abs @ T) @ T2) => (~ ((![U5 : type, V : type]: (~ ((typing @ (shift_type @ E @ zero_zero_nat @ U5) @ T @ V)))))))))). % abs_typeE
thf(fact_27_Suc__inject, axiom,
    ((![X : nat, Y : nat]: (((suc @ X) = (suc @ Y)) => (X = Y))))). % Suc_inject
thf(fact_28_n__not__Suc__n, axiom,
    ((![N : nat]: (~ ((N = (suc @ N))))))). % n_not_Suc_n
thf(fact_29_liftn__0, axiom,
    ((![T : dB, K : nat]: ((liftn @ zero_zero_nat @ T @ K) = T)))). % liftn_0
thf(fact_30_shift__commute, axiom,
    ((![E : nat > type, I : nat, U : type, T2 : type]: ((shift_type @ (shift_type @ E @ I @ U) @ zero_zero_nat @ T2) = (shift_type @ (shift_type @ E @ zero_zero_nat @ T2) @ (suc @ I) @ U))))). % shift_commute
thf(fact_31_not0__implies__Suc, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (?[M : nat]: (N = (suc @ M))))))). % not0_implies_Suc
thf(fact_32_old_Onat_Oinducts, axiom,
    ((![P : nat > $o, Nat : nat]: ((P @ zero_zero_nat) => ((![Nat3 : nat]: ((P @ Nat3) => (P @ (suc @ Nat3)))) => (P @ Nat)))))). % old.nat.inducts
thf(fact_33_old_Onat_Oexhaust, axiom,
    ((![Y : nat]: ((~ ((Y = zero_zero_nat))) => (~ ((![Nat3 : nat]: (~ ((Y = (suc @ Nat3))))))))))). % old.nat.exhaust
thf(fact_34_Zero__not__Suc, axiom,
    ((![M2 : nat]: (~ ((zero_zero_nat = (suc @ M2))))))). % Zero_not_Suc
thf(fact_35_Zero__neq__Suc, axiom,
    ((![M2 : nat]: (~ ((zero_zero_nat = (suc @ M2))))))). % Zero_neq_Suc
thf(fact_36_Suc__neq__Zero, axiom,
    ((![M2 : nat]: (~ (((suc @ M2) = zero_zero_nat)))))). % Suc_neq_Zero
thf(fact_37_zero__induct, axiom,
    ((![P : nat > $o, K : nat]: ((P @ K) => ((![N2 : nat]: ((P @ (suc @ N2)) => (P @ N2))) => (P @ zero_zero_nat)))))). % zero_induct
thf(fact_38_diff__induct, axiom,
    ((![P : nat > nat > $o, M2 : nat, N : nat]: ((![X4 : nat]: (P @ X4 @ zero_zero_nat)) => ((![Y4 : nat]: (P @ zero_zero_nat @ (suc @ Y4))) => ((![X4 : nat, Y4 : nat]: ((P @ X4 @ Y4) => (P @ (suc @ X4) @ (suc @ Y4)))) => (P @ M2 @ N))))))). % diff_induct
thf(fact_39_nat__induct, axiom,
    ((![P : nat > $o, N : nat]: ((P @ zero_zero_nat) => ((![N2 : nat]: ((P @ N2) => (P @ (suc @ N2)))) => (P @ N)))))). % nat_induct
thf(fact_40_nat_OdiscI, axiom,
    ((![Nat : nat, X2 : nat]: ((Nat = (suc @ X2)) => (~ ((Nat = zero_zero_nat))))))). % nat.discI
thf(fact_41_old_Onat_Odistinct_I1_J, axiom,
    ((![Nat2 : nat]: (~ ((zero_zero_nat = (suc @ Nat2))))))). % old.nat.distinct(1)
thf(fact_42_old_Onat_Odistinct_I2_J, axiom,
    ((![Nat2 : nat]: (~ (((suc @ Nat2) = zero_zero_nat)))))). % old.nat.distinct(2)
thf(fact_43_nat_Odistinct_I1_J, axiom,
    ((![X2 : nat]: (~ ((zero_zero_nat = (suc @ X2))))))). % nat.distinct(1)
thf(fact_44_substn__subst__n, axiom,
    ((substn = (^[T5 : dB]: (^[S3 : dB]: (^[N3 : nat]: (subst @ T5 @ (liftn @ N3 @ S3 @ zero_zero_nat) @ N3))))))). % substn_subst_n
thf(fact_45_Abs, axiom,
    ((![Env : nat > type, T2 : type, T : dB, U : type]: ((typing @ (shift_type @ Env @ zero_zero_nat @ T2) @ T @ U) => (typing @ Env @ (abs @ T) @ (fun @ T2 @ U)))))). % Abs
thf(fact_46_typing__elims_I3_J, axiom,
    ((![E : nat > type, T : dB, T2 : type]: ((typing @ E @ (abs @ T) @ T2) => (~ ((![T6 : type, U5 : type]: ((T2 = (fun @ T6 @ U5)) => (~ ((typing @ (shift_type @ E @ zero_zero_nat @ T6) @ T @ U5))))))))))). % typing_elims(3)
thf(fact_47_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_48_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_49_zero__natural_Orsp, axiom,
    ((zero_zero_nat = zero_zero_nat))). % zero_natural.rsp
thf(fact_50_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_51_type__induct, axiom,
    ((![P : type > $o, T2 : type]: ((![T6 : type]: ((![T1 : type, T22 : type]: ((T6 = (fun @ T1 @ T22)) => (P @ T1))) => ((![T1 : type, T22 : type]: ((T6 = (fun @ T1 @ T22)) => (P @ T22))) => (P @ T6)))) => (P @ T2))))). % type_induct
thf(fact_52_substn_Osimps_I2_J, axiom,
    ((![T : dB, U2 : dB, S : dB, K : nat]: ((substn @ (app @ T @ U2) @ S @ K) = (app @ (substn @ T @ S @ K) @ (substn @ U2 @ S @ K)))))). % substn.simps(2)
thf(fact_53_typing__elims_I2_J, axiom,
    ((![E : nat > type, T : dB, U2 : dB, T2 : type]: ((typing @ E @ (app @ T @ U2) @ T2) => (~ ((![T6 : type]: ((typing @ E @ T @ (fun @ T6 @ T2)) => (~ ((typing @ E @ U2 @ T6))))))))))). % typing_elims(2)
thf(fact_54_App, axiom,
    ((![Env : nat > type, S : dB, T2 : type, U : type, T : dB]: ((typing @ Env @ S @ (fun @ T2 @ U)) => ((typing @ Env @ T @ T2) => (typing @ Env @ (app @ S @ T) @ U)))))). % App
thf(fact_55_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_56_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_57_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_58_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_59_typing_Ocases, axiom,
    ((![A1 : nat > type, A2 : dB, A3 : type]: ((typing @ A1 @ A2 @ A3) => ((![X4 : nat]: ((A2 = (var @ X4)) => (~ (((A1 @ X4) = A3))))) => ((![T6 : type, T3 : dB]: ((A2 = (abs @ T3)) => (![U5 : type]: ((A3 = (fun @ T6 @ U5)) => (~ ((typing @ (shift_type @ A1 @ zero_zero_nat @ T6) @ T3 @ U5))))))) => (~ ((![S2 : dB, T6 : type, U5 : type, T3 : dB]: ((A2 = (app @ S2 @ T3)) => ((A3 = U5) => ((typing @ A1 @ S2 @ (fun @ T6 @ U5)) => (~ ((typing @ A1 @ T3 @ T6))))))))))))))). % typing.cases
thf(fact_60_typing_Osimps, axiom,
    ((typing = (^[A12 : nat > type]: (^[A22 : dB]: (^[A32 : type]: (((?[Env2 : nat > type]: (?[X5 : nat]: (?[T7 : type]: (((A12 = Env2)) & ((((A22 = (var @ X5))) & ((((A32 = T7)) & (((Env2 @ X5) = T7))))))))))) | ((((?[Env2 : nat > type]: (?[T7 : type]: (?[T5 : dB]: (?[U6 : type]: (((A12 = Env2)) & ((((A22 = (abs @ T5))) & ((((A32 = (fun @ T7 @ U6))) & ((typing @ (shift_type @ Env2 @ zero_zero_nat @ T7) @ T5 @ U6)))))))))))) | ((?[Env2 : nat > type]: (?[S3 : dB]: (?[T7 : type]: (?[U6 : type]: (?[T5 : dB]: (((A12 = Env2)) & ((((A22 = (app @ S3 @ T5))) & ((((A32 = U6)) & ((((typing @ Env2 @ S3 @ (fun @ T7 @ U6))) & ((typing @ Env2 @ T5 @ T7)))))))))))))))))))))))). % typing.simps
thf(fact_61_dB_Oinject_I1_J, axiom,
    ((![X1 : nat, Y1 : nat]: (((var @ X1) = (var @ Y1)) = (X1 = Y1))))). % dB.inject(1)
thf(fact_62_subst__eq, axiom,
    ((![K : nat, U2 : dB]: ((subst @ (var @ K) @ U2 @ K) = U2)))). % subst_eq
thf(fact_63_Var__apps__eq__Var__apps__conv, axiom,
    ((![M2 : nat, Rs : list_dB, N : nat, Ss : list_dB]: (((foldl_dB_dB @ app @ (var @ M2) @ Rs) = (foldl_dB_dB @ app @ (var @ N) @ Ss)) = (((M2 = N)) & ((Rs = Ss))))))). % Var_apps_eq_Var_apps_conv
thf(fact_64_dB_Odistinct_I1_J, axiom,
    ((![X1 : nat, X21 : dB, X22 : dB]: (~ (((var @ X1) = (app @ X21 @ X22))))))). % dB.distinct(1)
thf(fact_65_beta__cases_I1_J, axiom,
    ((![I : nat, T : dB]: (~ ((beta @ (var @ I) @ T)))))). % beta_cases(1)
thf(fact_66_dB_Odistinct_I3_J, axiom,
    ((![X1 : nat, X3 : dB]: (~ (((var @ X1) = (abs @ X3))))))). % dB.distinct(3)
thf(fact_67_typing_OVar, axiom,
    ((![Env : nat > type, X : nat, T2 : type]: (((Env @ X) = T2) => (typing @ Env @ (var @ X) @ T2))))). % typing.Var
thf(fact_68_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_69_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_70_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_71_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_72_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_73_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_74_var__app__type__eq, axiom,
    ((![E : nat > type, I : nat, Ts : list_dB, T2 : type, U : type]: ((typing @ E @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ T2) => ((typing @ E @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ U) => (T2 = U)))))). % var_app_type_eq
thf(fact_75_typing_Oinducts, axiom,
    ((![X1 : nat > type, X2 : dB, X3 : type, P : (nat > type) > dB > type > $o]: ((typing @ X1 @ X2 @ X3) => ((![Env3 : nat > type, X4 : nat, T6 : type]: (((Env3 @ X4) = T6) => (P @ Env3 @ (var @ X4) @ T6))) => ((![Env3 : nat > type, T6 : type, T3 : dB, U5 : type]: ((typing @ (shift_type @ Env3 @ zero_zero_nat @ T6) @ T3 @ U5) => ((P @ (shift_type @ Env3 @ zero_zero_nat @ T6) @ T3 @ U5) => (P @ Env3 @ (abs @ T3) @ (fun @ T6 @ U5))))) => ((![Env3 : nat > type, S2 : dB, T6 : type, U5 : type, T3 : dB]: ((typing @ Env3 @ S2 @ (fun @ T6 @ U5)) => ((P @ Env3 @ S2 @ (fun @ T6 @ U5)) => ((typing @ Env3 @ T3 @ T6) => ((P @ Env3 @ T3 @ T6) => (P @ Env3 @ (app @ S2 @ T3) @ U5)))))) => (P @ X1 @ X2 @ X3)))))))). % typing.inducts
thf(fact_76_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]: (?[S3 : dB]: (?[Ss2 : list_dB]: (((A = (foldl_dB_dB @ app @ (app @ (abs @ R2) @ S3) @ Ss2))) & ((((it @ (foldl_dB_dB @ app @ (subst @ R2 @ S3 @ zero_zero_nat) @ Ss2))) & ((it @ S3)))))))))))))))). % IT.simps
thf(fact_77_IT_Ocases, axiom,
    ((![A4 : dB]: ((it @ A4) => ((![Rs3 : list_dB]: ((?[N2 : nat]: (A4 = (foldl_dB_dB @ app @ (var @ N2) @ Rs3))) => (~ ((listsp_dB @ it @ Rs3))))) => ((![R3 : dB]: ((A4 = (abs @ R3)) => (~ ((it @ R3))))) => (~ ((![R3 : dB, S2 : dB, Ss3 : list_dB]: ((A4 = (foldl_dB_dB @ app @ (app @ (abs @ R3) @ S2) @ Ss3)) => ((it @ (foldl_dB_dB @ app @ (subst @ R3 @ S2 @ zero_zero_nat) @ Ss3)) => (~ ((it @ S2)))))))))))))). % IT.cases
thf(fact_78_head__Var__reduction, axiom,
    ((![N : nat, Rs : list_dB, V2 : dB]: ((beta @ (foldl_dB_dB @ app @ (var @ N) @ Rs) @ V2) => (?[Ss3 : list_dB]: ((step1_dB @ beta @ Rs @ Ss3) & (V2 = (foldl_dB_dB @ app @ (var @ N) @ Ss3)))))))). % head_Var_reduction
thf(fact_79_dB_Osize__gen_I1_J, axiom,
    ((![X1 : nat]: ((size_dB @ (var @ X1)) = zero_zero_nat)))). % dB.size_gen(1)
thf(fact_80_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_81_IT_OVar, axiom,
    ((![Rs : list_dB, N : nat]: ((listsp_dB @ it @ Rs) => (it @ (foldl_dB_dB @ app @ (var @ N) @ Rs)))))). % IT.Var
thf(fact_82_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)))))) => (~ ((![T3 : dB]: ((R = (abs @ T3)) => (![U3 : dB, Us : list_dB]: ((Rs = (cons_dB @ U3 @ Us)) => (~ ((S = (foldl_dB_dB @ app @ (subst @ T3 @ U3 @ zero_zero_nat) @ Us)))))))))))))))). % apps_betasE
thf(fact_83_dB_Osize__gen_I3_J, axiom,
    ((![X3 : dB]: ((size_dB @ (abs @ X3)) = (plus_plus_nat @ (size_dB @ X3) @ (suc @ zero_zero_nat)))))). % dB.size_gen(3)
thf(fact_84_dB_Osize__gen_I2_J, axiom,
    ((![X21 : dB, X22 : dB]: ((size_dB @ (app @ X21 @ X22)) = (plus_plus_nat @ (plus_plus_nat @ (size_dB @ X21) @ (size_dB @ X22)) @ (suc @ zero_zero_nat)))))). % dB.size_gen(2)
thf(fact_85_add__left__cancel, axiom,
    ((![A4 : nat, B : nat, C : nat]: (((plus_plus_nat @ A4 @ B) = (plus_plus_nat @ A4 @ C)) = (B = C))))). % add_left_cancel
thf(fact_86_add__right__cancel, axiom,
    ((![B : nat, A4 : nat, C : nat]: (((plus_plus_nat @ B @ A4) = (plus_plus_nat @ C @ A4)) = (B = C))))). % add_right_cancel
thf(fact_87_zero__eq__add__iff__both__eq__0, axiom,
    ((![X : nat, Y : nat]: ((zero_zero_nat = (plus_plus_nat @ X @ Y)) = (((X = zero_zero_nat)) & ((Y = zero_zero_nat))))))). % zero_eq_add_iff_both_eq_0
thf(fact_88_add__eq__0__iff__both__eq__0, axiom,
    ((![X : nat, Y : nat]: (((plus_plus_nat @ X @ Y) = zero_zero_nat) = (((X = zero_zero_nat)) & ((Y = zero_zero_nat))))))). % add_eq_0_iff_both_eq_0
thf(fact_89_add__cancel__right__right, axiom,
    ((![A4 : nat, B : nat]: ((A4 = (plus_plus_nat @ A4 @ B)) = (B = zero_zero_nat))))). % add_cancel_right_right
thf(fact_90_add__cancel__right__left, axiom,
    ((![A4 : nat, B : nat]: ((A4 = (plus_plus_nat @ B @ A4)) = (B = zero_zero_nat))))). % add_cancel_right_left
thf(fact_91_add__cancel__left__right, axiom,
    ((![A4 : nat, B : nat]: (((plus_plus_nat @ A4 @ B) = A4) = (B = zero_zero_nat))))). % add_cancel_left_right
thf(fact_92_add__cancel__left__left, axiom,
    ((![B : nat, A4 : nat]: (((plus_plus_nat @ B @ A4) = A4) = (B = zero_zero_nat))))). % add_cancel_left_left
thf(fact_93_add_Oright__neutral, axiom,
    ((![A4 : nat]: ((plus_plus_nat @ A4 @ zero_zero_nat) = A4)))). % add.right_neutral
thf(fact_94_add_Oleft__neutral, axiom,
    ((![A4 : nat]: ((plus_plus_nat @ zero_zero_nat @ A4) = A4)))). % add.left_neutral
thf(fact_95_add__is__0, axiom,
    ((![M2 : nat, N : nat]: (((plus_plus_nat @ M2 @ N) = zero_zero_nat) = (((M2 = zero_zero_nat)) & ((N = zero_zero_nat))))))). % add_is_0
thf(fact_96_Nat_Oadd__0__right, axiom,
    ((![M2 : nat]: ((plus_plus_nat @ M2 @ zero_zero_nat) = M2)))). % Nat.add_0_right
thf(fact_97_add__Suc__right, axiom,
    ((![M2 : nat, N : nat]: ((plus_plus_nat @ M2 @ (suc @ N)) = (suc @ (plus_plus_nat @ M2 @ N)))))). % add_Suc_right
thf(fact_98_ab__semigroup__add__class_Oadd__ac_I1_J, axiom,
    ((![A4 : nat, B : nat, C : nat]: ((plus_plus_nat @ (plus_plus_nat @ A4 @ B) @ C) = (plus_plus_nat @ A4 @ (plus_plus_nat @ B @ C)))))). % ab_semigroup_add_class.add_ac(1)
thf(fact_99_add__mono__thms__linordered__semiring_I4_J, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: (((I = J) & (K = L)) => ((plus_plus_nat @ I @ K) = (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_semiring(4)
thf(fact_100_group__cancel_Oadd1, axiom,
    ((![A5 : nat, K : nat, A4 : nat, B : nat]: ((A5 = (plus_plus_nat @ K @ A4)) => ((plus_plus_nat @ A5 @ B) = (plus_plus_nat @ K @ (plus_plus_nat @ A4 @ B))))))). % group_cancel.add1
thf(fact_101_group__cancel_Oadd2, axiom,
    ((![B2 : nat, K : nat, B : nat, A4 : nat]: ((B2 = (plus_plus_nat @ K @ B)) => ((plus_plus_nat @ A4 @ B2) = (plus_plus_nat @ K @ (plus_plus_nat @ A4 @ B))))))). % group_cancel.add2
thf(fact_102_add_Oassoc, axiom,
    ((![A4 : nat, B : nat, C : nat]: ((plus_plus_nat @ (plus_plus_nat @ A4 @ B) @ C) = (plus_plus_nat @ A4 @ (plus_plus_nat @ B @ C)))))). % add.assoc
thf(fact_103_add_Ocommute, axiom,
    ((plus_plus_nat = (^[A : nat]: (^[B3 : nat]: (plus_plus_nat @ B3 @ A)))))). % add.commute
thf(fact_104_add_Oleft__commute, axiom,
    ((![B : nat, A4 : nat, C : nat]: ((plus_plus_nat @ B @ (plus_plus_nat @ A4 @ C)) = (plus_plus_nat @ A4 @ (plus_plus_nat @ B @ C)))))). % add.left_commute
thf(fact_105_add__left__imp__eq, axiom,
    ((![A4 : nat, B : nat, C : nat]: (((plus_plus_nat @ A4 @ B) = (plus_plus_nat @ A4 @ C)) => (B = C))))). % add_left_imp_eq
thf(fact_106_add__right__imp__eq, axiom,
    ((![B : nat, A4 : nat, C : nat]: (((plus_plus_nat @ B @ A4) = (plus_plus_nat @ C @ A4)) => (B = C))))). % add_right_imp_eq
thf(fact_107_add__Suc__shift, axiom,
    ((![M2 : nat, N : nat]: ((plus_plus_nat @ (suc @ M2) @ N) = (plus_plus_nat @ M2 @ (suc @ N)))))). % add_Suc_shift
thf(fact_108_nat__arith_Osuc1, axiom,
    ((![A5 : nat, K : nat, A4 : nat]: ((A5 = (plus_plus_nat @ K @ A4)) => ((suc @ A5) = (plus_plus_nat @ K @ (suc @ A4))))))). % nat_arith.suc1
thf(fact_109_add__Suc, axiom,
    ((![M2 : nat, N : nat]: ((plus_plus_nat @ (suc @ M2) @ N) = (suc @ (plus_plus_nat @ M2 @ N)))))). % add_Suc
thf(fact_110_plus__nat_Oadd__0, axiom,
    ((![N : nat]: ((plus_plus_nat @ zero_zero_nat @ N) = N)))). % plus_nat.add_0
thf(fact_111_add__eq__self__zero, axiom,
    ((![M2 : nat, N : nat]: (((plus_plus_nat @ M2 @ N) = M2) => (N = zero_zero_nat))))). % add_eq_self_zero
thf(fact_112_add_Ocomm__neutral, axiom,
    ((![A4 : nat]: ((plus_plus_nat @ A4 @ zero_zero_nat) = A4)))). % add.comm_neutral
thf(fact_113_comm__monoid__add__class_Oadd__0, axiom,
    ((![A4 : nat]: ((plus_plus_nat @ zero_zero_nat @ A4) = A4)))). % comm_monoid_add_class.add_0
thf(fact_114_add__is__1, axiom,
    ((![M2 : nat, N : nat]: (((plus_plus_nat @ M2 @ N) = (suc @ zero_zero_nat)) = (((((M2 = (suc @ zero_zero_nat))) & ((N = zero_zero_nat)))) | ((((M2 = zero_zero_nat)) & ((N = (suc @ zero_zero_nat)))))))))). % add_is_1
thf(fact_115_one__is__add, axiom,
    ((![M2 : nat, N : nat]: (((suc @ zero_zero_nat) = (plus_plus_nat @ M2 @ N)) = (((((M2 = (suc @ zero_zero_nat))) & ((N = zero_zero_nat)))) | ((((M2 = zero_zero_nat)) & ((N = (suc @ zero_zero_nat)))))))))). % one_is_add
thf(fact_116_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_117_type_Osize__gen_I2_J, axiom,
    ((![X21 : type, X22 : type]: ((size_type @ (fun @ X21 @ X22)) = (plus_plus_nat @ (plus_plus_nat @ (size_type @ X21) @ (size_type @ X22)) @ (suc @ zero_zero_nat)))))). % type.size_gen(2)
thf(fact_118_type_Osize_I4_J, axiom,
    ((![X21 : type, X22 : type]: ((size_size_type @ (fun @ X21 @ X22)) = (plus_plus_nat @ (plus_plus_nat @ (size_size_type @ X21) @ (size_size_type @ X22)) @ (suc @ zero_zero_nat)))))). % type.size(4)
thf(fact_119_size__neq__size__imp__neq, axiom,
    ((![X : type, Y : type]: ((~ (((size_size_type @ X) = (size_size_type @ Y)))) => (~ ((X = Y))))))). % size_neq_size_imp_neq
thf(fact_120_size__neq__size__imp__neq, axiom,
    ((![X : dB, Y : dB]: ((~ (((size_size_dB @ X) = (size_size_dB @ Y)))) => (~ ((X = Y))))))). % size_neq_size_imp_neq
thf(fact_121_size__neq__size__imp__neq, axiom,
    ((![X : list_dB, Y : list_dB]: ((~ (((size_size_list_dB @ X) = (size_size_list_dB @ Y)))) => (~ ((X = Y))))))). % size_neq_size_imp_neq
thf(fact_122_Cons__step1E, axiom,
    ((![R : dB > dB > $o, Ys : list_dB, X : dB, Xs : list_dB]: ((step1_dB @ R @ Ys @ (cons_dB @ X @ Xs)) => ((![Y4 : dB]: ((Ys = (cons_dB @ Y4 @ Xs)) => (~ ((R @ Y4 @ X))))) => (~ ((![Zs : list_dB]: ((Ys = (cons_dB @ X @ Zs)) => (~ ((step1_dB @ R @ Zs @ Xs)))))))))))). % Cons_step1E
thf(fact_123_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_124_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_125_Suc__length__conv, axiom,
    ((![N : nat, Xs : list_dB]: (((suc @ N) = (size_size_list_dB @ Xs)) = (?[Y5 : dB]: (?[Ys2 : list_dB]: (((Xs = (cons_dB @ Y5 @ Ys2))) & (((size_size_list_dB @ Ys2) = N))))))))). % Suc_length_conv
thf(fact_126_length__Suc__conv, axiom,
    ((![Xs : list_dB, N : nat]: (((size_size_list_dB @ Xs) = (suc @ N)) = (?[Y5 : dB]: (?[Ys2 : list_dB]: (((Xs = (cons_dB @ Y5 @ Ys2))) & (((size_size_list_dB @ Ys2) = N))))))))). % length_Suc_conv
thf(fact_127_list_Osize_I4_J, axiom,
    ((![X21 : dB, X22 : list_dB]: ((size_size_list_dB @ (cons_dB @ X21 @ X22)) = (plus_plus_nat @ (size_size_list_dB @ X22) @ (suc @ zero_zero_nat)))))). % list.size(4)
thf(fact_128_dB_Osize_I4_J, axiom,
    ((![X1 : nat]: ((size_size_dB @ (var @ X1)) = zero_zero_nat)))). % dB.size(4)
thf(fact_129_dB_Osize_I5_J, axiom,
    ((![X21 : dB, X22 : dB]: ((size_size_dB @ (app @ X21 @ X22)) = (plus_plus_nat @ (plus_plus_nat @ (size_size_dB @ X21) @ (size_size_dB @ X22)) @ (suc @ zero_zero_nat)))))). % dB.size(5)
thf(fact_130_dB_Osize_I6_J, axiom,
    ((![X3 : dB]: ((size_size_dB @ (abs @ X3)) = (plus_plus_nat @ (size_size_dB @ X3) @ (suc @ zero_zero_nat)))))). % dB.size(6)
thf(fact_131_foldl__Cons, axiom,
    ((![F : nat > nat > nat, A4 : nat, X : nat, Xs : list_nat]: ((foldl_nat_nat @ F @ A4 @ (cons_nat @ X @ Xs)) = (foldl_nat_nat @ F @ (F @ A4 @ X) @ Xs))))). % foldl_Cons
thf(fact_132_foldl__Cons, axiom,
    ((![F : dB > dB > dB, A4 : dB, X : dB, Xs : list_dB]: ((foldl_dB_dB @ F @ A4 @ (cons_dB @ X @ Xs)) = (foldl_dB_dB @ F @ (F @ A4 @ X) @ Xs))))). % foldl_Cons
thf(fact_133_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_134_Euclid__induct, axiom,
    ((![P : nat > nat > $o, A4 : nat, B : nat]: ((![A6 : nat, B4 : nat]: ((P @ A6 @ B4) = (P @ B4 @ A6))) => ((![A6 : nat]: (P @ A6 @ zero_zero_nat)) => ((![A6 : nat, B4 : nat]: ((P @ A6 @ B4) => (P @ A6 @ (plus_plus_nat @ A6 @ B4)))) => (P @ A4 @ B))))))). % Euclid_induct
thf(fact_135_length__Cons, axiom,
    ((![X : dB, Xs : list_dB]: ((size_size_list_dB @ (cons_dB @ X @ Xs)) = (suc @ (size_size_list_dB @ Xs)))))). % length_Cons
thf(fact_136_verit__sum__simplify, axiom,
    ((![A4 : nat]: ((plus_plus_nat @ A4 @ zero_zero_nat) = A4)))). % verit_sum_simplify
thf(fact_137_add__0__iff, axiom,
    ((![B : nat, A4 : nat]: ((B = (plus_plus_nat @ B @ A4)) = (A4 = zero_zero_nat))))). % add_0_iff
thf(fact_138_size__apps, axiom,
    ((![R : dB, Rs : list_dB]: ((size_size_dB @ (foldl_dB_dB @ app @ R @ Rs)) = (plus_plus_nat @ (plus_plus_nat @ (size_size_dB @ R) @ (foldl_nat_nat @ plus_plus_nat @ zero_zero_nat @ (map_dB_nat @ size_size_dB @ Rs))) @ (size_size_list_dB @ Rs)))))). % size_apps
thf(fact_139_subst__Abs, axiom,
    ((![T : dB, S : dB, K : nat]: ((subst @ (abs @ T) @ S @ K) = (abs @ (subst @ T @ (lift @ S @ zero_zero_nat) @ (plus_plus_nat @ K @ one_one_nat))))))). % subst_Abs
thf(fact_140_one__natural_Orsp, axiom,
    ((one_one_nat = one_one_nat))). % one_natural.rsp
thf(fact_141_one__reorient, axiom,
    ((![X : nat]: ((one_one_nat = X) = (X = one_one_nat))))). % one_reorient

% Conjectures (2)
thf(conj_0, hypothesis,
    ((it @ t))).
thf(conj_1, conjecture,
    ((it @ (lift @ t @ i)))).
