% 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_266__5212272_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:38:46.209

% 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_OAtom, type,
    atom : 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_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_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_c_List_Oproduct__lists_001t__Lambda__OdB, type,
    product_lists_dB : list_list_dB > list_list_dB).
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_T, type,
    t : type).
thf(sy_v_Ta____, type,
    ta : type).
thf(sy_v_U____, type,
    u : type).
thf(sy_v_e, type,
    e : nat > type).
thf(sy_v_ea____, type,
    ea : nat > type).
thf(sy_v_s____, type,
    s : dB).
thf(sy_v_t, type,
    t2 : dB).
thf(sy_v_ta____, type,
    ta2 : dB).

% Relevant facts (131)
thf(fact_0_assms, axiom,
    ((typing @ e @ t2 @ t))). % assms
thf(fact_1_App_Ohyps_I1_J, axiom,
    ((typing @ ea @ s @ (fun @ ta @ u)))). % App.hyps(1)
thf(fact_2_App_Ohyps_I3_J, axiom,
    ((typing @ ea @ ta2 @ ta))). % App.hyps(3)
thf(fact_3_shift__eq, axiom,
    ((![I : nat, J : nat, E : nat > type, T : type]: ((I = J) => ((shift_type @ E @ I @ T @ J) = T))))). % shift_eq
thf(fact_4_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_5_dB_Oinject_I1_J, axiom,
    ((![X1 : nat, Y1 : nat]: (((var @ X1) = (var @ Y1)) = (X1 = Y1))))). % dB.inject(1)
thf(fact_6_typing__elims_I1_J, axiom,
    ((![E : nat > type, I : nat, T : type]: ((typing @ E @ (var @ I) @ T) => ((E @ I) = T))))). % typing_elims(1)
thf(fact_7_typing_OVar, axiom,
    ((![Env : nat > type, X : nat, T : type]: (((Env @ X) = T) => (typing @ Env @ (var @ X) @ T))))). % typing.Var
thf(fact_8_type__induct, axiom,
    ((![P : type > $o, T : type]: ((![T2 : type]: ((![T1 : type, T22 : type]: ((T2 = (fun @ T1 @ T22)) => (P @ T1))) => ((![T1 : type, T22 : type]: ((T2 = (fun @ T1 @ T22)) => (P @ T22))) => (P @ T2)))) => (P @ T))))). % type_induct
thf(fact_9_typing__elims_I3_J, axiom,
    ((![E : nat > type, T3 : dB, T : type]: ((typing @ E @ (abs @ T3) @ T) => (~ ((![T2 : type, U : type]: ((T = (fun @ T2 @ U)) => (~ ((typing @ (shift_type @ E @ zero_zero_nat @ T2) @ T3 @ U))))))))))). % typing_elims(3)
thf(fact_10_Abs, axiom,
    ((![Env : nat > type, T : type, T3 : dB, U2 : type]: ((typing @ (shift_type @ Env @ zero_zero_nat @ T) @ T3 @ U2) => (typing @ Env @ (abs @ T3) @ (fun @ T @ U2)))))). % Abs
thf(fact_11_zero__natural_Orsp, axiom,
    ((zero_zero_nat = zero_zero_nat))). % zero_natural.rsp
thf(fact_12_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_13_dB_Oinject_I3_J, axiom,
    ((![X3 : dB, Y3 : dB]: (((abs @ X3) = (abs @ Y3)) = (X3 = Y3))))). % dB.inject(3)
thf(fact_14_App_Ohyps_I4_J, axiom,
    ((it @ ta2))). % App.hyps(4)
thf(fact_15_App_Ohyps_I2_J, axiom,
    ((it @ s))). % App.hyps(2)
thf(fact_16_dB_Odistinct_I3_J, axiom,
    ((![X1 : nat, X3 : dB]: (~ (((var @ X1) = (abs @ X3))))))). % dB.distinct(3)
thf(fact_17_abs__typeE, axiom,
    ((![E : nat > type, T3 : dB, T : type]: ((typing @ E @ (abs @ T3) @ T) => (~ ((![U : type, V : type]: (~ ((typing @ (shift_type @ E @ zero_zero_nat @ U) @ T3 @ V)))))))))). % abs_typeE
thf(fact_18_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, T2 : type]: (((Env2 @ X4) = T2) => (P @ Env2 @ (var @ X4) @ T2))) => ((![Env2 : nat > type, T2 : type, T4 : dB, U : type]: ((typing @ (shift_type @ Env2 @ zero_zero_nat @ T2) @ T4 @ U) => ((P @ (shift_type @ Env2 @ zero_zero_nat @ T2) @ T4 @ U) => (P @ Env2 @ (abs @ T4) @ (fun @ T2 @ U))))) => ((![Env2 : nat > type, S : dB, T2 : type, U : type, T4 : dB]: ((typing @ Env2 @ S @ (fun @ T2 @ U)) => ((P @ Env2 @ S @ (fun @ T2 @ U)) => ((typing @ Env2 @ T4 @ T2) => ((P @ Env2 @ T4 @ T2) => (P @ Env2 @ (app @ S @ T4) @ U)))))) => (P @ X1 @ X2 @ X3)))))))). % typing.inducts
thf(fact_19_typing_Osimps, axiom,
    ((typing = (^[A1 : nat > type]: (^[A2 : dB]: (^[A3 : type]: (((?[Env3 : nat > type]: (?[X5 : nat]: (?[T5 : type]: (((A1 = Env3)) & ((((A2 = (var @ X5))) & ((((A3 = T5)) & (((Env3 @ X5) = T5))))))))))) | ((((?[Env3 : nat > type]: (?[T5 : type]: (?[T6 : dB]: (?[U3 : type]: (((A1 = Env3)) & ((((A2 = (abs @ T6))) & ((((A3 = (fun @ T5 @ U3))) & ((typing @ (shift_type @ Env3 @ zero_zero_nat @ T5) @ T6 @ U3)))))))))))) | ((?[Env3 : nat > type]: (?[S2 : dB]: (?[T5 : type]: (?[U3 : type]: (?[T6 : dB]: (((A1 = Env3)) & ((((A2 = (app @ S2 @ T6))) & ((((A3 = U3)) & ((((typing @ Env3 @ S2 @ (fun @ T5 @ U3))) & ((typing @ Env3 @ T6 @ T5)))))))))))))))))))))))). % typing.simps
thf(fact_20_typing_Ocases, axiom,
    ((![A12 : nat > type, A22 : dB, A32 : type]: ((typing @ A12 @ A22 @ A32) => ((![X4 : nat]: ((A22 = (var @ X4)) => (~ (((A12 @ X4) = A32))))) => ((![T2 : type, T4 : dB]: ((A22 = (abs @ T4)) => (![U : type]: ((A32 = (fun @ T2 @ U)) => (~ ((typing @ (shift_type @ A12 @ zero_zero_nat @ T2) @ T4 @ U))))))) => (~ ((![S : dB, T2 : type, U : type, T4 : dB]: ((A22 = (app @ S @ T4)) => ((A32 = U) => ((typing @ A12 @ S @ (fun @ T2 @ U)) => (~ ((typing @ A12 @ T4 @ T2))))))))))))))). % typing.cases
thf(fact_21_dB_Osize__gen_I1_J, axiom,
    ((![X1 : nat]: ((size_dB @ (var @ X1)) = zero_zero_nat)))). % dB.size_gen(1)
thf(fact_22_lift__type, axiom,
    ((![E : nat > type, T3 : dB, T : type, I : nat, U2 : type]: ((typing @ E @ T3 @ T) => (typing @ (shift_type @ E @ I @ U2) @ (lift @ T3 @ I) @ T))))). % lift_type
thf(fact_23_dB_Osize_I4_J, axiom,
    ((![X1 : nat]: ((size_size_dB @ (var @ X1)) = zero_zero_nat)))). % dB.size(4)
thf(fact_24_liftn__0, axiom,
    ((![T3 : dB, K : nat]: ((liftn @ zero_zero_nat @ T3 @ K) = T3)))). % liftn_0
thf(fact_25_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_26__092_060open_062IT_A_Ilift_At_A0_J_092_060close_062, axiom,
    ((it @ (lift @ ta2 @ zero_zero_nat)))). % \<open>IT (lift t 0)\<close>
thf(fact_27__092_060open_062IT_A_IVar_A0_A_092_060degree_062_Alift_At_A0_J_092_060close_062, axiom,
    ((it @ (app @ (var @ zero_zero_nat) @ (lift @ ta2 @ zero_zero_nat))))). % \<open>IT (Var 0 \<degree> lift t 0)\<close>
thf(fact_28_lift__IT, axiom,
    ((![T3 : dB, I : nat]: ((it @ T3) => (it @ (lift @ T3 @ I)))))). % lift_IT
thf(fact_29_lift_Osimps_I2_J, axiom,
    ((![S3 : dB, T3 : dB, K : nat]: ((lift @ (app @ S3 @ T3) @ K) = (app @ (lift @ S3 @ K) @ (lift @ T3 @ K)))))). % lift.simps(2)
thf(fact_30_liftn_Osimps_I2_J, axiom,
    ((![N : nat, S3 : dB, T3 : dB, K : nat]: ((liftn @ N @ (app @ S3 @ T3) @ K) = (app @ (liftn @ N @ S3 @ K) @ (liftn @ N @ T3 @ K)))))). % liftn.simps(2)
thf(fact_31_app__Var__IT, axiom,
    ((![T3 : dB, I : nat]: ((it @ T3) => (it @ (app @ T3 @ (var @ I))))))). % app_Var_IT
thf(fact_32_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_33_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_34_dB_Odistinct_I5_J, axiom,
    ((![X21 : dB, X22 : dB, X3 : dB]: (~ (((app @ X21 @ X22) = (abs @ X3))))))). % dB.distinct(5)
thf(fact_35_dB_Odistinct_I1_J, axiom,
    ((![X1 : nat, X21 : dB, X22 : dB]: (~ (((var @ X1) = (app @ X21 @ X22))))))). % dB.distinct(1)
thf(fact_36_Var__IT, axiom,
    ((![N : nat]: (it @ (var @ N))))). % Var_IT
thf(fact_37_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_38_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_39_typing__elims_I2_J, axiom,
    ((![E : nat > type, T3 : dB, U4 : dB, T : type]: ((typing @ E @ (app @ T3 @ U4) @ T) => (~ ((![T2 : type]: ((typing @ E @ T3 @ (fun @ T2 @ T)) => (~ ((typing @ E @ U4 @ T2))))))))))). % typing_elims(2)
thf(fact_40_typing_OApp, axiom,
    ((![Env : nat > type, S3 : dB, T : type, U2 : type, T3 : dB]: ((typing @ Env @ S3 @ (fun @ T @ U2)) => ((typing @ Env @ T3 @ T) => (typing @ Env @ (app @ S3 @ T3) @ U2)))))). % typing.App
thf(fact_41_Lambda, axiom,
    ((![R : dB]: ((it @ R) => (it @ (abs @ R)))))). % Lambda
thf(fact_42_subst__type__IT, axiom,
    ((![T3 : dB, E : nat > type, I : nat, U2 : type, T : type, U4 : dB]: ((it @ T3) => ((typing @ (shift_type @ E @ I @ U2) @ T3 @ T) => ((it @ U4) => ((typing @ E @ U4 @ U2) => (it @ (subst @ T3 @ U4 @ I))))))))). % subst_type_IT
thf(fact_43_liftn__lift, axiom,
    ((![N : nat, T3 : dB, K : nat]: ((liftn @ (suc @ N) @ T3 @ K) = (lift @ (liftn @ N @ T3 @ K) @ K))))). % liftn_lift
thf(fact_44_var__app__type__eq, axiom,
    ((![E : nat > type, I : nat, Ts : list_dB, T : type, U2 : type]: ((typing @ E @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ T) => ((typing @ E @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ U2) => (T = U2)))))). % var_app_type_eq
thf(fact_45__092_060open_062IT_A_IVar_A0_A_092_060degree_062_092_060degree_062_A_091lift_At_A0_093_J_092_060close_062, axiom,
    ((it @ (foldl_dB_dB @ app @ (var @ zero_zero_nat) @ (cons_dB @ (lift @ ta2 @ zero_zero_nat) @ nil_dB))))). % \<open>IT (Var 0 \<degree>\<degree> [lift t 0])\<close>
thf(fact_46_shift__commute, axiom,
    ((![E : nat > type, I : nat, U2 : type, T : type]: ((shift_type @ (shift_type @ E @ I @ U2) @ zero_zero_nat @ T) = (shift_type @ (shift_type @ E @ zero_zero_nat @ T) @ (suc @ I) @ U2))))). % shift_commute
thf(fact_47_old_Onat_Oinject, axiom,
    ((![Nat : nat, Nat2 : nat]: (((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))). % old.nat.inject
thf(fact_48_nat_Oinject, axiom,
    ((![X2 : nat, Y2 : nat]: (((suc @ X2) = (suc @ Y2)) = (X2 = Y2))))). % nat.inject
thf(fact_49_type_Oinject_I1_J, axiom,
    ((![X1 : nat, Y1 : nat]: (((atom @ X1) = (atom @ Y1)) = (X1 = Y1))))). % type.inject(1)
thf(fact_50_subst__eq, axiom,
    ((![K : nat, U4 : dB]: ((subst @ (var @ K) @ U4 @ K) = U4)))). % subst_eq
thf(fact_51_subst__lift, axiom,
    ((![T3 : dB, K : nat, S3 : dB]: ((subst @ (lift @ T3 @ K) @ S3 @ K) = T3)))). % subst_lift
thf(fact_52__092_060open_062Var_A0_A_092_060degree_062_092_060degree_062_A_091lift_At_A0_093_A_061_AVar_A0_A_092_060degree_062_Alift_At_A0_092_060close_062, axiom,
    (((foldl_dB_dB @ app @ (var @ zero_zero_nat) @ (cons_dB @ (lift @ ta2 @ zero_zero_nat) @ nil_dB)) = (app @ (var @ zero_zero_nat) @ (lift @ ta2 @ zero_zero_nat))))). % \<open>Var 0 \<degree>\<degree> [lift t 0] = Var 0 \<degree> lift t 0\<close>
thf(fact_53_n__not__Suc__n, axiom,
    ((![N : nat]: (~ ((N = (suc @ N))))))). % n_not_Suc_n
thf(fact_54_Suc__inject, axiom,
    ((![X : nat, Y : nat]: (((suc @ X) = (suc @ Y)) => (X = Y))))). % Suc_inject
thf(fact_55_type_Osize_I3_J, axiom,
    ((![X1 : nat]: ((size_size_type @ (atom @ X1)) = zero_zero_nat)))). % type.size(3)
thf(fact_56_not0__implies__Suc, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (?[M : nat]: (N = (suc @ M))))))). % not0_implies_Suc
thf(fact_57_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_58_old_Onat_Oexhaust, axiom,
    ((![Y : nat]: ((~ ((Y = zero_zero_nat))) => (~ ((![Nat3 : nat]: (~ ((Y = (suc @ Nat3))))))))))). % old.nat.exhaust
thf(fact_59_Zero__not__Suc, axiom,
    ((![M2 : nat]: (~ ((zero_zero_nat = (suc @ M2))))))). % Zero_not_Suc
thf(fact_60_Zero__neq__Suc, axiom,
    ((![M2 : nat]: (~ ((zero_zero_nat = (suc @ M2))))))). % Zero_neq_Suc
thf(fact_61_Suc__neq__Zero, axiom,
    ((![M2 : nat]: (~ (((suc @ M2) = zero_zero_nat)))))). % Suc_neq_Zero
thf(fact_62_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_63_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_64_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_65_nat_OdiscI, axiom,
    ((![Nat : nat, X2 : nat]: ((Nat = (suc @ X2)) => (~ ((Nat = zero_zero_nat))))))). % nat.discI
thf(fact_66_old_Onat_Odistinct_I1_J, axiom,
    ((![Nat2 : nat]: (~ ((zero_zero_nat = (suc @ Nat2))))))). % old.nat.distinct(1)
thf(fact_67_old_Onat_Odistinct_I2_J, axiom,
    ((![Nat2 : nat]: (~ (((suc @ Nat2) = zero_zero_nat)))))). % old.nat.distinct(2)
thf(fact_68_nat_Odistinct_I1_J, axiom,
    ((![X2 : nat]: (~ ((zero_zero_nat = (suc @ X2))))))). % nat.distinct(1)
thf(fact_69_subst__App, axiom,
    ((![T3 : dB, U4 : dB, S3 : dB, K : nat]: ((subst @ (app @ T3 @ U4) @ S3 @ K) = (app @ (subst @ T3 @ S3 @ K) @ (subst @ U4 @ S3 @ K)))))). % subst_App
thf(fact_70_Beta, axiom,
    ((![R : dB, S3 : dB, Ss : list_dB]: ((it @ (foldl_dB_dB @ app @ (subst @ R @ S3 @ zero_zero_nat) @ Ss)) => ((it @ S3) => (it @ (foldl_dB_dB @ app @ (app @ (abs @ R) @ S3) @ Ss))))))). % Beta
thf(fact_71_subst__Var__IT, axiom,
    ((![R : dB, I : nat, J : nat]: ((it @ R) => (it @ (subst @ R @ (var @ I) @ J)))))). % subst_Var_IT
thf(fact_72_type_Oexhaust, axiom,
    ((![Y : type]: ((![X12 : nat]: (~ ((Y = (atom @ X12))))) => (~ ((![X212 : type, X222 : type]: (~ ((Y = (fun @ X212 @ X222))))))))))). % type.exhaust
thf(fact_73_type_Oinduct, axiom,
    ((![P : type > $o, Type : type]: ((![X4 : nat]: (P @ (atom @ X4))) => ((![X1a : type, X23 : type]: ((P @ X1a) => ((P @ X23) => (P @ (fun @ X1a @ X23))))) => (P @ Type)))))). % type.induct
thf(fact_74_type_Odistinct_I1_J, axiom,
    ((![X1 : nat, X21 : type, X22 : type]: (~ (((atom @ X1) = (fun @ X21 @ X22))))))). % type.distinct(1)
thf(fact_75_subst__lemma, axiom,
    ((![E : nat > type, T3 : dB, T : type, E2 : nat > type, U4 : dB, U2 : type, I : nat]: ((typing @ E @ T3 @ T) => ((typing @ E2 @ U4 @ U2) => ((E = (shift_type @ E2 @ I @ U2)) => (typing @ E2 @ (subst @ T3 @ U4 @ I) @ T))))))). % subst_lemma
thf(fact_76_Var__eq__apps__conv, axiom,
    ((![M2 : nat, S3 : dB, Ss : list_dB]: (((var @ M2) = (foldl_dB_dB @ app @ S3 @ Ss)) = ((((var @ M2) = S3)) & ((Ss = nil_dB))))))). % Var_eq_apps_conv
thf(fact_77_apps__eq__Abs__conv, axiom,
    ((![S3 : dB, Ss : list_dB, R : dB]: (((foldl_dB_dB @ app @ S3 @ Ss) = (abs @ R)) = (((S3 = (abs @ R))) & ((Ss = nil_dB))))))). % apps_eq_Abs_conv
thf(fact_78_Abs__eq__apps__conv, axiom,
    ((![R : dB, S3 : dB, Ss : list_dB]: (((abs @ R) = (foldl_dB_dB @ app @ S3 @ Ss)) = ((((abs @ R) = S3)) & ((Ss = nil_dB))))))). % Abs_eq_apps_conv
thf(fact_79_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_80_Abs__apps__eq__Abs__apps__conv, axiom,
    ((![R : dB, Rs : list_dB, S3 : dB, Ss : list_dB]: (((foldl_dB_dB @ app @ (abs @ R) @ Rs) = (foldl_dB_dB @ app @ (abs @ S3) @ Ss)) = (((R = S3)) & ((Rs = Ss))))))). % Abs_apps_eq_Abs_apps_conv
thf(fact_81_apps__eq__tail__conv, axiom,
    ((![R : dB, Ts : list_dB, S3 : dB]: (((foldl_dB_dB @ app @ R @ Ts) = (foldl_dB_dB @ app @ S3 @ Ts)) = (R = S3))))). % apps_eq_tail_conv
thf(fact_82_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_83_Abs__App__neq__Var__apps, axiom,
    ((![S3 : dB, T3 : dB, N : nat, Ss : list_dB]: (~ (((app @ (abs @ S3) @ T3) = (foldl_dB_dB @ app @ (var @ N) @ Ss))))))). % Abs_App_neq_Var_apps
thf(fact_84_ex__head__tail, axiom,
    ((![T3 : dB]: (?[Ts2 : list_dB, H : dB]: ((T3 = (foldl_dB_dB @ app @ H @ Ts2)) & ((?[N2 : nat]: (H = (var @ N2))) | (?[U5 : dB]: (H = (abs @ U5))))))))). % ex_head_tail
thf(fact_85__092_060open_062listsp_AIT_A_091lift_At_A0_093_092_060close_062, axiom,
    ((listsp_dB @ it @ (cons_dB @ (lift @ ta2 @ zero_zero_nat) @ nil_dB)))). % \<open>listsp IT [lift t 0]\<close>
thf(fact_86_length__0__conv, axiom,
    ((![Xs : list_dB]: (((size_size_list_dB @ Xs) = zero_zero_nat) = (Xs = nil_dB))))). % length_0_conv
thf(fact_87_substn__subst__n, axiom,
    ((substn = (^[T6 : dB]: (^[S2 : dB]: (^[N3 : nat]: (subst @ T6 @ (liftn @ N3 @ S2 @ zero_zero_nat) @ N3))))))). % substn_subst_n
thf(fact_88_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_89_listsp__simps_I1_J, axiom,
    ((![A : dB > $o]: (listsp_dB @ A @ nil_dB)))). % listsp_simps(1)
thf(fact_90_listsp_Oinducts, axiom,
    ((![A : dB > $o, X : list_dB, P : list_dB > $o]: ((listsp_dB @ A @ X) => ((P @ nil_dB) => ((![A4 : dB, L : list_dB]: ((A @ A4) => ((listsp_dB @ A @ L) => ((P @ L) => (P @ (cons_dB @ A4 @ L)))))) => (P @ X))))))). % listsp.inducts
thf(fact_91_listsp_Osimps, axiom,
    ((listsp_dB = (^[A5 : dB > $o]: (^[A6 : list_dB]: (((A6 = nil_dB)) | ((?[B : dB]: (?[L2 : list_dB]: (((A6 = (cons_dB @ B @ L2))) & ((((A5 @ B)) & ((listsp_dB @ A5 @ L2)))))))))))))). % listsp.simps
thf(fact_92_listsp_Ocases, axiom,
    ((![A : dB > $o, A7 : list_dB]: ((listsp_dB @ A @ A7) => ((~ ((A7 = nil_dB))) => (~ ((![A4 : dB, L : list_dB]: ((A7 = (cons_dB @ A4 @ L)) => ((A @ A4) => (~ ((listsp_dB @ A @ L))))))))))))). % listsp.cases
thf(fact_93_listsp_OCons, axiom,
    ((![A : dB > $o, A7 : dB, L3 : list_dB]: ((A @ A7) => ((listsp_dB @ A @ L3) => (listsp_dB @ A @ (cons_dB @ A7 @ L3))))))). % listsp.Cons
thf(fact_94_listsp_ONil, axiom,
    ((![A : dB > $o]: (listsp_dB @ A @ nil_dB)))). % listsp.Nil
thf(fact_95_listspE, axiom,
    ((![A : dB > $o, X : dB, L3 : list_dB]: ((listsp_dB @ A @ (cons_dB @ X @ L3)) => (~ (((A @ X) => (~ ((listsp_dB @ A @ L3)))))))))). % listspE
thf(fact_96_listsp__simps_I2_J, axiom,
    ((![A : dB > $o, X : dB, Xs : list_dB]: ((listsp_dB @ A @ (cons_dB @ X @ Xs)) = (((A @ X)) & ((listsp_dB @ A @ Xs))))))). % listsp_simps(2)
thf(fact_97_substn_Osimps_I2_J, axiom,
    ((![T3 : dB, U4 : dB, S3 : dB, K : nat]: ((substn @ (app @ T3 @ U4) @ S3 @ K) = (app @ (substn @ T3 @ S3 @ K) @ (substn @ U4 @ S3 @ K)))))). % substn.simps(2)
thf(fact_98_not__Cons__self2, axiom,
    ((![X : dB, Xs : list_dB]: (~ (((cons_dB @ X @ Xs) = Xs)))))). % not_Cons_self2
thf(fact_99_substn__subst__0, axiom,
    ((![T3 : dB, S3 : dB]: ((substn @ T3 @ S3 @ zero_zero_nat) = (subst @ T3 @ S3 @ zero_zero_nat))))). % substn_subst_0
thf(fact_100_IT_OVar, axiom,
    ((![Rs : list_dB, N : nat]: ((listsp_dB @ it @ Rs) => (it @ (foldl_dB_dB @ app @ (var @ N) @ Rs)))))). % IT.Var
thf(fact_101_length__Suc__conv, axiom,
    ((![Xs : list_dB, N : nat]: (((size_size_list_dB @ Xs) = (suc @ N)) = (?[Y5 : dB]: (?[Ys : list_dB]: (((Xs = (cons_dB @ Y5 @ Ys))) & (((size_size_list_dB @ Ys) = N))))))))). % length_Suc_conv
thf(fact_102_Suc__length__conv, axiom,
    ((![N : nat, Xs : list_dB]: (((suc @ N) = (size_size_list_dB @ Xs)) = (?[Y5 : dB]: (?[Ys : list_dB]: (((Xs = (cons_dB @ Y5 @ Ys))) & (((size_size_list_dB @ Ys) = N))))))))). % Suc_length_conv
thf(fact_103_list__induct3, axiom,
    ((![Xs : list_dB, Ys2 : list_dB, Zs : list_dB, P : list_dB > list_dB > list_dB > $o]: (((size_size_list_dB @ Xs) = (size_size_list_dB @ Ys2)) => (((size_size_list_dB @ Ys2) = (size_size_list_dB @ Zs)) => ((P @ nil_dB @ nil_dB @ nil_dB) => ((![X4 : dB, Xs2 : list_dB, Y4 : dB, Ys3 : list_dB, Z : dB, Zs2 : list_dB]: (((size_size_list_dB @ Xs2) = (size_size_list_dB @ Ys3)) => (((size_size_list_dB @ Ys3) = (size_size_list_dB @ Zs2)) => ((P @ Xs2 @ Ys3 @ Zs2) => (P @ (cons_dB @ X4 @ Xs2) @ (cons_dB @ Y4 @ Ys3) @ (cons_dB @ Z @ Zs2)))))) => (P @ Xs @ Ys2 @ Zs)))))))). % list_induct3
thf(fact_104_list__induct2, axiom,
    ((![Xs : list_dB, Ys2 : list_dB, P : list_dB > list_dB > $o]: (((size_size_list_dB @ Xs) = (size_size_list_dB @ Ys2)) => ((P @ nil_dB @ nil_dB) => ((![X4 : dB, Xs2 : list_dB, Y4 : dB, Ys3 : list_dB]: (((size_size_list_dB @ Xs2) = (size_size_list_dB @ Ys3)) => ((P @ Xs2 @ Ys3) => (P @ (cons_dB @ X4 @ Xs2) @ (cons_dB @ Y4 @ Ys3))))) => (P @ Xs @ Ys2))))))). % list_induct2
thf(fact_105_list_Odistinct_I1_J, axiom,
    ((![X21 : dB, X22 : list_dB]: (~ ((nil_dB = (cons_dB @ X21 @ X22))))))). % list.distinct(1)
thf(fact_106_list_OdiscI, axiom,
    ((![List : list_dB, X21 : dB, X22 : list_dB]: ((List = (cons_dB @ X21 @ X22)) => (~ ((List = nil_dB))))))). % list.discI
thf(fact_107_list_Oexhaust, axiom,
    ((![Y : list_dB]: ((~ ((Y = nil_dB))) => (~ ((![X212 : dB, X222 : list_dB]: (~ ((Y = (cons_dB @ X212 @ X222))))))))))). % list.exhaust
thf(fact_108_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_109_neq__Nil__conv, axiom,
    ((![Xs : list_dB]: ((~ ((Xs = nil_dB))) = (?[Y5 : dB]: (?[Ys : list_dB]: (Xs = (cons_dB @ Y5 @ Ys)))))))). % neq_Nil_conv
thf(fact_110_list__induct2_H, axiom,
    ((![P : list_dB > list_dB > $o, Xs : list_dB, Ys2 : list_dB]: ((P @ nil_dB @ nil_dB) => ((![X4 : dB, Xs2 : list_dB]: (P @ (cons_dB @ X4 @ Xs2) @ nil_dB)) => ((![Y4 : dB, Ys3 : list_dB]: (P @ nil_dB @ (cons_dB @ Y4 @ Ys3))) => ((![X4 : dB, Xs2 : list_dB, Y4 : dB, Ys3 : list_dB]: ((P @ Xs2 @ Ys3) => (P @ (cons_dB @ X4 @ Xs2) @ (cons_dB @ Y4 @ Ys3)))) => (P @ Xs @ Ys2)))))))). % list_induct2'
thf(fact_111_splice_Oinduct, axiom,
    ((![P : list_dB > list_dB > $o, A0 : list_dB, A12 : list_dB]: ((![X_1 : list_dB]: (P @ nil_dB @ X_1)) => ((![X4 : dB, Xs2 : list_dB, Ys3 : list_dB]: ((P @ Ys3 @ Xs2) => (P @ (cons_dB @ X4 @ Xs2) @ Ys3))) => (P @ A0 @ A12)))))). % splice.induct
thf(fact_112_induct__list012, axiom,
    ((![P : list_dB > $o, Xs : list_dB]: ((P @ nil_dB) => ((![X4 : dB]: (P @ (cons_dB @ X4 @ nil_dB))) => ((![X4 : dB, Y4 : dB, Zs2 : list_dB]: ((P @ Zs2) => ((P @ (cons_dB @ Y4 @ Zs2)) => (P @ (cons_dB @ X4 @ (cons_dB @ Y4 @ Zs2)))))) => (P @ Xs))))))). % induct_list012
thf(fact_113_shuffles_Oinduct, axiom,
    ((![P : list_dB > list_dB > $o, A0 : list_dB, A12 : list_dB]: ((![X_1 : list_dB]: (P @ nil_dB @ X_1)) => ((![Xs2 : list_dB]: (P @ Xs2 @ nil_dB)) => ((![X4 : dB, Xs2 : list_dB, Y4 : dB, Ys3 : list_dB]: ((P @ Xs2 @ (cons_dB @ Y4 @ Ys3)) => ((P @ (cons_dB @ X4 @ Xs2) @ Ys3) => (P @ (cons_dB @ X4 @ Xs2) @ (cons_dB @ Y4 @ Ys3))))) => (P @ A0 @ A12))))))). % shuffles.induct
thf(fact_114_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_115_remdups__adj_Ocases, axiom,
    ((![X : list_dB]: ((~ ((X = nil_dB))) => ((![X4 : dB]: (~ ((X = (cons_dB @ X4 @ nil_dB))))) => (~ ((![X4 : dB, Y4 : dB, Xs2 : list_dB]: (~ ((X = (cons_dB @ X4 @ (cons_dB @ Y4 @ Xs2))))))))))))). % remdups_adj.cases
thf(fact_116_sorted__wrt_Oinduct, axiom,
    ((![P : (dB > dB > $o) > list_dB > $o, A0 : dB > dB > $o, A12 : list_dB]: ((![P2 : dB > dB > $o]: (P @ P2 @ nil_dB)) => ((![P2 : dB > dB > $o, X4 : dB, Ys3 : list_dB]: ((P @ P2 @ Ys3) => (P @ P2 @ (cons_dB @ X4 @ Ys3)))) => (P @ A0 @ A12)))))). % sorted_wrt.induct
thf(fact_117_remdups__adj_Oinduct, axiom,
    ((![P : list_dB > $o, A0 : list_dB]: ((P @ nil_dB) => ((![X4 : dB]: (P @ (cons_dB @ X4 @ nil_dB))) => ((![X4 : dB, Y4 : dB, Xs2 : list_dB]: (((X4 = Y4) => (P @ (cons_dB @ X4 @ Xs2))) => (((~ ((X4 = Y4))) => (P @ (cons_dB @ Y4 @ Xs2))) => (P @ (cons_dB @ X4 @ (cons_dB @ Y4 @ Xs2)))))) => (P @ A0))))))). % remdups_adj.induct
thf(fact_118_successively_Oinduct, axiom,
    ((![P : (dB > dB > $o) > list_dB > $o, A0 : dB > dB > $o, A12 : 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, Y4 : dB, Xs2 : list_dB]: ((P @ P2 @ (cons_dB @ Y4 @ Xs2)) => (P @ P2 @ (cons_dB @ X4 @ (cons_dB @ Y4 @ Xs2))))) => (P @ A0 @ A12))))))). % successively.induct
thf(fact_119_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_120_map__tailrec__rev_Oinduct, axiom,
    ((![P : (dB > dB) > list_dB > list_dB > $o, A0 : dB > dB, A12 : list_dB, A22 : list_dB]: ((![F : dB > dB, X_1 : list_dB]: (P @ F @ nil_dB @ X_1)) => ((![F : dB > dB, A4 : dB, As : list_dB, Bs : list_dB]: ((P @ F @ As @ (cons_dB @ (F @ A4) @ Bs)) => (P @ F @ (cons_dB @ A4 @ As) @ Bs))) => (P @ A0 @ A12 @ A22)))))). % map_tailrec_rev.induct
thf(fact_121_list_Osize_I3_J, axiom,
    (((size_size_list_dB @ nil_dB) = zero_zero_nat))). % list.size(3)
thf(fact_122_foldl__Cons, axiom,
    ((![F2 : dB > dB > dB, A7 : dB, X : dB, Xs : list_dB]: ((foldl_dB_dB @ F2 @ A7 @ (cons_dB @ X @ Xs)) = (foldl_dB_dB @ F2 @ (F2 @ A7 @ X) @ Xs))))). % foldl_Cons
thf(fact_123_foldl__Nil, axiom,
    ((![F2 : dB > dB > dB, A7 : dB]: ((foldl_dB_dB @ F2 @ A7 @ nil_dB) = A7)))). % foldl_Nil
thf(fact_124_IT_Osimps, axiom,
    ((it = (^[A6 : dB]: (((?[Rs2 : list_dB]: (?[N3 : nat]: (((A6 = (foldl_dB_dB @ app @ (var @ N3) @ Rs2))) & ((listsp_dB @ it @ Rs2)))))) | ((((?[R2 : dB]: (((A6 = (abs @ R2))) & ((it @ R2))))) | ((?[R2 : dB]: (?[S2 : dB]: (?[Ss2 : list_dB]: (((A6 = (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_125_IT_Ocases, axiom,
    ((![A7 : dB]: ((it @ A7) => ((![Rs3 : list_dB]: ((?[N2 : nat]: (A7 = (foldl_dB_dB @ app @ (var @ N2) @ Rs3))) => (~ ((listsp_dB @ it @ Rs3))))) => ((![R3 : dB]: ((A7 = (abs @ R3)) => (~ ((it @ R3))))) => (~ ((![R3 : dB, S : dB, Ss3 : list_dB]: ((A7 = (foldl_dB_dB @ app @ (app @ (abs @ R3) @ S) @ Ss3)) => ((it @ (foldl_dB_dB @ app @ (subst @ R3 @ S @ zero_zero_nat) @ Ss3)) => (~ ((it @ S)))))))))))))). % IT.cases
thf(fact_126_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_127_type_Osize__gen_I1_J, axiom,
    ((![X1 : nat]: ((size_type @ (atom @ X1)) = zero_zero_nat)))). % type.size_gen(1)
thf(fact_128_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_129_n__lists_Osimps_I1_J, axiom,
    ((![Xs : list_dB]: ((n_lists_dB @ zero_zero_nat @ Xs) = (cons_list_dB @ nil_dB @ nil_list_dB))))). % n_lists.simps(1)
thf(fact_130_product__lists_Osimps_I1_J, axiom,
    (((product_lists_dB @ nil_list_dB) = (cons_list_dB @ nil_dB @ nil_list_dB)))). % product_lists.simps(1)

% Conjectures (1)
thf(conj_0, conjecture,
    ((typing @ (shift_type @ ea @ zero_zero_nat @ (fun @ ta @ u)) @ (var @ zero_zero_nat) @ (fun @ ta @ u)))).
