% 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_63__5209872_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:37:10.737

% Could-be-implicit typings (5)
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).
thf(ty_n_t__Int__Oint, type,
    int : $tType).

% Explicit typings (26)
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint, type,
    uminus_uminus_int : int > int).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint, type,
    zero_zero_int : int).
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_Int_Onat, type,
    nat2 : int > nat).
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_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_Olist_OCons_001t__Lambda__OdB, type,
    cons_dB : dB > list_dB > list_dB).
thf(sy_c_List_Olistsp_001t__Lambda__OdB, type,
    listsp_dB : (dB > $o) > list_dB > $o).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint, type,
    semiri2019852685at_int : nat > int).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat, type,
    semiri1382578993at_nat : nat > nat).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint, type,
    ord_less_int : int > int > $o).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat, type,
    ord_less_nat : nat > nat > $o).
thf(sy_v_n, type,
    n : nat).

% Relevant facts (148)
thf(fact_0_dB_Oinject_I1_J, axiom,
    ((![X1 : nat, Y1 : nat]: (((var @ X1) = (var @ Y1)) = (X1 = Y1))))). % dB.inject(1)
thf(fact_1_subst__Var__IT, axiom,
    ((![R : dB, I : nat, J : nat]: ((it @ R) => (it @ (subst @ R @ (var @ I) @ J)))))). % subst_Var_IT
thf(fact_2_lift__IT, axiom,
    ((![T : dB, I : nat]: ((it @ T) => (it @ (lift @ T @ I)))))). % lift_IT
thf(fact_3_subst__eq, axiom,
    ((![K : nat, U : dB]: ((subst @ (var @ K) @ U @ K) = U)))). % subst_eq
thf(fact_4_Lambda, axiom,
    ((![R : dB]: ((it @ R) => (it @ (abs @ R)))))). % Lambda
thf(fact_5_dB_Odistinct_I3_J, axiom,
    ((![X1 : nat, X3 : dB]: (~ (((var @ X1) = (abs @ X3))))))). % dB.distinct(3)
thf(fact_6_beta__cases_I1_J, axiom,
    ((![I : nat, T : dB]: (~ ((beta @ (var @ I) @ T)))))). % beta_cases(1)
thf(fact_7_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_8_typing_OVar, axiom,
    ((![Env : nat > type, X : nat, T2 : type]: (((Env @ X) = T2) => (typing @ Env @ (var @ X) @ T2))))). % typing.Var
thf(fact_9_dB_Odistinct_I1_J, axiom,
    ((![X1 : nat, X21 : dB, X22 : dB]: (~ (((var @ X1) = (app @ X21 @ X22))))))). % dB.distinct(1)
thf(fact_10_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_11_dB_Oinject_I3_J, axiom,
    ((![X3 : dB, Y3 : dB]: (((abs @ X3) = (abs @ Y3)) = (X3 = Y3))))). % dB.inject(3)
thf(fact_12_subst__lift, axiom,
    ((![T : dB, K : nat, S : dB]: ((subst @ (lift @ T @ K) @ S @ K) = T)))). % subst_lift
thf(fact_13_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_14_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_15_dB_Odistinct_I5_J, axiom,
    ((![X21 : dB, X22 : dB, X3 : dB]: (~ (((app @ X21 @ X22) = (abs @ X3))))))). % dB.distinct(5)
thf(fact_16_abs, axiom,
    ((![S : dB, T : dB]: ((beta @ S @ T) => (beta @ (abs @ S) @ (abs @ T)))))). % abs
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_appR, axiom,
    ((![S : dB, T : dB, U : dB]: ((beta @ S @ T) => (beta @ (app @ U @ S) @ (app @ U @ T)))))). % appR
thf(fact_19_dB_Oinduct, axiom,
    ((![P : dB > $o, DB : dB]: ((![X2 : nat]: (P @ (var @ X2))) => ((![X1a : dB, X23 : dB]: ((P @ X1a) => ((P @ X23) => (P @ (app @ X1a @ X23))))) => ((![X2 : dB]: ((P @ X2) => (P @ (abs @ X2)))) => (P @ DB))))))). % dB.induct
thf(fact_20_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_21_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_22_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_23_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_24_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_25_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)))))) => ((![T3 : dB]: ((U = (app @ T3 @ T)) => (~ ((beta @ S @ T3))))) => (~ ((![T3 : dB]: ((U = (app @ S @ T3)) => (~ ((beta @ T @ T3))))))))))))). % beta_cases(3)
thf(fact_26_beta, axiom,
    ((![S : dB, T : dB]: (beta @ (app @ (abs @ S) @ T) @ (subst @ S @ T @ zero_zero_nat))))). % beta
thf(fact_27_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, U2 : dB]: ((A1 = (app @ S2 @ U2)) => ((A2 = (app @ T3 @ U2)) => (~ ((beta @ S2 @ T3)))))) => ((![S2 : dB, T3 : dB, U2 : dB]: ((A1 = (app @ U2 @ S2)) => ((A2 = (app @ U2 @ T3)) => (~ ((beta @ S2 @ T3)))))) => (~ ((![S2 : dB]: ((A1 = (abs @ S2)) => (![T3 : dB]: ((A2 = (abs @ T3)) => (~ ((beta @ S2 @ T3)))))))))))))))). % beta.cases
thf(fact_28_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]: (?[U3 : dB]: (((A12 = (app @ S3 @ U3))) & ((((A22 = (app @ T5 @ U3))) & ((beta @ S3 @ T5))))))))) | ((((?[S3 : dB]: (?[T5 : dB]: (?[U3 : dB]: (((A12 = (app @ U3 @ S3))) & ((((A22 = (app @ U3 @ T5))) & ((beta @ S3 @ T5))))))))) | ((?[S3 : dB]: (?[T5 : dB]: (((A12 = (abs @ S3))) & ((((A22 = (abs @ T5))) & ((beta @ S3 @ T5)))))))))))))))))). % beta.simps
thf(fact_29_beta_Oinducts, axiom,
    ((![X1 : dB, X24 : dB, P : dB > dB > $o]: ((beta @ X1 @ X24) => ((![S2 : dB, T3 : dB]: (P @ (app @ (abs @ S2) @ T3) @ (subst @ S2 @ T3 @ zero_zero_nat))) => ((![S2 : dB, T3 : dB, U2 : dB]: ((beta @ S2 @ T3) => ((P @ S2 @ T3) => (P @ (app @ S2 @ U2) @ (app @ T3 @ U2))))) => ((![S2 : dB, T3 : dB, U2 : dB]: ((beta @ S2 @ T3) => ((P @ S2 @ T3) => (P @ (app @ U2 @ S2) @ (app @ U2 @ T3))))) => ((![S2 : dB, T3 : dB]: ((beta @ S2 @ T3) => ((P @ S2 @ T3) => (P @ (abs @ S2) @ (abs @ T3))))) => (P @ X1 @ X24))))))))). % beta.inducts
thf(fact_30_lift__type, axiom,
    ((![E : nat > type, T : dB, T2 : type, I : nat, U4 : type]: ((typing @ E @ T @ T2) => (typing @ (shift_type @ E @ I @ U4) @ (lift @ T @ I) @ T2))))). % lift_type
thf(fact_31_subst__lemma, axiom,
    ((![E : nat > type, T : dB, T2 : type, E2 : nat > type, U : dB, U4 : type, I : nat]: ((typing @ E @ T @ T2) => ((typing @ E2 @ U @ U4) => ((E = (shift_type @ E2 @ I @ U4)) => (typing @ E2 @ (subst @ T @ U @ I) @ T2))))))). % subst_lemma
thf(fact_32_var__app__type__eq, axiom,
    ((![E : nat > type, I : nat, Ts : list_dB, T2 : type, U4 : type]: ((typing @ E @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ T2) => ((typing @ E @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ U4) => (T2 = U4)))))). % var_app_type_eq
thf(fact_33_subst__lt, axiom,
    ((![J : nat, I : nat, U : dB]: ((ord_less_nat @ J @ I) => ((subst @ (var @ J) @ U @ I) = (var @ J)))))). % subst_lt
thf(fact_34_typing__elims_I2_J, axiom,
    ((![E : nat > type, T : dB, U : dB, T2 : type]: ((typing @ E @ (app @ T @ U) @ T2) => (~ ((![T6 : type]: ((typing @ E @ T @ (fun @ T6 @ T2)) => (~ ((typing @ E @ U @ T6))))))))))). % typing_elims(2)
thf(fact_35_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_36_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_37_shift__gt, axiom,
    ((![J : nat, I : nat, E : nat > type, T2 : type]: ((ord_less_nat @ J @ I) => ((shift_type @ E @ I @ T2 @ J) = (E @ J)))))). % shift_gt
thf(fact_38_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_39_Abs, axiom,
    ((![Env : nat > type, T2 : type, T : dB, U4 : type]: ((typing @ (shift_type @ Env @ zero_zero_nat @ T2) @ T @ U4) => (typing @ Env @ (abs @ T) @ (fun @ T2 @ U4)))))). % Abs
thf(fact_40_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_41_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_42_typing_Oinducts, axiom,
    ((![X1 : nat > type, X24 : dB, X3 : type, P : (nat > type) > dB > type > $o]: ((typing @ X1 @ X24 @ X3) => ((![Env2 : nat > type, X2 : nat, T6 : type]: (((Env2 @ X2) = T6) => (P @ Env2 @ (var @ X2) @ T6))) => ((![Env2 : nat > type, T6 : type, T3 : dB, U5 : type]: ((typing @ (shift_type @ Env2 @ zero_zero_nat @ T6) @ T3 @ U5) => ((P @ (shift_type @ Env2 @ zero_zero_nat @ T6) @ T3 @ U5) => (P @ Env2 @ (abs @ T3) @ (fun @ T6 @ U5))))) => ((![Env2 : nat > type, S2 : dB, T6 : type, U5 : type, T3 : dB]: ((typing @ Env2 @ S2 @ (fun @ T6 @ U5)) => ((P @ Env2 @ S2 @ (fun @ T6 @ U5)) => ((typing @ Env2 @ T3 @ T6) => ((P @ Env2 @ T3 @ T6) => (P @ Env2 @ (app @ S2 @ T3) @ U5)))))) => (P @ X1 @ X24 @ X3)))))))). % typing.inducts
thf(fact_43_typing_Osimps, axiom,
    ((typing = (^[A12 : nat > type]: (^[A22 : dB]: (^[A3 : type]: (((?[Env3 : nat > type]: (?[X4 : nat]: (?[T7 : type]: (((A12 = Env3)) & ((((A22 = (var @ X4))) & ((((A3 = T7)) & (((Env3 @ X4) = T7))))))))))) | ((((?[Env3 : nat > type]: (?[T7 : type]: (?[T5 : dB]: (?[U6 : type]: (((A12 = Env3)) & ((((A22 = (abs @ T5))) & ((((A3 = (fun @ T7 @ U6))) & ((typing @ (shift_type @ Env3 @ zero_zero_nat @ T7) @ T5 @ U6)))))))))))) | ((?[Env3 : nat > type]: (?[S3 : dB]: (?[T7 : type]: (?[U6 : type]: (?[T5 : dB]: (((A12 = Env3)) & ((((A22 = (app @ S3 @ T5))) & ((((A3 = U6)) & ((((typing @ Env3 @ S3 @ (fun @ T7 @ U6))) & ((typing @ Env3 @ T5 @ T7)))))))))))))))))))))))). % typing.simps
thf(fact_44_typing_Ocases, axiom,
    ((![A1 : nat > type, A2 : dB, A32 : type]: ((typing @ A1 @ A2 @ A32) => ((![X2 : nat]: ((A2 = (var @ X2)) => (~ (((A1 @ X2) = A32))))) => ((![T6 : type, T3 : dB]: ((A2 = (abs @ T3)) => (![U5 : type]: ((A32 = (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)) => ((A32 = U5) => ((typing @ A1 @ S2 @ (fun @ T6 @ U5)) => (~ ((typing @ A1 @ T3 @ T6))))))))))))))). % typing.cases
thf(fact_45_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_46_App, axiom,
    ((![Env : nat > type, S : dB, T2 : type, U4 : type, T : dB]: ((typing @ Env @ S @ (fun @ T2 @ U4)) => ((typing @ Env @ T @ T2) => (typing @ Env @ (app @ S @ T) @ U4)))))). % App
thf(fact_47_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_48_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_49_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_50_bot__nat__0_Onot__eq__extremum, axiom,
    ((![A : nat]: ((~ ((A = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ A))))). % bot_nat_0.not_eq_extremum
thf(fact_51_less__nat__zero__code, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_52_neq0__conv, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N))))). % neq0_conv
thf(fact_53_nat__neq__iff, axiom,
    ((![M : nat, N : nat]: ((~ ((M = N))) = (((ord_less_nat @ M @ N)) | ((ord_less_nat @ N @ M))))))). % nat_neq_iff
thf(fact_54_less__not__refl, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_not_refl
thf(fact_55_less__not__refl2, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ N @ M) => (~ ((M = N))))))). % less_not_refl2
thf(fact_56_less__not__refl3, axiom,
    ((![S : nat, T : nat]: ((ord_less_nat @ S @ T) => (~ ((S = T))))))). % less_not_refl3
thf(fact_57_less__irrefl__nat, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_irrefl_nat
thf(fact_58_nat__less__induct, axiom,
    ((![P : nat > $o, N : nat]: ((![N2 : nat]: ((![M2 : nat]: ((ord_less_nat @ M2 @ N2) => (P @ M2))) => (P @ N2))) => (P @ N))))). % nat_less_induct
thf(fact_59_infinite__descent, axiom,
    ((![P : nat > $o, N : nat]: ((![N2 : nat]: ((~ ((P @ N2))) => (?[M2 : nat]: ((ord_less_nat @ M2 @ N2) & (~ ((P @ M2))))))) => (P @ N))))). % infinite_descent
thf(fact_60_linorder__neqE__nat, axiom,
    ((![X : nat, Y : nat]: ((~ ((X = Y))) => ((~ ((ord_less_nat @ X @ Y))) => (ord_less_nat @ Y @ X)))))). % linorder_neqE_nat
thf(fact_61_gr0I, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr0I
thf(fact_62_not__gr0, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr0
thf(fact_63_not__less0, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less0
thf(fact_64_less__zeroE, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_zeroE
thf(fact_65_gr__implies__not0, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not0
thf(fact_66_infinite__descent0, axiom,
    ((![P : nat > $o, N : nat]: ((P @ zero_zero_nat) => ((![N2 : nat]: ((ord_less_nat @ zero_zero_nat @ N2) => ((~ ((P @ N2))) => (?[M2 : nat]: ((ord_less_nat @ M2 @ N2) & (~ ((P @ M2)))))))) => (P @ N)))))). % infinite_descent0
thf(fact_67_bot__nat__0_Oextremum__strict, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ zero_zero_nat)))))). % bot_nat_0.extremum_strict
thf(fact_68_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_69_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_70_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_71_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_72_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_73_IT_Ocases, axiom,
    ((![A : dB]: ((it @ A) => ((![Rs2 : list_dB]: ((?[N2 : nat]: (A = (foldl_dB_dB @ app @ (var @ N2) @ Rs2))) => (~ ((listsp_dB @ it @ Rs2))))) => ((![R2 : dB]: ((A = (abs @ R2)) => (~ ((it @ R2))))) => (~ ((![R2 : dB, S2 : dB, Ss2 : list_dB]: ((A = (foldl_dB_dB @ app @ (app @ (abs @ R2) @ S2) @ Ss2)) => ((it @ (foldl_dB_dB @ app @ (subst @ R2 @ S2 @ zero_zero_nat) @ Ss2)) => (~ ((it @ S2)))))))))))))). % IT.cases
thf(fact_74_IT_Osimps, axiom,
    ((it = (^[A4 : dB]: (((?[Rs3 : list_dB]: (?[N3 : nat]: (((A4 = (foldl_dB_dB @ app @ (var @ N3) @ Rs3))) & ((listsp_dB @ it @ Rs3)))))) | ((((?[R3 : dB]: (((A4 = (abs @ R3))) & ((it @ R3))))) | ((?[R3 : dB]: (?[S3 : dB]: (?[Ss3 : list_dB]: (((A4 = (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_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_76_zero__reorient, axiom,
    ((![X : int]: ((zero_zero_int = X) = (X = zero_zero_int))))). % zero_reorient
thf(fact_77_IT_OVar, axiom,
    ((![Rs : list_dB, N : nat]: ((listsp_dB @ it @ Rs) => (it @ (foldl_dB_dB @ app @ (var @ N) @ Rs)))))). % IT.Var
thf(fact_78_zero__less__iff__neq__zero, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) = (~ ((N = zero_zero_nat))))))). % zero_less_iff_neq_zero
thf(fact_79_gr__implies__not__zero, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not_zero
thf(fact_80_not__less__zero, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less_zero
thf(fact_81_gr__zeroI, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr_zeroI
thf(fact_82_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_nat @ zero_zero_nat @ zero_zero_nat))))). % less_numeral_extra(3)
thf(fact_83_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_int @ zero_zero_int @ zero_zero_int))))). % less_numeral_extra(3)
thf(fact_84_head__Var__reduction, axiom,
    ((![N : nat, Rs : list_dB, V2 : dB]: ((beta @ (foldl_dB_dB @ app @ (var @ N) @ Rs) @ V2) => (?[Ss2 : list_dB]: ((step1_dB @ beta @ Rs @ Ss2) & (V2 = (foldl_dB_dB @ app @ (var @ N) @ Ss2)))))))). % head_Var_reduction
thf(fact_85_dB_Osize__gen_I1_J, axiom,
    ((![X1 : nat]: ((size_dB @ (var @ X1)) = zero_zero_nat)))). % dB.size_gen(1)
thf(fact_86_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_87_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)) => (![U2 : dB, Us : list_dB]: ((Rs = (cons_dB @ U2 @ Us)) => (~ ((S = (foldl_dB_dB @ app @ (subst @ T3 @ U2 @ zero_zero_nat) @ Us)))))))))))))))). % apps_betasE
thf(fact_88_of__nat__0__less__iff, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ (semiri1382578993at_nat @ N)) = (ord_less_nat @ zero_zero_nat @ N))))). % of_nat_0_less_iff
thf(fact_89_of__nat__0__less__iff, axiom,
    ((![N : nat]: ((ord_less_int @ zero_zero_int @ (semiri2019852685at_int @ N)) = (ord_less_nat @ zero_zero_nat @ N))))). % of_nat_0_less_iff
thf(fact_90_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_91_of__nat__eq__iff, axiom,
    ((![M : nat, N : nat]: (((semiri2019852685at_int @ M) = (semiri2019852685at_int @ N)) = (M = N))))). % of_nat_eq_iff
thf(fact_92_of__nat__eq__0__iff, axiom,
    ((![M : nat]: (((semiri1382578993at_nat @ M) = zero_zero_nat) = (M = zero_zero_nat))))). % of_nat_eq_0_iff
thf(fact_93_of__nat__eq__0__iff, axiom,
    ((![M : nat]: (((semiri2019852685at_int @ M) = zero_zero_int) = (M = zero_zero_nat))))). % of_nat_eq_0_iff
thf(fact_94_of__nat__0__eq__iff, axiom,
    ((![N : nat]: ((zero_zero_nat = (semiri1382578993at_nat @ N)) = (zero_zero_nat = N))))). % of_nat_0_eq_iff
thf(fact_95_of__nat__0__eq__iff, axiom,
    ((![N : nat]: ((zero_zero_int = (semiri2019852685at_int @ N)) = (zero_zero_nat = N))))). % of_nat_0_eq_iff
thf(fact_96_of__nat__0, axiom,
    (((semiri1382578993at_nat @ zero_zero_nat) = zero_zero_nat))). % of_nat_0
thf(fact_97_of__nat__0, axiom,
    (((semiri2019852685at_int @ zero_zero_nat) = zero_zero_int))). % of_nat_0
thf(fact_98_of__nat__less__iff, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ (semiri1382578993at_nat @ M) @ (semiri1382578993at_nat @ N)) = (ord_less_nat @ M @ N))))). % of_nat_less_iff
thf(fact_99_of__nat__less__iff, axiom,
    ((![M : nat, N : nat]: ((ord_less_int @ (semiri2019852685at_int @ M) @ (semiri2019852685at_int @ N)) = (ord_less_nat @ M @ N))))). % of_nat_less_iff
thf(fact_100_of__nat__less__0__iff, axiom,
    ((![M : nat]: (~ ((ord_less_nat @ (semiri1382578993at_nat @ M) @ zero_zero_nat)))))). % of_nat_less_0_iff
thf(fact_101_of__nat__less__0__iff, axiom,
    ((![M : nat]: (~ ((ord_less_int @ (semiri2019852685at_int @ M) @ zero_zero_int)))))). % of_nat_less_0_iff
thf(fact_102_of__nat__less__imp__less, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ (semiri1382578993at_nat @ M) @ (semiri1382578993at_nat @ N)) => (ord_less_nat @ M @ N))))). % of_nat_less_imp_less
thf(fact_103_of__nat__less__imp__less, axiom,
    ((![M : nat, N : nat]: ((ord_less_int @ (semiri2019852685at_int @ M) @ (semiri2019852685at_int @ N)) => (ord_less_nat @ M @ N))))). % of_nat_less_imp_less
thf(fact_104_less__imp__of__nat__less, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (ord_less_nat @ (semiri1382578993at_nat @ M) @ (semiri1382578993at_nat @ N)))))). % less_imp_of_nat_less
thf(fact_105_less__imp__of__nat__less, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (ord_less_int @ (semiri2019852685at_int @ M) @ (semiri2019852685at_int @ N)))))). % less_imp_of_nat_less
thf(fact_106_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_107_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_108_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_109_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_110_liftn__0, axiom,
    ((![T : dB, K : nat]: ((liftn @ zero_zero_nat @ T @ K) = T)))). % liftn_0
thf(fact_111_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_112_listsp_OCons, axiom,
    ((![A5 : dB > $o, A : dB, L : list_dB]: ((A5 @ A) => ((listsp_dB @ A5 @ L) => (listsp_dB @ A5 @ (cons_dB @ A @ L))))))). % listsp.Cons
thf(fact_113_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_114_foldl__Cons, axiom,
    ((![F : dB > dB > dB, A : dB, X : dB, Xs : list_dB]: ((foldl_dB_dB @ F @ A @ (cons_dB @ X @ Xs)) = (foldl_dB_dB @ F @ (F @ A @ X) @ Xs))))). % foldl_Cons
thf(fact_115_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_116_zero__less__imp__eq__int, axiom,
    ((![K : int]: ((ord_less_int @ zero_zero_int @ K) => (?[N2 : nat]: ((ord_less_nat @ zero_zero_nat @ N2) & (K = (semiri2019852685at_int @ N2)))))))). % zero_less_imp_eq_int
thf(fact_117_pos__int__cases, axiom,
    ((![K : int]: ((ord_less_int @ zero_zero_int @ K) => (~ ((![N2 : nat]: ((K = (semiri2019852685at_int @ N2)) => (~ ((ord_less_nat @ zero_zero_nat @ N2))))))))))). % pos_int_cases
thf(fact_118_nat__int__comparison_I2_J, axiom,
    ((ord_less_nat = (^[A4 : nat]: (^[B : nat]: (ord_less_int @ (semiri2019852685at_int @ A4) @ (semiri2019852685at_int @ B))))))). % nat_int_comparison(2)
thf(fact_119_int__ops_I1_J, axiom,
    (((semiri2019852685at_int @ zero_zero_nat) = zero_zero_int))). % int_ops(1)
thf(fact_120_verit__comp__simplify1_I1_J, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ A)))))). % verit_comp_simplify1(1)
thf(fact_121_verit__comp__simplify1_I1_J, axiom,
    ((![A : int]: (~ ((ord_less_int @ A @ A)))))). % verit_comp_simplify1(1)
thf(fact_122_neg__int__cases, axiom,
    ((![K : int]: ((ord_less_int @ K @ zero_zero_int) => (~ ((![N2 : nat]: ((K = (uminus_uminus_int @ (semiri2019852685at_int @ N2))) => (~ ((ord_less_nat @ zero_zero_nat @ N2))))))))))). % neg_int_cases
thf(fact_123_zero__less__nat__eq, axiom,
    ((![Z : int]: ((ord_less_nat @ zero_zero_nat @ (nat2 @ Z)) = (ord_less_int @ zero_zero_int @ Z))))). % zero_less_nat_eq
thf(fact_124_neg__equal__iff__equal, axiom,
    ((![A : int, B2 : int]: (((uminus_uminus_int @ A) = (uminus_uminus_int @ B2)) = (A = B2))))). % neg_equal_iff_equal
thf(fact_125_add_Oinverse__inverse, axiom,
    ((![A : int]: ((uminus_uminus_int @ (uminus_uminus_int @ A)) = A)))). % add.inverse_inverse
thf(fact_126_neg__equal__zero, axiom,
    ((![A : int]: (((uminus_uminus_int @ A) = A) = (A = zero_zero_int))))). % neg_equal_zero
thf(fact_127_equal__neg__zero, axiom,
    ((![A : int]: ((A = (uminus_uminus_int @ A)) = (A = zero_zero_int))))). % equal_neg_zero
thf(fact_128_neg__equal__0__iff__equal, axiom,
    ((![A : int]: (((uminus_uminus_int @ A) = zero_zero_int) = (A = zero_zero_int))))). % neg_equal_0_iff_equal
thf(fact_129_neg__0__equal__iff__equal, axiom,
    ((![A : int]: ((zero_zero_int = (uminus_uminus_int @ A)) = (zero_zero_int = A))))). % neg_0_equal_iff_equal
thf(fact_130_add_Oinverse__neutral, axiom,
    (((uminus_uminus_int @ zero_zero_int) = zero_zero_int))). % add.inverse_neutral
thf(fact_131_neg__less__iff__less, axiom,
    ((![B2 : int, A : int]: ((ord_less_int @ (uminus_uminus_int @ B2) @ (uminus_uminus_int @ A)) = (ord_less_int @ A @ B2))))). % neg_less_iff_less
thf(fact_132_less__neg__neg, axiom,
    ((![A : int]: ((ord_less_int @ A @ (uminus_uminus_int @ A)) = (ord_less_int @ A @ zero_zero_int))))). % less_neg_neg
thf(fact_133_neg__less__pos, axiom,
    ((![A : int]: ((ord_less_int @ (uminus_uminus_int @ A) @ A) = (ord_less_int @ zero_zero_int @ A))))). % neg_less_pos
thf(fact_134_neg__0__less__iff__less, axiom,
    ((![A : int]: ((ord_less_int @ zero_zero_int @ (uminus_uminus_int @ A)) = (ord_less_int @ A @ zero_zero_int))))). % neg_0_less_iff_less
thf(fact_135_neg__less__0__iff__less, axiom,
    ((![A : int]: ((ord_less_int @ (uminus_uminus_int @ A) @ zero_zero_int) = (ord_less_int @ zero_zero_int @ A))))). % neg_less_0_iff_less
thf(fact_136_negative__eq__positive, axiom,
    ((![N : nat, M : nat]: (((uminus_uminus_int @ (semiri2019852685at_int @ N)) = (semiri2019852685at_int @ M)) = (((N = zero_zero_nat)) & ((M = zero_zero_nat))))))). % negative_eq_positive
thf(fact_137_zless__nat__conj, axiom,
    ((![W : int, Z : int]: ((ord_less_nat @ (nat2 @ W) @ (nat2 @ Z)) = (((ord_less_int @ zero_zero_int @ Z)) & ((ord_less_int @ W @ Z))))))). % zless_nat_conj
thf(fact_138_nat__zminus__int, axiom,
    ((![N : nat]: ((nat2 @ (uminus_uminus_int @ (semiri2019852685at_int @ N))) = zero_zero_nat)))). % nat_zminus_int
thf(fact_139_verit__negate__coefficient_I2_J, axiom,
    ((![A : int, B2 : int]: ((ord_less_int @ A @ B2) => (ord_less_int @ (uminus_uminus_int @ B2) @ (uminus_uminus_int @ A)))))). % verit_negate_coefficient(2)
thf(fact_140_nat__zero__as__int, axiom,
    ((zero_zero_nat = (nat2 @ zero_zero_int)))). % nat_zero_as_int
thf(fact_141_minus__equation__iff, axiom,
    ((![A : int, B2 : int]: (((uminus_uminus_int @ A) = B2) = ((uminus_uminus_int @ B2) = A))))). % minus_equation_iff
thf(fact_142_equation__minus__iff, axiom,
    ((![A : int, B2 : int]: ((A = (uminus_uminus_int @ B2)) = (B2 = (uminus_uminus_int @ A)))))). % equation_minus_iff
thf(fact_143_minus__less__iff, axiom,
    ((![A : int, B2 : int]: ((ord_less_int @ (uminus_uminus_int @ A) @ B2) = (ord_less_int @ (uminus_uminus_int @ B2) @ A))))). % minus_less_iff
thf(fact_144_less__minus__iff, axiom,
    ((![A : int, B2 : int]: ((ord_less_int @ A @ (uminus_uminus_int @ B2)) = (ord_less_int @ B2 @ (uminus_uminus_int @ A)))))). % less_minus_iff
thf(fact_145_nat__mono__iff, axiom,
    ((![Z : int, W : int]: ((ord_less_int @ zero_zero_int @ Z) => ((ord_less_nat @ (nat2 @ W) @ (nat2 @ Z)) = (ord_less_int @ W @ Z)))))). % nat_mono_iff
thf(fact_146_zless__nat__eq__int__zless, axiom,
    ((![M : nat, Z : int]: ((ord_less_nat @ M @ (nat2 @ Z)) = (ord_less_int @ (semiri2019852685at_int @ M) @ Z))))). % zless_nat_eq_int_zless
thf(fact_147_int__cases4, axiom,
    ((![M : int]: ((![N2 : nat]: (~ ((M = (semiri2019852685at_int @ N2))))) => (~ ((![N2 : nat]: ((ord_less_nat @ zero_zero_nat @ N2) => (~ ((M = (uminus_uminus_int @ (semiri2019852685at_int @ N2))))))))))))). % int_cases4

% Conjectures (1)
thf(conj_0, conjecture,
    ((it @ (var @ n)))).
