% 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_38__5209738_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:36:59.572

% 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 (24)
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_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_Osubst, type,
    subst : 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_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_i, type,
    i : nat).
thf(sy_v_j, type,
    j : nat).
thf(sy_v_r, type,
    r : dB).

% Relevant facts (182)
thf(fact_0_subst__eq, axiom,
    ((![K : nat, U : dB]: ((subst @ (var @ K) @ U @ K) = U)))). % subst_eq
thf(fact_1_dB_Oinject_I1_J, axiom,
    ((![X1 : nat, Y1 : nat]: (((var @ X1) = (var @ Y1)) = (X1 = Y1))))). % dB.inject(1)
thf(fact_2_lift__IT, axiom,
    ((![T : dB, I : nat]: ((it @ T) => (it @ (lift @ T @ I)))))). % lift_IT
thf(fact_3_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_4_subst__lift, axiom,
    ((![T : dB, K : nat, S : dB]: ((subst @ (lift @ T @ K) @ S @ K) = T)))). % subst_lift
thf(fact_5_Lambda, axiom,
    ((![R : dB]: ((it @ R) => (it @ (abs @ R)))))). % Lambda
thf(fact_6_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_7_dB_Odistinct_I3_J, axiom,
    ((![X1 : nat, X3 : dB]: (~ (((var @ X1) = (abs @ X3))))))). % dB.distinct(3)
thf(fact_8_beta__cases_I1_J, axiom,
    ((![I : nat, T : dB]: (~ ((beta @ (var @ I) @ T)))))). % beta_cases(1)
thf(fact_9_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_10_dB_Oinject_I3_J, axiom,
    ((![X3 : dB, Y3 : dB]: (((abs @ X3) = (abs @ Y3)) = (X3 = Y3))))). % dB.inject(3)
thf(fact_11_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_12_abs, axiom,
    ((![S : dB, T : dB]: ((beta @ S @ T) => (beta @ (abs @ S) @ (abs @ T)))))). % abs
thf(fact_13_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_14_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_15_typing_OVar, axiom,
    ((![Env : nat > type, X : nat, T2 : type]: (((Env @ X) = T2) => (typing @ Env @ (var @ X) @ T2))))). % typing.Var
thf(fact_16_lift__type, axiom,
    ((![E : nat > type, T : dB, T2 : type, I : nat, U2 : type]: ((typing @ E @ T @ T2) => (typing @ (shift_type @ E @ I @ U2) @ (lift @ T @ I) @ T2))))). % lift_type
thf(fact_17_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_18_less__not__refl, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_not_refl
thf(fact_19_less__not__refl2, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ N @ M) => (~ ((M = N))))))). % less_not_refl2
thf(fact_20_less__not__refl3, axiom,
    ((![S : nat, T : nat]: ((ord_less_nat @ S @ T) => (~ ((S = T))))))). % less_not_refl3
thf(fact_21_less__irrefl__nat, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_irrefl_nat
thf(fact_22_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_23_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_24_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_25_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_26_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_27_subst__lemma, axiom,
    ((![E : nat > type, T : dB, T2 : type, E2 : nat > type, U : dB, U2 : type, I : nat]: ((typing @ E @ T @ T2) => ((typing @ E2 @ U @ U2) => ((E = (shift_type @ E2 @ I @ U2)) => (typing @ E2 @ (subst @ T @ U @ I) @ T2))))))). % subst_lemma
thf(fact_28_abs__typeE, axiom,
    ((![E : nat > type, T : dB, T2 : type]: ((typing @ E @ (abs @ T) @ T2) => (~ ((![U3 : type, V : type]: (~ ((typing @ (shift_type @ E @ zero_zero_nat @ U3) @ T @ V)))))))))). % abs_typeE
thf(fact_29_linorder__neqE__linordered__idom, axiom,
    ((![X : int, Y : int]: ((~ ((X = Y))) => ((~ ((ord_less_int @ X @ Y))) => (ord_less_int @ Y @ X)))))). % linorder_neqE_linordered_idom
thf(fact_30_dual__order_Ostrict__implies__not__eq, axiom,
    ((![B : nat, A : nat]: ((ord_less_nat @ B @ A) => (~ ((A = B))))))). % dual_order.strict_implies_not_eq
thf(fact_31_dual__order_Ostrict__implies__not__eq, axiom,
    ((![B : int, A : int]: ((ord_less_int @ B @ A) => (~ ((A = B))))))). % dual_order.strict_implies_not_eq
thf(fact_32_order_Ostrict__implies__not__eq, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((A = B))))))). % order.strict_implies_not_eq
thf(fact_33_order_Ostrict__implies__not__eq, axiom,
    ((![A : int, B : int]: ((ord_less_int @ A @ B) => (~ ((A = B))))))). % order.strict_implies_not_eq
thf(fact_34_neq0__conv, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N))))). % neq0_conv
thf(fact_35_less__nat__zero__code, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_36_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_37_gr0I, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr0I
thf(fact_38_not__gr0, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr0
thf(fact_39_not__less0, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less0
thf(fact_40_less__zeroE, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_zeroE
thf(fact_41_gr__implies__not0, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not0
thf(fact_42_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_43_bot__nat__0_Oextremum__strict, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ zero_zero_nat)))))). % bot_nat_0.extremum_strict
thf(fact_44_ord__eq__less__subst, axiom,
    ((![A : nat, F : nat > nat, B : nat, C : nat]: ((A = (F @ B)) => ((ord_less_nat @ B @ C) => ((![X2 : nat, Y2 : nat]: ((ord_less_nat @ X2 @ Y2) => (ord_less_nat @ (F @ X2) @ (F @ Y2)))) => (ord_less_nat @ A @ (F @ C)))))))). % ord_eq_less_subst
thf(fact_45_ord__eq__less__subst, axiom,
    ((![A : int, F : nat > int, B : nat, C : nat]: ((A = (F @ B)) => ((ord_less_nat @ B @ C) => ((![X2 : nat, Y2 : nat]: ((ord_less_nat @ X2 @ Y2) => (ord_less_int @ (F @ X2) @ (F @ Y2)))) => (ord_less_int @ A @ (F @ C)))))))). % ord_eq_less_subst
thf(fact_46_ord__eq__less__subst, axiom,
    ((![A : nat, F : int > nat, B : int, C : int]: ((A = (F @ B)) => ((ord_less_int @ B @ C) => ((![X2 : int, Y2 : int]: ((ord_less_int @ X2 @ Y2) => (ord_less_nat @ (F @ X2) @ (F @ Y2)))) => (ord_less_nat @ A @ (F @ C)))))))). % ord_eq_less_subst
thf(fact_47_ord__eq__less__subst, axiom,
    ((![A : int, F : int > int, B : int, C : int]: ((A = (F @ B)) => ((ord_less_int @ B @ C) => ((![X2 : int, Y2 : int]: ((ord_less_int @ X2 @ Y2) => (ord_less_int @ (F @ X2) @ (F @ Y2)))) => (ord_less_int @ A @ (F @ C)))))))). % ord_eq_less_subst
thf(fact_48_ord__less__eq__subst, axiom,
    ((![A : nat, B : nat, F : nat > nat, C : nat]: ((ord_less_nat @ A @ B) => (((F @ B) = C) => ((![X2 : nat, Y2 : nat]: ((ord_less_nat @ X2 @ Y2) => (ord_less_nat @ (F @ X2) @ (F @ Y2)))) => (ord_less_nat @ (F @ A) @ C))))))). % ord_less_eq_subst
thf(fact_49_ord__less__eq__subst, axiom,
    ((![A : nat, B : nat, F : nat > int, C : int]: ((ord_less_nat @ A @ B) => (((F @ B) = C) => ((![X2 : nat, Y2 : nat]: ((ord_less_nat @ X2 @ Y2) => (ord_less_int @ (F @ X2) @ (F @ Y2)))) => (ord_less_int @ (F @ A) @ C))))))). % ord_less_eq_subst
thf(fact_50_ord__less__eq__subst, axiom,
    ((![A : int, B : int, F : int > nat, C : nat]: ((ord_less_int @ A @ B) => (((F @ B) = C) => ((![X2 : int, Y2 : int]: ((ord_less_int @ X2 @ Y2) => (ord_less_nat @ (F @ X2) @ (F @ Y2)))) => (ord_less_nat @ (F @ A) @ C))))))). % ord_less_eq_subst
thf(fact_51_ord__less__eq__subst, axiom,
    ((![A : int, B : int, F : int > int, C : int]: ((ord_less_int @ A @ B) => (((F @ B) = C) => ((![X2 : int, Y2 : int]: ((ord_less_int @ X2 @ Y2) => (ord_less_int @ (F @ X2) @ (F @ Y2)))) => (ord_less_int @ (F @ A) @ C))))))). % ord_less_eq_subst
thf(fact_52_order__less__subst1, axiom,
    ((![A : nat, F : nat > nat, B : nat, C : nat]: ((ord_less_nat @ A @ (F @ B)) => ((ord_less_nat @ B @ C) => ((![X2 : nat, Y2 : nat]: ((ord_less_nat @ X2 @ Y2) => (ord_less_nat @ (F @ X2) @ (F @ Y2)))) => (ord_less_nat @ A @ (F @ C)))))))). % order_less_subst1
thf(fact_53_order__less__subst1, axiom,
    ((![A : nat, F : int > nat, B : int, C : int]: ((ord_less_nat @ A @ (F @ B)) => ((ord_less_int @ B @ C) => ((![X2 : int, Y2 : int]: ((ord_less_int @ X2 @ Y2) => (ord_less_nat @ (F @ X2) @ (F @ Y2)))) => (ord_less_nat @ A @ (F @ C)))))))). % order_less_subst1
thf(fact_54_order__less__subst1, axiom,
    ((![A : int, F : nat > int, B : nat, C : nat]: ((ord_less_int @ A @ (F @ B)) => ((ord_less_nat @ B @ C) => ((![X2 : nat, Y2 : nat]: ((ord_less_nat @ X2 @ Y2) => (ord_less_int @ (F @ X2) @ (F @ Y2)))) => (ord_less_int @ A @ (F @ C)))))))). % order_less_subst1
thf(fact_55_order__less__subst1, axiom,
    ((![A : int, F : int > int, B : int, C : int]: ((ord_less_int @ A @ (F @ B)) => ((ord_less_int @ B @ C) => ((![X2 : int, Y2 : int]: ((ord_less_int @ X2 @ Y2) => (ord_less_int @ (F @ X2) @ (F @ Y2)))) => (ord_less_int @ A @ (F @ C)))))))). % order_less_subst1
thf(fact_56_order__less__subst2, axiom,
    ((![A : nat, B : nat, F : nat > nat, C : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ (F @ B) @ C) => ((![X2 : nat, Y2 : nat]: ((ord_less_nat @ X2 @ Y2) => (ord_less_nat @ (F @ X2) @ (F @ Y2)))) => (ord_less_nat @ (F @ A) @ C))))))). % order_less_subst2
thf(fact_57_order__less__subst2, axiom,
    ((![A : nat, B : nat, F : nat > int, C : int]: ((ord_less_nat @ A @ B) => ((ord_less_int @ (F @ B) @ C) => ((![X2 : nat, Y2 : nat]: ((ord_less_nat @ X2 @ Y2) => (ord_less_int @ (F @ X2) @ (F @ Y2)))) => (ord_less_int @ (F @ A) @ C))))))). % order_less_subst2
thf(fact_58_order__less__subst2, axiom,
    ((![A : int, B : int, F : int > nat, C : nat]: ((ord_less_int @ A @ B) => ((ord_less_nat @ (F @ B) @ C) => ((![X2 : int, Y2 : int]: ((ord_less_int @ X2 @ Y2) => (ord_less_nat @ (F @ X2) @ (F @ Y2)))) => (ord_less_nat @ (F @ A) @ C))))))). % order_less_subst2
thf(fact_59_order__less__subst2, axiom,
    ((![A : int, B : int, F : int > int, C : int]: ((ord_less_int @ A @ B) => ((ord_less_int @ (F @ B) @ C) => ((![X2 : int, Y2 : int]: ((ord_less_int @ X2 @ Y2) => (ord_less_int @ (F @ X2) @ (F @ Y2)))) => (ord_less_int @ (F @ A) @ C))))))). % order_less_subst2
thf(fact_60_lt__ex, axiom,
    ((![X : int]: (?[Y2 : int]: (ord_less_int @ Y2 @ X))))). % lt_ex
thf(fact_61_gt__ex, axiom,
    ((![X : nat]: (?[X_1 : nat]: (ord_less_nat @ X @ X_1))))). % gt_ex
thf(fact_62_gt__ex, axiom,
    ((![X : int]: (?[X_1 : int]: (ord_less_int @ X @ X_1))))). % gt_ex
thf(fact_63_neqE, axiom,
    ((![X : nat, Y : nat]: ((~ ((X = Y))) => ((~ ((ord_less_nat @ X @ Y))) => (ord_less_nat @ Y @ X)))))). % neqE
thf(fact_64_neqE, axiom,
    ((![X : int, Y : int]: ((~ ((X = Y))) => ((~ ((ord_less_int @ X @ Y))) => (ord_less_int @ Y @ X)))))). % neqE
thf(fact_65_neq__iff, axiom,
    ((![X : nat, Y : nat]: ((~ ((X = Y))) = (((ord_less_nat @ X @ Y)) | ((ord_less_nat @ Y @ X))))))). % neq_iff
thf(fact_66_neq__iff, axiom,
    ((![X : int, Y : int]: ((~ ((X = Y))) = (((ord_less_int @ X @ Y)) | ((ord_less_int @ Y @ X))))))). % neq_iff
thf(fact_67_order_Oasym, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((ord_less_nat @ B @ A))))))). % order.asym
thf(fact_68_order_Oasym, axiom,
    ((![A : int, B : int]: ((ord_less_int @ A @ B) => (~ ((ord_less_int @ B @ A))))))). % order.asym
thf(fact_69_less__imp__neq, axiom,
    ((![X : nat, Y : nat]: ((ord_less_nat @ X @ Y) => (~ ((X = Y))))))). % less_imp_neq
thf(fact_70_less__imp__neq, axiom,
    ((![X : int, Y : int]: ((ord_less_int @ X @ Y) => (~ ((X = Y))))))). % less_imp_neq
thf(fact_71_less__asym, axiom,
    ((![X : nat, Y : nat]: ((ord_less_nat @ X @ Y) => (~ ((ord_less_nat @ Y @ X))))))). % less_asym
thf(fact_72_less__asym, axiom,
    ((![X : int, Y : int]: ((ord_less_int @ X @ Y) => (~ ((ord_less_int @ Y @ X))))))). % less_asym
thf(fact_73_less__asym_H, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((ord_less_nat @ B @ A))))))). % less_asym'
thf(fact_74_less__asym_H, axiom,
    ((![A : int, B : int]: ((ord_less_int @ A @ B) => (~ ((ord_less_int @ B @ A))))))). % less_asym'
thf(fact_75_less__trans, axiom,
    ((![X : nat, Y : nat, Z : nat]: ((ord_less_nat @ X @ Y) => ((ord_less_nat @ Y @ Z) => (ord_less_nat @ X @ Z)))))). % less_trans
thf(fact_76_less__trans, axiom,
    ((![X : int, Y : int, Z : int]: ((ord_less_int @ X @ Y) => ((ord_less_int @ Y @ Z) => (ord_less_int @ X @ Z)))))). % less_trans
thf(fact_77_less__linear, axiom,
    ((![X : nat, Y : nat]: ((ord_less_nat @ X @ Y) | ((X = Y) | (ord_less_nat @ Y @ X)))))). % less_linear
thf(fact_78_less__linear, axiom,
    ((![X : int, Y : int]: ((ord_less_int @ X @ Y) | ((X = Y) | (ord_less_int @ Y @ X)))))). % less_linear
thf(fact_79_less__irrefl, axiom,
    ((![X : nat]: (~ ((ord_less_nat @ X @ X)))))). % less_irrefl
thf(fact_80_less__irrefl, axiom,
    ((![X : int]: (~ ((ord_less_int @ X @ X)))))). % less_irrefl
thf(fact_81_ord__eq__less__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((A = B) => ((ord_less_nat @ B @ C) => (ord_less_nat @ A @ C)))))). % ord_eq_less_trans
thf(fact_82_ord__eq__less__trans, axiom,
    ((![A : int, B : int, C : int]: ((A = B) => ((ord_less_int @ B @ C) => (ord_less_int @ A @ C)))))). % ord_eq_less_trans
thf(fact_83_ord__less__eq__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => ((B = C) => (ord_less_nat @ A @ C)))))). % ord_less_eq_trans
thf(fact_84_ord__less__eq__trans, axiom,
    ((![A : int, B : int, C : int]: ((ord_less_int @ A @ B) => ((B = C) => (ord_less_int @ A @ C)))))). % ord_less_eq_trans
thf(fact_85_dual__order_Oasym, axiom,
    ((![B : nat, A : nat]: ((ord_less_nat @ B @ A) => (~ ((ord_less_nat @ A @ B))))))). % dual_order.asym
thf(fact_86_dual__order_Oasym, axiom,
    ((![B : int, A : int]: ((ord_less_int @ B @ A) => (~ ((ord_less_int @ A @ B))))))). % dual_order.asym
thf(fact_87_less__imp__not__eq, axiom,
    ((![X : nat, Y : nat]: ((ord_less_nat @ X @ Y) => (~ ((X = Y))))))). % less_imp_not_eq
thf(fact_88_less__imp__not__eq, axiom,
    ((![X : int, Y : int]: ((ord_less_int @ X @ Y) => (~ ((X = Y))))))). % less_imp_not_eq
thf(fact_89_less__not__sym, axiom,
    ((![X : nat, Y : nat]: ((ord_less_nat @ X @ Y) => (~ ((ord_less_nat @ Y @ X))))))). % less_not_sym
thf(fact_90_less__not__sym, axiom,
    ((![X : int, Y : int]: ((ord_less_int @ X @ Y) => (~ ((ord_less_int @ Y @ X))))))). % less_not_sym
thf(fact_91_less__induct, axiom,
    ((![P : nat > $o, A : nat]: ((![X2 : nat]: ((![Y4 : nat]: ((ord_less_nat @ Y4 @ X2) => (P @ Y4))) => (P @ X2))) => (P @ A))))). % less_induct
thf(fact_92_antisym__conv3, axiom,
    ((![Y : nat, X : nat]: ((~ ((ord_less_nat @ Y @ X))) => ((~ ((ord_less_nat @ X @ Y))) = (X = Y)))))). % antisym_conv3
thf(fact_93_antisym__conv3, axiom,
    ((![Y : int, X : int]: ((~ ((ord_less_int @ Y @ X))) => ((~ ((ord_less_int @ X @ Y))) = (X = Y)))))). % antisym_conv3
thf(fact_94_less__imp__not__eq2, axiom,
    ((![X : nat, Y : nat]: ((ord_less_nat @ X @ Y) => (~ ((Y = X))))))). % less_imp_not_eq2
thf(fact_95_less__imp__not__eq2, axiom,
    ((![X : int, Y : int]: ((ord_less_int @ X @ Y) => (~ ((Y = X))))))). % less_imp_not_eq2
thf(fact_96_less__imp__triv, axiom,
    ((![X : nat, Y : nat, P : $o]: ((ord_less_nat @ X @ Y) => ((ord_less_nat @ Y @ X) => P))))). % less_imp_triv
thf(fact_97_less__imp__triv, axiom,
    ((![X : int, Y : int, P : $o]: ((ord_less_int @ X @ Y) => ((ord_less_int @ Y @ X) => P))))). % less_imp_triv
thf(fact_98_linorder__cases, axiom,
    ((![X : nat, Y : nat]: ((~ ((ord_less_nat @ X @ Y))) => ((~ ((X = Y))) => (ord_less_nat @ Y @ X)))))). % linorder_cases
thf(fact_99_linorder__cases, axiom,
    ((![X : int, Y : int]: ((~ ((ord_less_int @ X @ Y))) => ((~ ((X = Y))) => (ord_less_int @ Y @ X)))))). % linorder_cases
thf(fact_100_dual__order_Oirrefl, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ A)))))). % dual_order.irrefl
thf(fact_101_dual__order_Oirrefl, axiom,
    ((![A : int]: (~ ((ord_less_int @ A @ A)))))). % dual_order.irrefl
thf(fact_102_order_Ostrict__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ B @ C) => (ord_less_nat @ A @ C)))))). % order.strict_trans
thf(fact_103_order_Ostrict__trans, axiom,
    ((![A : int, B : int, C : int]: ((ord_less_int @ A @ B) => ((ord_less_int @ B @ C) => (ord_less_int @ A @ C)))))). % order.strict_trans
thf(fact_104_less__imp__not__less, axiom,
    ((![X : nat, Y : nat]: ((ord_less_nat @ X @ Y) => (~ ((ord_less_nat @ Y @ X))))))). % less_imp_not_less
thf(fact_105_less__imp__not__less, axiom,
    ((![X : int, Y : int]: ((ord_less_int @ X @ Y) => (~ ((ord_less_int @ Y @ X))))))). % less_imp_not_less
thf(fact_106_exists__least__iff, axiom,
    (((^[P2 : nat > $o]: (?[X4 : nat]: (P2 @ X4))) = (^[P3 : nat > $o]: (?[N3 : nat]: (((P3 @ N3)) & ((![M3 : nat]: (((ord_less_nat @ M3 @ N3)) => ((~ ((P3 @ M3))))))))))))). % exists_least_iff
thf(fact_107_linorder__less__wlog, axiom,
    ((![P : nat > nat > $o, A : nat, B : nat]: ((![A2 : nat, B2 : nat]: ((ord_less_nat @ A2 @ B2) => (P @ A2 @ B2))) => ((![A2 : nat]: (P @ A2 @ A2)) => ((![A2 : nat, B2 : nat]: ((P @ B2 @ A2) => (P @ A2 @ B2))) => (P @ A @ B))))))). % linorder_less_wlog
thf(fact_108_linorder__less__wlog, axiom,
    ((![P : int > int > $o, A : int, B : int]: ((![A2 : int, B2 : int]: ((ord_less_int @ A2 @ B2) => (P @ A2 @ B2))) => ((![A2 : int]: (P @ A2 @ A2)) => ((![A2 : int, B2 : int]: ((P @ B2 @ A2) => (P @ A2 @ B2))) => (P @ A @ B))))))). % linorder_less_wlog
thf(fact_109_dual__order_Ostrict__trans, axiom,
    ((![B : nat, A : nat, C : nat]: ((ord_less_nat @ B @ A) => ((ord_less_nat @ C @ B) => (ord_less_nat @ C @ A)))))). % dual_order.strict_trans
thf(fact_110_dual__order_Ostrict__trans, axiom,
    ((![B : int, A : int, C : int]: ((ord_less_int @ B @ A) => ((ord_less_int @ C @ B) => (ord_less_int @ C @ A)))))). % dual_order.strict_trans
thf(fact_111_not__less__iff__gr__or__eq, axiom,
    ((![X : nat, Y : nat]: ((~ ((ord_less_nat @ X @ Y))) = (((ord_less_nat @ Y @ X)) | ((X = Y))))))). % not_less_iff_gr_or_eq
thf(fact_112_not__less__iff__gr__or__eq, axiom,
    ((![X : int, Y : int]: ((~ ((ord_less_int @ X @ Y))) = (((ord_less_int @ Y @ X)) | ((X = Y))))))). % not_less_iff_gr_or_eq
thf(fact_113_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_114_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_115_gr__implies__not__zero, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not_zero
thf(fact_116_not__less__zero, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less_zero
thf(fact_117_gr__zeroI, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr_zeroI
thf(fact_118_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_119_zero__reorient, axiom,
    ((![X : int]: ((zero_zero_int = X) = (X = zero_zero_int))))). % zero_reorient
thf(fact_120_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_nat @ zero_zero_nat @ zero_zero_nat))))). % less_numeral_extra(3)
thf(fact_121_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_int @ zero_zero_int @ zero_zero_int))))). % less_numeral_extra(3)
thf(fact_122_Abs, axiom,
    ((![Env : nat > type, T2 : type, T : dB, U2 : type]: ((typing @ (shift_type @ Env @ zero_zero_nat @ T2) @ T @ U2) => (typing @ Env @ (abs @ T) @ (fun @ T2 @ U2)))))). % Abs
thf(fact_123_typing__elims_I3_J, axiom,
    ((![E : nat > type, T : dB, T2 : type]: ((typing @ E @ (abs @ T) @ T2) => (~ ((![T5 : type, U3 : type]: ((T2 = (fun @ T5 @ U3)) => (~ ((typing @ (shift_type @ E @ zero_zero_nat @ T5) @ T @ U3))))))))))). % typing_elims(3)
thf(fact_124_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_125_type__induct, axiom,
    ((![P : type > $o, T2 : type]: ((![T5 : type]: ((![T1 : type, T22 : type]: ((T5 = (fun @ T1 @ T22)) => (P @ T1))) => ((![T1 : type, T22 : type]: ((T5 = (fun @ T1 @ T22)) => (P @ T22))) => (P @ T5)))) => (P @ T2))))). % type_induct
thf(fact_126_typing_Ocases, axiom,
    ((![A1 : nat > type, A22 : dB, A3 : type]: ((typing @ A1 @ A22 @ A3) => ((![X2 : nat]: ((A22 = (var @ X2)) => (~ (((A1 @ X2) = A3))))) => ((![T5 : type, T3 : dB]: ((A22 = (abs @ T3)) => (![U3 : type]: ((A3 = (fun @ T5 @ U3)) => (~ ((typing @ (shift_type @ A1 @ zero_zero_nat @ T5) @ T3 @ U3))))))) => (~ ((![S2 : dB, T5 : type, U3 : type, T3 : dB]: ((A22 = (app @ S2 @ T3)) => ((A3 = U3) => ((typing @ A1 @ S2 @ (fun @ T5 @ U3)) => (~ ((typing @ A1 @ T3 @ T5))))))))))))))). % typing.cases
thf(fact_127_typing_Osimps, axiom,
    ((typing = (^[A12 : nat > type]: (^[A23 : dB]: (^[A32 : type]: (((?[Env2 : nat > type]: (?[X5 : nat]: (?[T6 : type]: (((A12 = Env2)) & ((((A23 = (var @ X5))) & ((((A32 = T6)) & (((Env2 @ X5) = T6))))))))))) | ((((?[Env2 : nat > type]: (?[T6 : type]: (?[T7 : dB]: (?[U4 : type]: (((A12 = Env2)) & ((((A23 = (abs @ T7))) & ((((A32 = (fun @ T6 @ U4))) & ((typing @ (shift_type @ Env2 @ zero_zero_nat @ T6) @ T7 @ U4)))))))))))) | ((?[Env2 : nat > type]: (?[S3 : dB]: (?[T6 : type]: (?[U4 : type]: (?[T7 : dB]: (((A12 = Env2)) & ((((A23 = (app @ S3 @ T7))) & ((((A32 = U4)) & ((((typing @ Env2 @ S3 @ (fun @ T6 @ U4))) & ((typing @ Env2 @ T7 @ T6)))))))))))))))))))))))). % typing.simps
thf(fact_128_typing_Oinducts, axiom,
    ((![X1 : nat > type, X23 : dB, X3 : type, P : (nat > type) > dB > type > $o]: ((typing @ X1 @ X23 @ X3) => ((![Env3 : nat > type, X2 : nat, T5 : type]: (((Env3 @ X2) = T5) => (P @ Env3 @ (var @ X2) @ T5))) => ((![Env3 : nat > type, T5 : type, T3 : dB, U3 : type]: ((typing @ (shift_type @ Env3 @ zero_zero_nat @ T5) @ T3 @ U3) => ((P @ (shift_type @ Env3 @ zero_zero_nat @ T5) @ T3 @ U3) => (P @ Env3 @ (abs @ T3) @ (fun @ T5 @ U3))))) => ((![Env3 : nat > type, S2 : dB, T5 : type, U3 : type, T3 : dB]: ((typing @ Env3 @ S2 @ (fun @ T5 @ U3)) => ((P @ Env3 @ S2 @ (fun @ T5 @ U3)) => ((typing @ Env3 @ T3 @ T5) => ((P @ Env3 @ T3 @ T5) => (P @ Env3 @ (app @ S2 @ T3) @ U3)))))) => (P @ X1 @ X23 @ X3)))))))). % typing.inducts
thf(fact_129_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_130_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_131_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_132_dB_Odistinct_I1_J, axiom,
    ((![X1 : nat, X21 : dB, X22 : dB]: (~ (((var @ X1) = (app @ X21 @ X22))))))). % dB.distinct(1)
thf(fact_133_dB_Odistinct_I5_J, axiom,
    ((![X21 : dB, X22 : dB, X3 : dB]: (~ (((app @ X21 @ X22) = (abs @ X3))))))). % dB.distinct(5)
thf(fact_134_appL, axiom,
    ((![S : dB, T : dB, U : dB]: ((beta @ S @ T) => (beta @ (app @ S @ U) @ (app @ T @ U)))))). % appL
thf(fact_135_appR, axiom,
    ((![S : dB, T : dB, U : dB]: ((beta @ S @ T) => (beta @ (app @ U @ S) @ (app @ U @ T)))))). % appR
thf(fact_136_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_137_dB_Oinduct, axiom,
    ((![P : dB > $o, DB : dB]: ((![X2 : nat]: (P @ (var @ X2))) => ((![X1a : dB, X24 : dB]: ((P @ X1a) => ((P @ X24) => (P @ (app @ X1a @ X24))))) => ((![X2 : dB]: ((P @ X2) => (P @ (abs @ X2)))) => (P @ DB))))))). % dB.induct
thf(fact_138_typing__elims_I2_J, axiom,
    ((![E : nat > type, T : dB, U : dB, T2 : type]: ((typing @ E @ (app @ T @ U) @ T2) => (~ ((![T5 : type]: ((typing @ E @ T @ (fun @ T5 @ T2)) => (~ ((typing @ E @ U @ T5))))))))))). % typing_elims(2)
thf(fact_139_App, axiom,
    ((![Env : nat > type, S : dB, T2 : type, U2 : type, T : dB]: ((typing @ Env @ S @ (fun @ T2 @ U2)) => ((typing @ Env @ T @ T2) => (typing @ Env @ (app @ S @ T) @ U2)))))). % App
thf(fact_140_beta_Oinducts, axiom,
    ((![X1 : dB, X23 : dB, P : dB > dB > $o]: ((beta @ X1 @ X23) => ((![S2 : dB, T3 : dB]: (P @ (app @ (abs @ S2) @ T3) @ (subst @ S2 @ T3 @ zero_zero_nat))) => ((![S2 : dB, T3 : dB, U5 : dB]: ((beta @ S2 @ T3) => ((P @ S2 @ T3) => (P @ (app @ S2 @ U5) @ (app @ T3 @ U5))))) => ((![S2 : dB, T3 : dB, U5 : dB]: ((beta @ S2 @ T3) => ((P @ S2 @ T3) => (P @ (app @ U5 @ S2) @ (app @ U5 @ T3))))) => ((![S2 : dB, T3 : dB]: ((beta @ S2 @ T3) => ((P @ S2 @ T3) => (P @ (abs @ S2) @ (abs @ T3))))) => (P @ X1 @ X23))))))))). % beta.inducts
thf(fact_141_beta_Osimps, axiom,
    ((beta = (^[A12 : dB]: (^[A23 : dB]: (((?[S3 : dB]: (?[T7 : dB]: (((A12 = (app @ (abs @ S3) @ T7))) & ((A23 = (subst @ S3 @ T7 @ zero_zero_nat))))))) | ((((?[S3 : dB]: (?[T7 : dB]: (?[U6 : dB]: (((A12 = (app @ S3 @ U6))) & ((((A23 = (app @ T7 @ U6))) & ((beta @ S3 @ T7))))))))) | ((((?[S3 : dB]: (?[T7 : dB]: (?[U6 : dB]: (((A12 = (app @ U6 @ S3))) & ((((A23 = (app @ U6 @ T7))) & ((beta @ S3 @ T7))))))))) | ((?[S3 : dB]: (?[T7 : dB]: (((A12 = (abs @ S3))) & ((((A23 = (abs @ T7))) & ((beta @ S3 @ T7)))))))))))))))))). % beta.simps
thf(fact_142_beta_Ocases, axiom,
    ((![A1 : dB, A22 : dB]: ((beta @ A1 @ A22) => ((![S2 : dB, T3 : dB]: ((A1 = (app @ (abs @ S2) @ T3)) => (~ ((A22 = (subst @ S2 @ T3 @ zero_zero_nat)))))) => ((![S2 : dB, T3 : dB, U5 : dB]: ((A1 = (app @ S2 @ U5)) => ((A22 = (app @ T3 @ U5)) => (~ ((beta @ S2 @ T3)))))) => ((![S2 : dB, T3 : dB, U5 : dB]: ((A1 = (app @ U5 @ S2)) => ((A22 = (app @ U5 @ T3)) => (~ ((beta @ S2 @ T3)))))) => (~ ((![S2 : dB]: ((A1 = (abs @ S2)) => (![T3 : dB]: ((A22 = (abs @ T3)) => (~ ((beta @ S2 @ T3)))))))))))))))). % beta.cases
thf(fact_143_beta, axiom,
    ((![S : dB, T : dB]: (beta @ (app @ (abs @ S) @ T) @ (subst @ S @ T @ zero_zero_nat))))). % beta
thf(fact_144_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_145_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_146_dB_Osize__gen_I1_J, axiom,
    ((![X1 : nat]: ((size_dB @ (var @ X1)) = zero_zero_nat)))). % dB.size_gen(1)
thf(fact_147_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_148_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_149_of__nat__eq__iff, axiom,
    ((![M : nat, N : nat]: (((semiri2019852685at_int @ M) = (semiri2019852685at_int @ N)) = (M = N))))). % of_nat_eq_iff
thf(fact_150_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_151_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_152_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_153_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_154_of__nat__0, axiom,
    (((semiri1382578993at_nat @ zero_zero_nat) = zero_zero_nat))). % of_nat_0
thf(fact_155_of__nat__0, axiom,
    (((semiri2019852685at_int @ zero_zero_nat) = zero_zero_int))). % of_nat_0
thf(fact_156_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_157_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_158_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_159_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_160_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_161_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_162_of__nat__less__0__iff, axiom,
    ((![M : nat]: (~ ((ord_less_nat @ (semiri1382578993at_nat @ M) @ zero_zero_nat)))))). % of_nat_less_0_iff
thf(fact_163_of__nat__less__0__iff, axiom,
    ((![M : nat]: (~ ((ord_less_int @ (semiri2019852685at_int @ M) @ zero_zero_int)))))). % of_nat_less_0_iff
thf(fact_164_var__app__type__eq, axiom,
    ((![E : nat > type, I : nat, Ts : list_dB, T2 : type, U2 : type]: ((typing @ E @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ T2) => ((typing @ E @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ U2) => (T2 = U2)))))). % var_app_type_eq
thf(fact_165_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_166_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_167_ex__head__tail, axiom,
    ((![T : dB]: (?[Ts2 : list_dB, H : dB]: ((T = (foldl_dB_dB @ app @ H @ Ts2)) & ((?[N2 : nat]: (H = (var @ N2))) | (?[U5 : dB]: (H = (abs @ U5))))))))). % ex_head_tail
thf(fact_168_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_169_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_170_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_171_IT_Osimps, axiom,
    ((it = (^[A4 : dB]: (((?[Rs2 : list_dB]: (?[N3 : nat]: (((A4 = (foldl_dB_dB @ app @ (var @ N3) @ Rs2))) & ((listsp_dB @ it @ Rs2)))))) | ((((?[R2 : dB]: (((A4 = (abs @ R2))) & ((it @ R2))))) | ((?[R2 : dB]: (?[S3 : dB]: (?[Ss2 : list_dB]: (((A4 = (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_172_IT_OVar, axiom,
    ((![Rs : list_dB, N : nat]: ((listsp_dB @ it @ Rs) => (it @ (foldl_dB_dB @ app @ (var @ N) @ Rs)))))). % IT.Var
thf(fact_173_IT_Ocases, axiom,
    ((![A : dB]: ((it @ A) => ((![Rs3 : list_dB]: ((?[N2 : nat]: (A = (foldl_dB_dB @ app @ (var @ N2) @ Rs3))) => (~ ((listsp_dB @ it @ Rs3))))) => ((![R3 : dB]: ((A = (abs @ R3)) => (~ ((it @ R3))))) => (~ ((![R3 : dB, S2 : dB, Ss3 : list_dB]: ((A = (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_174_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_175_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_176_nat__int__comparison_I2_J, axiom,
    ((ord_less_nat = (^[A4 : nat]: (^[B3 : nat]: (ord_less_int @ (semiri2019852685at_int @ A4) @ (semiri2019852685at_int @ B3))))))). % nat_int_comparison(2)
thf(fact_177_int__ops_I1_J, axiom,
    (((semiri2019852685at_int @ zero_zero_nat) = zero_zero_int))). % int_ops(1)
thf(fact_178_verit__comp__simplify1_I1_J, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ A)))))). % verit_comp_simplify1(1)
thf(fact_179_verit__comp__simplify1_I1_J, axiom,
    ((![A : int]: (~ ((ord_less_int @ A @ A)))))). % verit_comp_simplify1(1)
thf(fact_180_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_181_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

% Conjectures (2)
thf(conj_0, hypothesis,
    ((it @ r))).
thf(conj_1, conjecture,
    ((it @ (subst @ r @ (var @ i) @ j)))).
