% 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_217__5211728_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:38:27.485

% Could-be-implicit typings (6)
thf(ty_n_t__List__Olist_It__LambdaType__Otype_J, type,
    list_type : $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).
thf(ty_n_t__Int__Oint, type,
    int : $tType).

% Explicit typings (33)
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_LambdaType_Otypings, type,
    typings : (nat > type) > list_dB > list_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_Osubst, type,
    subst : 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_Ofoldr_001t__LambdaType__Otype_001t__LambdaType__Otype, type,
    foldr_type_type : (type > type > type) > list_type > type > type).
thf(sy_c_List_Olist_Omap_001t__Lambda__OdB_001t__Lambda__OdB, type,
    map_dB_dB : (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_T_H1____, type,
    t_1 : type).
thf(sy_v_T_H____, type,
    t : type).
thf(sy_v_T____, type,
    t2 : type).
thf(sy_v_e1____, type,
    e1 : nat > type).
thf(sy_v_e____, type,
    e : nat > type).
thf(sy_v_i1____, type,
    i1 : nat).
thf(sy_v_i____, type,
    i : nat).
thf(sy_v_r____, type,
    r : dB).
thf(sy_v_t____, type,
    t3 : dB).
thf(sy_v_u1____, type,
    u1 : dB).
thf(sy_v_u____, type,
    u : dB).

% Relevant facts (140)
thf(fact_0__092_060open_062IT_At_092_060close_062, axiom,
    ((it @ t3))). % \<open>IT t\<close>
thf(fact_1_Lambda_Oprems_I2_J, axiom,
    ((it @ u1))). % Lambda.prems(2)
thf(fact_2_uIT, axiom,
    ((it @ u))). % uIT
thf(fact_3_uT, axiom,
    ((typing @ e @ u @ t2))). % uT
thf(fact_4_Lambda_Ohyps_I1_J, axiom,
    ((it @ r))). % Lambda.hyps(1)
thf(fact_5__092_060open_062e_092_060langle_062i_058T_092_060rangle_062_A_092_060turnstile_062_AAbs_Ar_A_058_AT_H_092_060close_062, axiom,
    ((typing @ (shift_type @ e @ i @ t2) @ (abs @ r) @ t))). % \<open>e\<langle>i:T\<rangle> \<turnstile> Abs r : T'\<close>
thf(fact_6_dB_Oinject_I3_J, axiom,
    ((![X3 : dB, Y3 : dB]: (((abs @ X3) = (abs @ Y3)) = (X3 = Y3))))). % dB.inject(3)
thf(fact_7_IT_OLambda, axiom,
    ((![R : dB]: ((it @ R) => (it @ (abs @ R)))))). % IT.Lambda
thf(fact_8_Lambda_Ohyps_I2_J, axiom,
    ((![E : nat > type, I : nat, T : type, U : dB]: ((typing @ (shift_type @ E @ I @ t2) @ r @ T) => ((it @ U) => ((typing @ E @ U @ t2) => (it @ (subst @ r @ U @ I)))))))). % Lambda.hyps(2)
thf(fact_9_MI1, axiom,
    ((![T1 : type, T2 : type, T3 : dB, E : nat > type, I : nat, T4 : type, U : dB]: ((t2 = (fun @ T1 @ T2)) => ((it @ T3) => ((typing @ (shift_type @ E @ I @ T1) @ T3 @ T4) => ((it @ U) => ((typing @ E @ U @ T1) => (it @ (subst @ T3 @ U @ I)))))))))). % MI1
thf(fact_10_MI2, axiom,
    ((![T1 : type, T2 : type, T3 : dB, E : nat > type, I : nat, T4 : type, U : dB]: ((t2 = (fun @ T1 @ T2)) => ((it @ T3) => ((typing @ (shift_type @ E @ I @ T2) @ T3 @ T4) => ((it @ U) => ((typing @ E @ U @ T2) => (it @ (subst @ T3 @ U @ I)))))))))). % MI2
thf(fact_11_subst__Var__IT, axiom,
    ((![R : dB, I : nat, J : nat]: ((it @ R) => (it @ (subst @ R @ (var @ I) @ J)))))). % subst_Var_IT
thf(fact_12_lift__IT, axiom,
    ((![T3 : dB, I : nat]: ((it @ T3) => (it @ (lift @ T3 @ I)))))). % lift_IT
thf(fact_13_Var__IT, axiom,
    ((![N : nat]: (it @ (var @ N))))). % Var_IT
thf(fact_14_subst__lemma, axiom,
    ((![E : nat > type, T3 : dB, T4 : type, E2 : nat > type, U : dB, U2 : type, I : nat]: ((typing @ E @ T3 @ T4) => ((typing @ E2 @ U @ U2) => ((E = (shift_type @ E2 @ I @ U2)) => (typing @ E2 @ (subst @ T3 @ U @ I) @ T4))))))). % subst_lemma
thf(fact_15_Lambda_Oprems_I1_J, axiom,
    ((typing @ (shift_type @ e1 @ i1 @ t2) @ (abs @ r) @ t_1))). % Lambda.prems(1)
thf(fact_16_Lambda_Oprems_I3_J, axiom,
    ((typing @ e1 @ u1 @ t2))). % Lambda.prems(3)
thf(fact_17_dB_Oinject_I1_J, axiom,
    ((![X1 : nat, Y1 : nat]: (((var @ X1) = (var @ Y1)) = (X1 = Y1))))). % dB.inject(1)
thf(fact_18_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_19_shift__eq, axiom,
    ((![I : nat, J : nat, E : nat > type, T4 : type]: ((I = J) => ((shift_type @ E @ I @ T4 @ J) = T4))))). % shift_eq
thf(fact_20_subst__eq, axiom,
    ((![K : nat, U : dB]: ((subst @ (var @ K) @ U @ K) = U)))). % subst_eq
thf(fact_21_subst__lift, axiom,
    ((![T3 : dB, K : nat, S : dB]: ((subst @ (lift @ T3 @ K) @ S @ K) = T3)))). % subst_lift
thf(fact_22_lift__type, axiom,
    ((![E : nat > type, T3 : dB, T4 : type, I : nat, U2 : type]: ((typing @ E @ T3 @ T4) => (typing @ (shift_type @ E @ I @ U2) @ (lift @ T3 @ I) @ T4))))). % lift_type
thf(fact_23_typing__elims_I1_J, axiom,
    ((![E : nat > type, I : nat, T4 : type]: ((typing @ E @ (var @ I) @ T4) => ((E @ I) = T4))))). % typing_elims(1)
thf(fact_24_typing_OVar, axiom,
    ((![Env : nat > type, X : nat, T4 : type]: (((Env @ X) = T4) => (typing @ Env @ (var @ X) @ T4))))). % typing.Var
thf(fact_25_type__induct, axiom,
    ((![P : type > $o, T4 : type]: ((![T5 : type]: ((![T12 : type, T22 : type]: ((T5 = (fun @ T12 @ T22)) => (P @ T12))) => ((![T12 : type, T22 : type]: ((T5 = (fun @ T12 @ T22)) => (P @ T22))) => (P @ T5)))) => (P @ T4))))). % type_induct
thf(fact_26_dB_Odistinct_I3_J, axiom,
    ((![X1 : nat, X3 : dB]: (~ (((var @ X1) = (abs @ X3))))))). % dB.distinct(3)
thf(fact_27_typing__elims_I3_J, axiom,
    ((![E : nat > type, T3 : dB, T4 : type]: ((typing @ E @ (abs @ T3) @ T4) => (~ ((![T5 : type, U3 : type]: ((T4 = (fun @ T5 @ U3)) => (~ ((typing @ (shift_type @ E @ zero_zero_nat @ T5) @ T3 @ U3))))))))))). % typing_elims(3)
thf(fact_28_Abs, axiom,
    ((![Env : nat > type, T4 : type, T3 : dB, U2 : type]: ((typing @ (shift_type @ Env @ zero_zero_nat @ T4) @ T3 @ U2) => (typing @ Env @ (abs @ T3) @ (fun @ T4 @ U2)))))). % Abs
thf(fact_29_abs__typeE, axiom,
    ((![E : nat > type, T3 : dB, T4 : type]: ((typing @ E @ (abs @ T3) @ T4) => (~ ((![U3 : type, V : type]: (~ ((typing @ (shift_type @ E @ zero_zero_nat @ U3) @ T3 @ V)))))))))). % abs_typeE
thf(fact_30_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_31_typing_Ocases, axiom,
    ((![A1 : nat > type, A2 : dB, A3 : type]: ((typing @ A1 @ A2 @ A3) => ((![X2 : nat]: ((A2 = (var @ X2)) => (~ (((A1 @ X2) = A3))))) => ((![T5 : type, T6 : dB]: ((A2 = (abs @ T6)) => (![U3 : type]: ((A3 = (fun @ T5 @ U3)) => (~ ((typing @ (shift_type @ A1 @ zero_zero_nat @ T5) @ T6 @ U3))))))) => (~ ((![S2 : dB, T5 : type, U3 : type, T6 : dB]: ((A2 = (app @ S2 @ T6)) => ((A3 = U3) => ((typing @ A1 @ S2 @ (fun @ T5 @ U3)) => (~ ((typing @ A1 @ T6 @ T5))))))))))))))). % typing.cases
thf(fact_32_typing_Osimps, axiom,
    ((typing = (^[A12 : nat > type]: (^[A22 : dB]: (^[A32 : type]: (((?[Env2 : nat > type]: (?[X4 : nat]: (?[T7 : type]: (((A12 = Env2)) & ((((A22 = (var @ X4))) & ((((A32 = T7)) & (((Env2 @ X4) = T7))))))))))) | ((((?[Env2 : nat > type]: (?[T7 : type]: (?[T8 : dB]: (?[U4 : type]: (((A12 = Env2)) & ((((A22 = (abs @ T8))) & ((((A32 = (fun @ T7 @ U4))) & ((typing @ (shift_type @ Env2 @ zero_zero_nat @ T7) @ T8 @ U4)))))))))))) | ((?[Env2 : nat > type]: (?[S3 : dB]: (?[T7 : type]: (?[U4 : type]: (?[T8 : dB]: (((A12 = Env2)) & ((((A22 = (app @ S3 @ T8))) & ((((A32 = U4)) & ((((typing @ Env2 @ S3 @ (fun @ T7 @ U4))) & ((typing @ Env2 @ T8 @ T7)))))))))))))))))))))))). % typing.simps
thf(fact_33_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, T6 : dB, U3 : type]: ((typing @ (shift_type @ Env3 @ zero_zero_nat @ T5) @ T6 @ U3) => ((P @ (shift_type @ Env3 @ zero_zero_nat @ T5) @ T6 @ U3) => (P @ Env3 @ (abs @ T6) @ (fun @ T5 @ U3))))) => ((![Env3 : nat > type, S2 : dB, T5 : type, U3 : type, T6 : dB]: ((typing @ Env3 @ S2 @ (fun @ T5 @ U3)) => ((P @ Env3 @ S2 @ (fun @ T5 @ U3)) => ((typing @ Env3 @ T6 @ T5) => ((P @ Env3 @ T6 @ T5) => (P @ Env3 @ (app @ S2 @ T6) @ U3)))))) => (P @ X1 @ X23 @ X3)))))))). % typing.inducts
thf(fact_34_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_35_shift__gt, axiom,
    ((![J : nat, I : nat, E : nat > type, T4 : type]: ((ord_less_nat @ J @ I) => ((shift_type @ E @ I @ T4 @ J) = (E @ J)))))). % shift_gt
thf(fact_36_dB_Odistinct_I5_J, axiom,
    ((![X21 : dB, X22 : dB, X3 : dB]: (~ (((app @ X21 @ X22) = (abs @ X3))))))). % dB.distinct(5)
thf(fact_37_dB_Odistinct_I1_J, axiom,
    ((![X1 : nat, X21 : dB, X22 : dB]: (~ (((var @ X1) = (app @ X21 @ X22))))))). % dB.distinct(1)
thf(fact_38_subst__App, axiom,
    ((![T3 : dB, U : dB, S : dB, K : nat]: ((subst @ (app @ T3 @ U) @ S @ K) = (app @ (subst @ T3 @ S @ K) @ (subst @ U @ S @ K)))))). % subst_App
thf(fact_39_lift_Osimps_I2_J, axiom,
    ((![S : dB, T3 : dB, K : nat]: ((lift @ (app @ S @ T3) @ K) = (app @ (lift @ S @ K) @ (lift @ T3 @ K)))))). % lift.simps(2)
thf(fact_40_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_41_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_42_app__Var__IT, axiom,
    ((![T3 : dB, I : nat]: ((it @ T3) => (it @ (app @ T3 @ (var @ I))))))). % app_Var_IT
thf(fact_43_App, axiom,
    ((![Env : nat > type, S : dB, T4 : type, U2 : type, T3 : dB]: ((typing @ Env @ S @ (fun @ T4 @ U2)) => ((typing @ Env @ T3 @ T4) => (typing @ Env @ (app @ S @ T3) @ U2)))))). % App
thf(fact_44_typing__elims_I2_J, axiom,
    ((![E : nat > type, T3 : dB, U : dB, T4 : type]: ((typing @ E @ (app @ T3 @ U) @ T4) => (~ ((![T5 : type]: ((typing @ E @ T3 @ (fun @ T5 @ T4)) => (~ ((typing @ E @ U @ T5))))))))))). % typing_elims(2)
thf(fact_45_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_46_less__nat__zero__code, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_47_neq0__conv, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N))))). % neq0_conv
thf(fact_48_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_49_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_50_zero__reorient, axiom,
    ((![X : int]: ((zero_zero_int = X) = (X = zero_zero_int))))). % zero_reorient
thf(fact_51_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_52_less__not__refl, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_not_refl
thf(fact_53_less__not__refl2, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ N @ M) => (~ ((M = N))))))). % less_not_refl2
thf(fact_54_less__not__refl3, axiom,
    ((![S : nat, T3 : nat]: ((ord_less_nat @ S @ T3) => (~ ((S = T3))))))). % less_not_refl3
thf(fact_55_less__irrefl__nat, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_irrefl_nat
thf(fact_56_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_57_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_58_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_59_gr__zeroI, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr_zeroI
thf(fact_60_not__less__zero, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less_zero
thf(fact_61_gr__implies__not__zero, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not_zero
thf(fact_62_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_63_gr0I, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr0I
thf(fact_64_not__gr0, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr0
thf(fact_65_not__less0, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less0
thf(fact_66_less__zeroE, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_zeroE
thf(fact_67_gr__implies__not0, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not0
thf(fact_68_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_69_bot__nat__0_Oextremum__strict, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ zero_zero_nat)))))). % bot_nat_0.extremum_strict
thf(fact_70_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_nat @ zero_zero_nat @ zero_zero_nat))))). % less_numeral_extra(3)
thf(fact_71_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_int @ zero_zero_int @ zero_zero_int))))). % less_numeral_extra(3)
thf(fact_72_dB_Osize__gen_I1_J, axiom,
    ((![X1 : nat]: ((size_dB @ (var @ X1)) = zero_zero_nat)))). % dB.size_gen(1)
thf(fact_73_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_74_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_75_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_76_of__nat__eq__iff, axiom,
    ((![M : nat, N : nat]: (((semiri2019852685at_int @ M) = (semiri2019852685at_int @ N)) = (M = N))))). % of_nat_eq_iff
thf(fact_77_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_78_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_79_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_80_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_81_of__nat__0, axiom,
    (((semiri1382578993at_nat @ zero_zero_nat) = zero_zero_nat))). % of_nat_0
thf(fact_82_of__nat__0, axiom,
    (((semiri2019852685at_int @ zero_zero_nat) = zero_zero_int))). % of_nat_0
thf(fact_83_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_84_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_85_of__nat__less__0__iff, axiom,
    ((![M : nat]: (~ ((ord_less_nat @ (semiri1382578993at_nat @ M) @ zero_zero_nat)))))). % of_nat_less_0_iff
thf(fact_86_of__nat__less__0__iff, axiom,
    ((![M : nat]: (~ ((ord_less_int @ (semiri2019852685at_int @ M) @ zero_zero_int)))))). % of_nat_less_0_iff
thf(fact_87_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_88_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_89_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_90_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_91_var__app__type__eq, axiom,
    ((![E : nat > type, I : nat, Ts : list_dB, T4 : type, U2 : type]: ((typing @ E @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ T4) => ((typing @ E @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ U2) => (T4 = U2)))))). % var_app_type_eq
thf(fact_92_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_93_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_94_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_95_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_96_Abs__App__neq__Var__apps, axiom,
    ((![S : dB, T3 : dB, N : nat, Ss : list_dB]: (~ (((app @ (abs @ S) @ T3) = (foldl_dB_dB @ app @ (var @ N) @ Ss))))))). % Abs_App_neq_Var_apps
thf(fact_97_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_98_IT_Oinducts, axiom,
    ((![X : dB, P : dB > $o]: ((it @ X) => ((![Rs2 : list_dB, N2 : nat]: ((listsp_dB @ (^[X4 : dB]: (((it @ X4)) & ((P @ X4)))) @ Rs2) => (P @ (foldl_dB_dB @ app @ (var @ N2) @ Rs2)))) => ((![R2 : dB]: ((it @ R2) => ((P @ R2) => (P @ (abs @ R2))))) => ((![R2 : dB, S2 : dB, Ss2 : list_dB]: ((it @ (foldl_dB_dB @ app @ (subst @ R2 @ S2 @ zero_zero_nat) @ Ss2)) => ((P @ (foldl_dB_dB @ app @ (subst @ R2 @ S2 @ zero_zero_nat) @ Ss2)) => ((it @ S2) => ((P @ S2) => (P @ (foldl_dB_dB @ app @ (app @ (abs @ R2) @ S2) @ Ss2))))))) => (P @ X)))))))). % IT.inducts
thf(fact_99_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_100_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_101_IT_OVar, axiom,
    ((![Rs : list_dB, N : nat]: ((listsp_dB @ it @ Rs) => (it @ (foldl_dB_dB @ app @ (var @ N) @ Rs)))))). % IT.Var
thf(fact_102_listsp__conj__eq, axiom,
    ((![A5 : dB > $o, B : dB > $o]: ((listsp_dB @ (^[X4 : dB]: (((A5 @ X4)) & ((B @ X4))))) = (^[X4 : list_dB]: (((listsp_dB @ A5 @ X4)) & ((listsp_dB @ B @ X4)))))))). % listsp_conj_eq
thf(fact_103_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_104_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_105_nat__less__as__int, axiom,
    ((ord_less_nat = (^[A4 : nat]: (^[B2 : nat]: (ord_less_int @ (semiri2019852685at_int @ A4) @ (semiri2019852685at_int @ B2))))))). % nat_less_as_int
thf(fact_106_nat__int__comparison_I2_J, axiom,
    ((ord_less_nat = (^[A4 : nat]: (^[B2 : nat]: (ord_less_int @ (semiri2019852685at_int @ A4) @ (semiri2019852685at_int @ B2))))))). % nat_int_comparison(2)
thf(fact_107_verit__comp__simplify1_I1_J, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ A)))))). % verit_comp_simplify1(1)
thf(fact_108_verit__comp__simplify1_I1_J, axiom,
    ((![A : int]: (~ ((ord_less_int @ A @ A)))))). % verit_comp_simplify1(1)
thf(fact_109_int__ops_I1_J, axiom,
    (((semiri2019852685at_int @ zero_zero_nat) = zero_zero_int))). % int_ops(1)
thf(fact_110_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_111_lifts__IT, axiom,
    ((![Ts : list_dB]: ((listsp_dB @ it @ Ts) => (listsp_dB @ it @ (map_dB_dB @ (^[T8 : dB]: (lift @ T8 @ zero_zero_nat)) @ Ts)))))). % lifts_IT
thf(fact_112_neg__equal__iff__equal, axiom,
    ((![A : int, B3 : int]: (((uminus_uminus_int @ A) = (uminus_uminus_int @ B3)) = (A = B3))))). % neg_equal_iff_equal
thf(fact_113_add_Oinverse__inverse, axiom,
    ((![A : int]: ((uminus_uminus_int @ (uminus_uminus_int @ A)) = A)))). % add.inverse_inverse
thf(fact_114_map__ident, axiom,
    (((map_dB_dB @ (^[X4 : dB]: X4)) = (^[Xs : list_dB]: Xs)))). % map_ident
thf(fact_115_neg__equal__zero, axiom,
    ((![A : int]: (((uminus_uminus_int @ A) = A) = (A = zero_zero_int))))). % neg_equal_zero
thf(fact_116_equal__neg__zero, axiom,
    ((![A : int]: ((A = (uminus_uminus_int @ A)) = (A = zero_zero_int))))). % equal_neg_zero
thf(fact_117_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_118_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_119_add_Oinverse__neutral, axiom,
    (((uminus_uminus_int @ zero_zero_int) = zero_zero_int))). % add.inverse_neutral
thf(fact_120_neg__less__iff__less, axiom,
    ((![B3 : int, A : int]: ((ord_less_int @ (uminus_uminus_int @ B3) @ (uminus_uminus_int @ A)) = (ord_less_int @ A @ B3))))). % neg_less_iff_less
thf(fact_121_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_122_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_123_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_124_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_125_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_126_subst__map, axiom,
    ((![T3 : dB, Ts : list_dB, U : dB, I : nat]: ((subst @ (foldl_dB_dB @ app @ T3 @ Ts) @ U @ I) = (foldl_dB_dB @ app @ (subst @ T3 @ U @ I) @ (map_dB_dB @ (^[T8 : dB]: (subst @ T8 @ U @ I)) @ Ts)))))). % subst_map
thf(fact_127_lift__map, axiom,
    ((![T3 : dB, Ts : list_dB, I : nat]: ((lift @ (foldl_dB_dB @ app @ T3 @ Ts) @ I) = (foldl_dB_dB @ app @ (lift @ T3 @ I) @ (map_dB_dB @ (^[T8 : dB]: (lift @ T8 @ I)) @ Ts)))))). % lift_map
thf(fact_128_verit__negate__coefficient_I2_J, axiom,
    ((![A : int, B3 : int]: ((ord_less_int @ A @ B3) => (ord_less_int @ (uminus_uminus_int @ B3) @ (uminus_uminus_int @ A)))))). % verit_negate_coefficient(2)
thf(fact_129_minus__equation__iff, axiom,
    ((![A : int, B3 : int]: (((uminus_uminus_int @ A) = B3) = ((uminus_uminus_int @ B3) = A))))). % minus_equation_iff
thf(fact_130_equation__minus__iff, axiom,
    ((![A : int, B3 : int]: ((A = (uminus_uminus_int @ B3)) = (B3 = (uminus_uminus_int @ A)))))). % equation_minus_iff
thf(fact_131_minus__less__iff, axiom,
    ((![A : int, B3 : int]: ((ord_less_int @ (uminus_uminus_int @ A) @ B3) = (ord_less_int @ (uminus_uminus_int @ B3) @ A))))). % minus_less_iff
thf(fact_132_less__minus__iff, axiom,
    ((![A : int, B3 : int]: ((ord_less_int @ A @ (uminus_uminus_int @ B3)) = (ord_less_int @ B3 @ (uminus_uminus_int @ A)))))). % less_minus_iff
thf(fact_133_list_Omap__ident, axiom,
    ((![T3 : list_dB]: ((map_dB_dB @ (^[X4 : dB]: X4) @ T3) = T3)))). % list.map_ident
thf(fact_134_foldl__map, axiom,
    ((![G : dB > dB > dB, A : dB, F : dB > dB, Xs2 : list_dB]: ((foldl_dB_dB @ G @ A @ (map_dB_dB @ F @ Xs2)) = (foldl_dB_dB @ (^[A4 : dB]: (^[X4 : dB]: (G @ A4 @ (F @ X4)))) @ A @ Xs2))))). % foldl_map
thf(fact_135_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
thf(fact_136_int__cases3, axiom,
    ((![K : int]: ((~ ((K = zero_zero_int))) => ((![N2 : nat]: ((K = (semiri2019852685at_int @ N2)) => (~ ((ord_less_nat @ zero_zero_nat @ N2))))) => (~ ((![N2 : nat]: ((K = (uminus_uminus_int @ (semiri2019852685at_int @ N2))) => (~ ((ord_less_nat @ zero_zero_nat @ N2)))))))))))). % int_cases3
thf(fact_137_substs__lemma, axiom,
    ((![E : nat > type, U : dB, T4 : type, I : nat, Ts : list_dB, Ts3 : list_type]: ((typing @ E @ U @ T4) => ((typings @ (shift_type @ E @ I @ T4) @ Ts @ Ts3) => (typings @ E @ (map_dB_dB @ (^[T8 : dB]: (subst @ T8 @ U @ I)) @ Ts) @ Ts3)))))). % substs_lemma
thf(fact_138_lift__types, axiom,
    ((![E : nat > type, Ts : list_dB, Ts3 : list_type, I : nat, U2 : type]: ((typings @ E @ Ts @ Ts3) => (typings @ (shift_type @ E @ I @ U2) @ (map_dB_dB @ (^[T8 : dB]: (lift @ T8 @ I)) @ Ts) @ Ts3))))). % lift_types
thf(fact_139_var__app__typesE, axiom,
    ((![E : nat > type, I : nat, Ts : list_dB, T4 : type]: ((typing @ E @ (foldl_dB_dB @ app @ (var @ I) @ Ts) @ T4) => (~ ((![Ts4 : list_type]: ((typing @ E @ (var @ I) @ (foldr_type_type @ fun @ Ts4 @ T4)) => (~ ((typings @ E @ Ts @ Ts4))))))))))). % var_app_typesE

% Conjectures (1)
thf(conj_0, conjecture,
    ((it @ (subst @ (abs @ r) @ u @ i)))).
