% 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_259__5212196_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:38:40.431

% Could-be-implicit typings (3)
thf(ty_n_t__LambdaType__Otype, type,
    type : $tType).
thf(ty_n_t__Lambda__OdB, type,
    dB : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).

% Explicit typings (25)
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat, type,
    plus_plus_nat : nat > nat > nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_InductTermi_OIT, type,
    it : dB > $o).
thf(sy_c_LambdaType_Oshift_001t__LambdaType__Otype, type,
    shift_type : (nat > type) > nat > type > nat > type).
thf(sy_c_LambdaType_Otype_OFun, type,
    fun : type > type > type).
thf(sy_c_LambdaType_Otyping, type,
    typing : (nat > type) > dB > type > $o).
thf(sy_c_Lambda_OdB_OAbs, type,
    abs : dB > dB).
thf(sy_c_Lambda_OdB_OApp, type,
    app : dB > dB > dB).
thf(sy_c_Lambda_OdB_OVar, type,
    var : nat > dB).
thf(sy_c_Lambda_OdB_Osize__dB, type,
    size_dB : dB > nat).
thf(sy_c_Lambda_Olift, type,
    lift : dB > nat > dB).
thf(sy_c_Lambda_Oliftn, type,
    liftn : nat > dB > nat > dB).
thf(sy_c_Lambda_Osubst, type,
    subst : dB > dB > nat > dB).
thf(sy_c_Lambda_Osubstn, type,
    substn : dB > dB > nat > dB).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat, type,
    semiri1382578993at_nat : nat > nat).
thf(sy_c_Nat_Osize__class_Osize_001t__Lambda__OdB, type,
    size_size_dB : dB > nat).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat, type,
    ord_less_nat : nat > nat > $o).
thf(sy_v_T, type,
    t : type).
thf(sy_v_Ta____, type,
    ta : type).
thf(sy_v_U____, type,
    u : type).
thf(sy_v_e, type,
    e : nat > type).
thf(sy_v_ea____, type,
    ea : nat > type).
thf(sy_v_s____, type,
    s : dB).
thf(sy_v_t, type,
    t2 : dB).
thf(sy_v_ta____, type,
    ta2 : dB).

% Relevant facts (133)
thf(fact_0_App_Ohyps_I4_J, axiom,
    ((it @ ta2))). % App.hyps(4)
thf(fact_1_App_Ohyps_I2_J, axiom,
    ((it @ s))). % App.hyps(2)
thf(fact_2_lift__IT, axiom,
    ((![T : dB, I : nat]: ((it @ T) => (it @ (lift @ T @ I)))))). % lift_IT
thf(fact_3_Var__IT, axiom,
    ((![N : nat]: (it @ (var @ N))))). % Var_IT
thf(fact_4_app__Var__IT, axiom,
    ((![T : dB, I : nat]: ((it @ T) => (it @ (app @ T @ (var @ I))))))). % app_Var_IT
thf(fact_5_subst__Var__IT, axiom,
    ((![R : dB, I : nat, J : nat]: ((it @ R) => (it @ (subst @ R @ (var @ I) @ J)))))). % subst_Var_IT
thf(fact_6_subst__lift, axiom,
    ((![T : dB, K : nat, S : dB]: ((subst @ (lift @ T @ K) @ S @ K) = T)))). % subst_lift
thf(fact_7_subst__eq, axiom,
    ((![K : nat, U : dB]: ((subst @ (var @ K) @ U @ K) = U)))). % subst_eq
thf(fact_8_dB_Oinject_I1_J, axiom,
    ((![X1 : nat, Y1 : nat]: (((var @ X1) = (var @ Y1)) = (X1 = Y1))))). % dB.inject(1)
thf(fact_9_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_10_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_11_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_12_dB_Odistinct_I1_J, axiom,
    ((![X1 : nat, X21 : dB, X22 : dB]: (~ (((var @ X1) = (app @ X21 @ X22))))))). % dB.distinct(1)
thf(fact_13_App_Ohyps_I3_J, axiom,
    ((typing @ ea @ ta2 @ ta))). % App.hyps(3)
thf(fact_14_zero__natural_Orsp, axiom,
    ((zero_zero_nat = zero_zero_nat))). % zero_natural.rsp
thf(fact_15_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_16_assms, axiom,
    ((typing @ e @ t2 @ t))). % assms
thf(fact_17_App_Ohyps_I1_J, axiom,
    ((typing @ ea @ s @ (fun @ ta @ u)))). % App.hyps(1)
thf(fact_18_typing_OVar, axiom,
    ((![Env : nat > type, X : nat, T2 : type]: (((Env @ X) = T2) => (typing @ Env @ (var @ X) @ T2))))). % typing.Var
thf(fact_19_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_20_dB_Osize__gen_I1_J, axiom,
    ((![X1 : nat]: ((size_dB @ (var @ X1)) = zero_zero_nat)))). % dB.size_gen(1)
thf(fact_21_subst__type__IT, axiom,
    ((![T : dB, E : nat > type, I : nat, U2 : type, T2 : type, U : dB]: ((it @ T) => ((typing @ (shift_type @ E @ I @ U2) @ T @ T2) => ((it @ U) => ((typing @ E @ U @ U2) => (it @ (subst @ T @ U @ I))))))))). % subst_type_IT
thf(fact_22_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_23_dB_Osize_I4_J, axiom,
    ((![X1 : nat]: ((size_size_dB @ (var @ X1)) = zero_zero_nat)))). % dB.size(4)
thf(fact_24_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_25_liftn__0, axiom,
    ((![T : dB, K : nat]: ((liftn @ zero_zero_nat @ T @ K) = T)))). % liftn_0
thf(fact_26_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_27_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_28_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_29_neq0__conv, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N))))). % neq0_conv
thf(fact_30_less__nat__zero__code, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_31_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_32_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_33_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_34_substn__subst__n, axiom,
    ((substn = (^[T3 : dB]: (^[S2 : dB]: (^[N2 : nat]: (subst @ T3 @ (liftn @ N2 @ S2 @ zero_zero_nat) @ N2))))))). % substn_subst_n
thf(fact_35_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_36_less__not__refl, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_not_refl
thf(fact_37_less__not__refl2, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ N @ M) => (~ ((M = N))))))). % less_not_refl2
thf(fact_38_less__not__refl3, axiom,
    ((![S : nat, T : nat]: ((ord_less_nat @ S @ T) => (~ ((S = T))))))). % less_not_refl3
thf(fact_39_less__irrefl__nat, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_irrefl_nat
thf(fact_40_nat__less__induct, axiom,
    ((![P : nat > $o, N : nat]: ((![N3 : nat]: ((![M2 : nat]: ((ord_less_nat @ M2 @ N3) => (P @ M2))) => (P @ N3))) => (P @ N))))). % nat_less_induct
thf(fact_41_infinite__descent, axiom,
    ((![P : nat > $o, N : nat]: ((![N3 : nat]: ((~ ((P @ N3))) => (?[M2 : nat]: ((ord_less_nat @ M2 @ N3) & (~ ((P @ M2))))))) => (P @ N))))). % infinite_descent
thf(fact_42_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_43_type__induct, axiom,
    ((![P : type > $o, T2 : type]: ((![T4 : type]: ((![T1 : type, T22 : type]: ((T4 = (fun @ T1 @ T22)) => (P @ T1))) => ((![T1 : type, T22 : type]: ((T4 = (fun @ T1 @ T22)) => (P @ T22))) => (P @ T4)))) => (P @ T2))))). % type_induct
thf(fact_44_size__neq__size__imp__neq, axiom,
    ((![X : dB, Y : dB]: ((~ (((size_size_dB @ X) = (size_size_dB @ Y)))) => (~ ((X = Y))))))). % size_neq_size_imp_neq
thf(fact_45_gr__zeroI, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr_zeroI
thf(fact_46_not__less__zero, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less_zero
thf(fact_47_gr__implies__not__zero, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not_zero
thf(fact_48_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_49_gr0I, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr0I
thf(fact_50_not__gr0, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr0
thf(fact_51_not__less0, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less0
thf(fact_52_less__zeroE, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_zeroE
thf(fact_53_gr__implies__not0, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not0
thf(fact_54_infinite__descent0, axiom,
    ((![P : nat > $o, N : nat]: ((P @ zero_zero_nat) => ((![N3 : nat]: ((ord_less_nat @ zero_zero_nat @ N3) => ((~ ((P @ N3))) => (?[M2 : nat]: ((ord_less_nat @ M2 @ N3) & (~ ((P @ M2)))))))) => (P @ N)))))). % infinite_descent0
thf(fact_55_bot__nat__0_Oextremum__strict, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ zero_zero_nat)))))). % bot_nat_0.extremum_strict
thf(fact_56_typing_OApp, 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)))))). % typing.App
thf(fact_57_typing__elims_I2_J, axiom,
    ((![E : nat > type, T : dB, U : dB, T2 : type]: ((typing @ E @ (app @ T @ U) @ T2) => (~ ((![T4 : type]: ((typing @ E @ T @ (fun @ T4 @ T2)) => (~ ((typing @ E @ U @ T4))))))))))). % typing_elims(2)
thf(fact_58_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_59_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_60_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_61_typing_Ocases, axiom,
    ((![A1 : nat > type, A2 : dB, A3 : type]: ((typing @ A1 @ A2 @ A3) => ((![X2 : nat]: ((A2 = (var @ X2)) => (~ (((A1 @ X2) = A3))))) => ((![T4 : type, T5 : dB]: ((A2 = (abs @ T5)) => (![U3 : type]: ((A3 = (fun @ T4 @ U3)) => (~ ((typing @ (shift_type @ A1 @ zero_zero_nat @ T4) @ T5 @ U3))))))) => (~ ((![S3 : dB, T4 : type, U3 : type, T5 : dB]: ((A2 = (app @ S3 @ T5)) => ((A3 = U3) => ((typing @ A1 @ S3 @ (fun @ T4 @ U3)) => (~ ((typing @ A1 @ T5 @ T4))))))))))))))). % typing.cases
thf(fact_62_typing_Osimps, axiom,
    ((typing = (^[A12 : nat > type]: (^[A22 : dB]: (^[A32 : type]: (((?[Env2 : nat > type]: (?[X3 : nat]: (?[T6 : type]: (((A12 = Env2)) & ((((A22 = (var @ X3))) & ((((A32 = T6)) & (((Env2 @ X3) = T6))))))))))) | ((((?[Env2 : nat > type]: (?[T6 : type]: (?[T3 : dB]: (?[U4 : type]: (((A12 = Env2)) & ((((A22 = (abs @ T3))) & ((((A32 = (fun @ T6 @ U4))) & ((typing @ (shift_type @ Env2 @ zero_zero_nat @ T6) @ T3 @ U4)))))))))))) | ((?[Env2 : nat > type]: (?[S2 : dB]: (?[T6 : type]: (?[U4 : type]: (?[T3 : dB]: (((A12 = Env2)) & ((((A22 = (app @ S2 @ T3))) & ((((A32 = U4)) & ((((typing @ Env2 @ S2 @ (fun @ T6 @ U4))) & ((typing @ Env2 @ T3 @ T6)))))))))))))))))))))))). % typing.simps
thf(fact_63_typing_Oinducts, axiom,
    ((![X1 : nat > type, X23 : dB, X32 : type, P : (nat > type) > dB > type > $o]: ((typing @ X1 @ X23 @ X32) => ((![Env3 : nat > type, X2 : nat, T4 : type]: (((Env3 @ X2) = T4) => (P @ Env3 @ (var @ X2) @ T4))) => ((![Env3 : nat > type, T4 : type, T5 : dB, U3 : type]: ((typing @ (shift_type @ Env3 @ zero_zero_nat @ T4) @ T5 @ U3) => ((P @ (shift_type @ Env3 @ zero_zero_nat @ T4) @ T5 @ U3) => (P @ Env3 @ (abs @ T5) @ (fun @ T4 @ U3))))) => ((![Env3 : nat > type, S3 : dB, T4 : type, U3 : type, T5 : dB]: ((typing @ Env3 @ S3 @ (fun @ T4 @ U3)) => ((P @ Env3 @ S3 @ (fun @ T4 @ U3)) => ((typing @ Env3 @ T5 @ T4) => ((P @ Env3 @ T5 @ T4) => (P @ Env3 @ (app @ S3 @ T5) @ U3)))))) => (P @ X1 @ X23 @ X32)))))))). % typing.inducts
thf(fact_64_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_nat @ zero_zero_nat @ zero_zero_nat))))). % less_numeral_extra(3)
thf(fact_65_typing__elims_I3_J, axiom,
    ((![E : nat > type, T : dB, T2 : type]: ((typing @ E @ (abs @ T) @ T2) => (~ ((![T4 : type, U3 : type]: ((T2 = (fun @ T4 @ U3)) => (~ ((typing @ (shift_type @ E @ zero_zero_nat @ T4) @ T @ U3))))))))))). % typing_elims(3)
thf(fact_66_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_67_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_68_liftn_Osimps_I1_J, axiom,
    ((![I : nat, K : nat, N : nat]: (((ord_less_nat @ I @ K) => ((liftn @ N @ (var @ I) @ K) = (var @ I))) & ((~ ((ord_less_nat @ I @ K))) => ((liftn @ N @ (var @ I) @ K) = (var @ (plus_plus_nat @ I @ N)))))))). % liftn.simps(1)
thf(fact_69_add__right__cancel, axiom,
    ((![B : nat, A : nat, C : nat]: (((plus_plus_nat @ B @ A) = (plus_plus_nat @ C @ A)) = (B = C))))). % add_right_cancel
thf(fact_70_add__left__cancel, axiom,
    ((![A : nat, B : nat, C : nat]: (((plus_plus_nat @ A @ B) = (plus_plus_nat @ A @ C)) = (B = C))))). % add_left_cancel
thf(fact_71_dB_Oinject_I3_J, axiom,
    ((![X32 : dB, Y3 : dB]: (((abs @ X32) = (abs @ Y3)) = (X32 = Y3))))). % dB.inject(3)
thf(fact_72_zero__eq__add__iff__both__eq__0, axiom,
    ((![X : nat, Y : nat]: ((zero_zero_nat = (plus_plus_nat @ X @ Y)) = (((X = zero_zero_nat)) & ((Y = zero_zero_nat))))))). % zero_eq_add_iff_both_eq_0
thf(fact_73_add__eq__0__iff__both__eq__0, axiom,
    ((![X : nat, Y : nat]: (((plus_plus_nat @ X @ Y) = zero_zero_nat) = (((X = zero_zero_nat)) & ((Y = zero_zero_nat))))))). % add_eq_0_iff_both_eq_0
thf(fact_74_add__cancel__right__right, axiom,
    ((![A : nat, B : nat]: ((A = (plus_plus_nat @ A @ B)) = (B = zero_zero_nat))))). % add_cancel_right_right
thf(fact_75_add__cancel__right__left, axiom,
    ((![A : nat, B : nat]: ((A = (plus_plus_nat @ B @ A)) = (B = zero_zero_nat))))). % add_cancel_right_left
thf(fact_76_add__cancel__left__right, axiom,
    ((![A : nat, B : nat]: (((plus_plus_nat @ A @ B) = A) = (B = zero_zero_nat))))). % add_cancel_left_right
thf(fact_77_add__cancel__left__left, axiom,
    ((![B : nat, A : nat]: (((plus_plus_nat @ B @ A) = A) = (B = zero_zero_nat))))). % add_cancel_left_left
thf(fact_78_add_Oright__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ A @ zero_zero_nat) = A)))). % add.right_neutral
thf(fact_79_add_Oleft__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ zero_zero_nat @ A) = A)))). % add.left_neutral
thf(fact_80_add__less__cancel__right, axiom,
    ((![A : nat, C : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ C)) = (ord_less_nat @ A @ B))))). % add_less_cancel_right
thf(fact_81_add__less__cancel__left, axiom,
    ((![C : nat, A : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ C @ A) @ (plus_plus_nat @ C @ B)) = (ord_less_nat @ A @ B))))). % add_less_cancel_left
thf(fact_82_Nat_Oadd__0__right, axiom,
    ((![M : nat]: ((plus_plus_nat @ M @ zero_zero_nat) = M)))). % Nat.add_0_right
thf(fact_83_add__is__0, axiom,
    ((![M : nat, N : nat]: (((plus_plus_nat @ M @ N) = zero_zero_nat) = (((M = zero_zero_nat)) & ((N = zero_zero_nat))))))). % add_is_0
thf(fact_84_nat__add__left__cancel__less, axiom,
    ((![K : nat, M : nat, N : nat]: ((ord_less_nat @ (plus_plus_nat @ K @ M) @ (plus_plus_nat @ K @ N)) = (ord_less_nat @ M @ N))))). % nat_add_left_cancel_less
thf(fact_85_less__add__same__cancel2, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ (plus_plus_nat @ B @ A)) = (ord_less_nat @ zero_zero_nat @ B))))). % less_add_same_cancel2
thf(fact_86_less__add__same__cancel1, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ (plus_plus_nat @ A @ B)) = (ord_less_nat @ zero_zero_nat @ B))))). % less_add_same_cancel1
thf(fact_87_add__less__same__cancel2, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ A @ B) @ B) = (ord_less_nat @ A @ zero_zero_nat))))). % add_less_same_cancel2
thf(fact_88_add__less__same__cancel1, axiom,
    ((![B : nat, A : nat]: ((ord_less_nat @ (plus_plus_nat @ B @ A) @ B) = (ord_less_nat @ A @ zero_zero_nat))))). % add_less_same_cancel1
thf(fact_89_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_90_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_91_of__nat__0, axiom,
    (((semiri1382578993at_nat @ zero_zero_nat) = zero_zero_nat))). % of_nat_0
thf(fact_92_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_93_of__nat__add, axiom,
    ((![M : nat, N : nat]: ((semiri1382578993at_nat @ (plus_plus_nat @ M @ N)) = (plus_plus_nat @ (semiri1382578993at_nat @ M) @ (semiri1382578993at_nat @ N)))))). % of_nat_add
thf(fact_94_add__gr__0, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ (plus_plus_nat @ M @ N)) = (((ord_less_nat @ zero_zero_nat @ M)) | ((ord_less_nat @ zero_zero_nat @ N))))))). % add_gr_0
thf(fact_95_add__right__imp__eq, axiom,
    ((![B : nat, A : nat, C : nat]: (((plus_plus_nat @ B @ A) = (plus_plus_nat @ C @ A)) => (B = C))))). % add_right_imp_eq
thf(fact_96_add__left__imp__eq, axiom,
    ((![A : nat, B : nat, C : nat]: (((plus_plus_nat @ A @ B) = (plus_plus_nat @ A @ C)) => (B = C))))). % add_left_imp_eq
thf(fact_97_add_Oleft__commute, axiom,
    ((![B : nat, A : nat, C : nat]: ((plus_plus_nat @ B @ (plus_plus_nat @ A @ C)) = (plus_plus_nat @ A @ (plus_plus_nat @ B @ C)))))). % add.left_commute
thf(fact_98_add_Ocommute, axiom,
    ((plus_plus_nat = (^[A4 : nat]: (^[B2 : nat]: (plus_plus_nat @ B2 @ A4)))))). % add.commute
thf(fact_99_add_Oassoc, axiom,
    ((![A : nat, B : nat, C : nat]: ((plus_plus_nat @ (plus_plus_nat @ A @ B) @ C) = (plus_plus_nat @ A @ (plus_plus_nat @ B @ C)))))). % add.assoc
thf(fact_100_group__cancel_Oadd2, axiom,
    ((![B3 : nat, K : nat, B : nat, A : nat]: ((B3 = (plus_plus_nat @ K @ B)) => ((plus_plus_nat @ A @ B3) = (plus_plus_nat @ K @ (plus_plus_nat @ A @ B))))))). % group_cancel.add2
thf(fact_101_group__cancel_Oadd1, axiom,
    ((![A5 : nat, K : nat, A : nat, B : nat]: ((A5 = (plus_plus_nat @ K @ A)) => ((plus_plus_nat @ A5 @ B) = (plus_plus_nat @ K @ (plus_plus_nat @ A @ B))))))). % group_cancel.add1
thf(fact_102_add__mono__thms__linordered__semiring_I4_J, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: (((I = J) & (K = L)) => ((plus_plus_nat @ I @ K) = (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_semiring(4)
thf(fact_103_ab__semigroup__add__class_Oadd__ac_I1_J, axiom,
    ((![A : nat, B : nat, C : nat]: ((plus_plus_nat @ (plus_plus_nat @ A @ B) @ C) = (plus_plus_nat @ A @ (plus_plus_nat @ B @ C)))))). % ab_semigroup_add_class.add_ac(1)
thf(fact_104_add_Ocomm__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ A @ zero_zero_nat) = A)))). % add.comm_neutral
thf(fact_105_comm__monoid__add__class_Oadd__0, axiom,
    ((![A : nat]: ((plus_plus_nat @ zero_zero_nat @ A) = A)))). % comm_monoid_add_class.add_0
thf(fact_106_add__less__imp__less__right, axiom,
    ((![A : nat, C : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ C)) => (ord_less_nat @ A @ B))))). % add_less_imp_less_right
thf(fact_107_add__less__imp__less__left, axiom,
    ((![C : nat, A : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ C @ A) @ (plus_plus_nat @ C @ B)) => (ord_less_nat @ A @ B))))). % add_less_imp_less_left
thf(fact_108_add__strict__right__mono, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => (ord_less_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ C)))))). % add_strict_right_mono
thf(fact_109_add__strict__left__mono, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => (ord_less_nat @ (plus_plus_nat @ C @ A) @ (plus_plus_nat @ C @ B)))))). % add_strict_left_mono
thf(fact_110_add__strict__mono, axiom,
    ((![A : nat, B : nat, C : nat, D : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ C @ D) => (ord_less_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ D))))))). % add_strict_mono
thf(fact_111_add__mono__thms__linordered__field_I1_J, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: (((ord_less_nat @ I @ J) & (K = L)) => (ord_less_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_field(1)
thf(fact_112_add__mono__thms__linordered__field_I2_J, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: (((I = J) & (ord_less_nat @ K @ L)) => (ord_less_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_field(2)
thf(fact_113_add__mono__thms__linordered__field_I5_J, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: (((ord_less_nat @ I @ J) & (ord_less_nat @ K @ L)) => (ord_less_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_field(5)
thf(fact_114_add__eq__self__zero, axiom,
    ((![M : nat, N : nat]: (((plus_plus_nat @ M @ N) = M) => (N = zero_zero_nat))))). % add_eq_self_zero
thf(fact_115_plus__nat_Oadd__0, axiom,
    ((![N : nat]: ((plus_plus_nat @ zero_zero_nat @ N) = N)))). % plus_nat.add_0
thf(fact_116_less__add__eq__less, axiom,
    ((![K : nat, L : nat, M : nat, N : nat]: ((ord_less_nat @ K @ L) => (((plus_plus_nat @ M @ L) = (plus_plus_nat @ K @ N)) => (ord_less_nat @ M @ N)))))). % less_add_eq_less
thf(fact_117_trans__less__add2, axiom,
    ((![I : nat, J : nat, M : nat]: ((ord_less_nat @ I @ J) => (ord_less_nat @ I @ (plus_plus_nat @ M @ J)))))). % trans_less_add2
thf(fact_118_trans__less__add1, axiom,
    ((![I : nat, J : nat, M : nat]: ((ord_less_nat @ I @ J) => (ord_less_nat @ I @ (plus_plus_nat @ J @ M)))))). % trans_less_add1
thf(fact_119_add__less__mono1, axiom,
    ((![I : nat, J : nat, K : nat]: ((ord_less_nat @ I @ J) => (ord_less_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ K)))))). % add_less_mono1
thf(fact_120_not__add__less2, axiom,
    ((![J : nat, I : nat]: (~ ((ord_less_nat @ (plus_plus_nat @ J @ I) @ I)))))). % not_add_less2
thf(fact_121_not__add__less1, axiom,
    ((![I : nat, J : nat]: (~ ((ord_less_nat @ (plus_plus_nat @ I @ J) @ I)))))). % not_add_less1
thf(fact_122_add__less__mono, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: ((ord_less_nat @ I @ J) => ((ord_less_nat @ K @ L) => (ord_less_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ L))))))). % add_less_mono
thf(fact_123_add__lessD1, axiom,
    ((![I : nat, J : nat, K : nat]: ((ord_less_nat @ (plus_plus_nat @ I @ J) @ K) => (ord_less_nat @ I @ K))))). % add_lessD1
thf(fact_124_dB_Odistinct_I5_J, axiom,
    ((![X21 : dB, X22 : dB, X32 : dB]: (~ (((app @ X21 @ X22) = (abs @ X32))))))). % dB.distinct(5)
thf(fact_125_dB_Odistinct_I3_J, axiom,
    ((![X1 : nat, X32 : dB]: (~ (((var @ X1) = (abs @ X32))))))). % dB.distinct(3)
thf(fact_126_of__nat__less__0__iff, axiom,
    ((![M : nat]: (~ ((ord_less_nat @ (semiri1382578993at_nat @ M) @ zero_zero_nat)))))). % of_nat_less_0_iff
thf(fact_127_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_128_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_129_pos__add__strict, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ zero_zero_nat @ A) => ((ord_less_nat @ B @ C) => (ord_less_nat @ B @ (plus_plus_nat @ A @ C))))))). % pos_add_strict
thf(fact_130_canonically__ordered__monoid__add__class_OlessE, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ B) => (~ ((![C2 : nat]: ((B = (plus_plus_nat @ A @ C2)) => (C2 = zero_zero_nat))))))))). % canonically_ordered_monoid_add_class.lessE
thf(fact_131_add__pos__pos, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ zero_zero_nat @ A) => ((ord_less_nat @ zero_zero_nat @ B) => (ord_less_nat @ zero_zero_nat @ (plus_plus_nat @ A @ B))))))). % add_pos_pos
thf(fact_132_add__neg__neg, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ zero_zero_nat) => ((ord_less_nat @ B @ zero_zero_nat) => (ord_less_nat @ (plus_plus_nat @ A @ B) @ zero_zero_nat)))))). % add_neg_neg

% Conjectures (1)
thf(conj_0, conjecture,
    ((it @ (subst @ (app @ (var @ zero_zero_nat) @ (lift @ ta2 @ zero_zero_nat)) @ s @ zero_zero_nat)))).
