% 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/FFT/prob_205__3224708_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:09:54.400

% Could-be-implicit typings (2)
thf(ty_n_t__Complex__Ocomplex, type,
    complex : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).

% Explicit typings (17)
thf(sy_c_FFT__Mirabelle__ulikgskiun_Oroot, type,
    fFT_Mirabelle_root : nat > complex).
thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex, type,
    one_one_complex : complex).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat, type,
    one_one_nat : nat).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Complex__Ocomplex, type,
    uminus1204672759omplex : complex > complex).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex, type,
    zero_zero_complex : complex).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_Num_Oneg__numeral__class_Odbl_001t__Complex__Ocomplex, type,
    neg_nu1648888445omplex : complex > complex).
thf(sy_c_Num_Oneg__numeral__class_Odbl__dec_001t__Complex__Ocomplex, type,
    neg_nu972282243omplex : complex > complex).
thf(sy_c_Num_Oneg__numeral__class_Odbl__inc_001t__Complex__Ocomplex, type,
    neg_nu484426047omplex : complex > complex).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat, type,
    ord_less_nat : nat > nat > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat, type,
    ord_less_eq_nat : nat > nat > $o).
thf(sy_c_Parity_Osemiring__bit__shifts__class_Otake__bit_001t__Nat__Onat, type,
    semiri967765622it_nat : nat > nat > nat).
thf(sy_c_Parity_Osemiring__bits__class_Obit_001t__Nat__Onat, type,
    semiri672524339it_nat : nat > nat > $o).
thf(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex, type,
    power_power_complex : complex > nat > complex).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat, type,
    power_power_nat : nat > nat > nat).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Complex__Ocomplex, type,
    dvd_dvd_complex : complex > complex > $o).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat, type,
    dvd_dvd_nat : nat > nat > $o).

% Relevant facts (118)
thf(fact_0_zero__neq__one, axiom,
    ((~ ((zero_zero_nat = one_one_nat))))). % zero_neq_one
thf(fact_1_zero__neq__one, axiom,
    ((~ ((zero_zero_complex = one_one_complex))))). % zero_neq_one
thf(fact_2_zero__natural_Orsp, axiom,
    ((zero_zero_nat = zero_zero_nat))). % zero_natural.rsp
thf(fact_3_one__reorient, axiom,
    ((![X : complex]: ((one_one_complex = X) = (X = one_one_complex))))). % one_reorient
thf(fact_4_one__reorient, axiom,
    ((![X : nat]: ((one_one_nat = X) = (X = one_one_nat))))). % one_reorient
thf(fact_5_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_6_zero__reorient, axiom,
    ((![X : complex]: ((zero_zero_complex = X) = (X = zero_zero_complex))))). % zero_reorient
thf(fact_7_root__unity, axiom,
    ((![N : nat]: ((power_power_complex @ (fFT_Mirabelle_root @ N) @ N) = one_one_complex)))). % root_unity
thf(fact_8_dbl__inc__simps_I2_J, axiom,
    (((neg_nu484426047omplex @ zero_zero_complex) = one_one_complex))). % dbl_inc_simps(2)
thf(fact_9_take__bit__of__1__eq__0__iff, axiom,
    ((![N : nat]: (((semiri967765622it_nat @ N @ one_one_nat) = zero_zero_nat) = (N = zero_zero_nat))))). % take_bit_of_1_eq_0_iff
thf(fact_10_take__bit__of__0, axiom,
    ((![N : nat]: ((semiri967765622it_nat @ N @ zero_zero_nat) = zero_zero_nat)))). % take_bit_of_0
thf(fact_11_take__bit__0, axiom,
    ((![A : nat]: ((semiri967765622it_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % take_bit_0
thf(fact_12_root__nonzero, axiom,
    ((![N : nat]: (~ (((fFT_Mirabelle_root @ N) = zero_zero_complex)))))). % root_nonzero
thf(fact_13_power__one, axiom,
    ((![N : nat]: ((power_power_complex @ one_one_complex @ N) = one_one_complex)))). % power_one
thf(fact_14_power__one, axiom,
    ((![N : nat]: ((power_power_nat @ one_one_nat @ N) = one_one_nat)))). % power_one
thf(fact_15_power__0__left, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((power_power_complex @ zero_zero_complex @ N) = one_one_complex)) & ((~ ((N = zero_zero_nat))) => ((power_power_complex @ zero_zero_complex @ N) = zero_zero_complex)))))). % power_0_left
thf(fact_16_power__0__left, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((power_power_nat @ zero_zero_nat @ N) = one_one_nat)) & ((~ ((N = zero_zero_nat))) => ((power_power_nat @ zero_zero_nat @ N) = zero_zero_nat)))))). % power_0_left
thf(fact_17_power__0, axiom,
    ((![A : complex]: ((power_power_complex @ A @ zero_zero_nat) = one_one_complex)))). % power_0
thf(fact_18_power__0, axiom,
    ((![A : nat]: ((power_power_nat @ A @ zero_zero_nat) = one_one_nat)))). % power_0
thf(fact_19_power__one__right, axiom,
    ((![A : complex]: ((power_power_complex @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_20_power__one__right, axiom,
    ((![A : nat]: ((power_power_nat @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_21_power__not__zero, axiom,
    ((![A : complex, N : nat]: ((~ ((A = zero_zero_complex))) => (~ (((power_power_complex @ A @ N) = zero_zero_complex))))))). % power_not_zero
thf(fact_22_power__not__zero, axiom,
    ((![A : nat, N : nat]: ((~ ((A = zero_zero_nat))) => (~ (((power_power_nat @ A @ N) = zero_zero_nat))))))). % power_not_zero
thf(fact_23_dbl__dec__simps_I3_J, axiom,
    (((neg_nu972282243omplex @ one_one_complex) = one_one_complex))). % dbl_dec_simps(3)
thf(fact_24_dbl__simps_I2_J, axiom,
    (((neg_nu1648888445omplex @ zero_zero_complex) = zero_zero_complex))). % dbl_simps(2)
thf(fact_25_one__natural_Orsp, axiom,
    ((one_one_nat = one_one_nat))). % one_natural.rsp
thf(fact_26_dbl__dec__simps_I2_J, axiom,
    (((neg_nu972282243omplex @ zero_zero_complex) = (uminus1204672759omplex @ one_one_complex)))). % dbl_dec_simps(2)
thf(fact_27_power__eq__0__iff, axiom,
    ((![A : complex, N : nat]: (((power_power_complex @ A @ N) = zero_zero_complex) = (((A = zero_zero_complex)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % power_eq_0_iff
thf(fact_28_power__eq__0__iff, axiom,
    ((![A : nat, N : nat]: (((power_power_nat @ A @ N) = zero_zero_nat) = (((A = zero_zero_nat)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % power_eq_0_iff
thf(fact_29_bit__1__iff, axiom,
    ((![N : nat]: ((semiri672524339it_nat @ one_one_nat @ N) = (((~ ((one_one_nat = zero_zero_nat)))) & ((N = zero_zero_nat))))))). % bit_1_iff
thf(fact_30_zero__power, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((power_power_complex @ zero_zero_complex @ N) = zero_zero_complex))))). % zero_power
thf(fact_31_zero__power, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((power_power_nat @ zero_zero_nat @ N) = zero_zero_nat))))). % zero_power
thf(fact_32_dbl__inc__simps_I4_J, axiom,
    (((neg_nu484426047omplex @ (uminus1204672759omplex @ one_one_complex)) = (uminus1204672759omplex @ one_one_complex)))). % dbl_inc_simps(4)
thf(fact_33_power__le__one, axiom,
    ((![A : nat, N : nat]: ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_eq_nat @ A @ one_one_nat) => (ord_less_eq_nat @ (power_power_nat @ A @ N) @ one_one_nat)))))). % power_le_one
thf(fact_34_is__unit__power__iff, axiom,
    ((![A : nat, N : nat]: ((dvd_dvd_nat @ (power_power_nat @ A @ N) @ one_one_nat) = (((dvd_dvd_nat @ A @ one_one_nat)) | ((N = zero_zero_nat))))))). % is_unit_power_iff
thf(fact_35_bot__nat__0_Oextremum, axiom,
    ((![A : nat]: (ord_less_eq_nat @ zero_zero_nat @ A)))). % bot_nat_0.extremum
thf(fact_36_le0, axiom,
    ((![N : nat]: (ord_less_eq_nat @ zero_zero_nat @ N)))). % le0
thf(fact_37_nat__dvd__1__iff__1, axiom,
    ((![M : nat]: ((dvd_dvd_nat @ M @ one_one_nat) = (M = one_one_nat))))). % nat_dvd_1_iff_1
thf(fact_38_le__zero__eq, axiom,
    ((![N : nat]: ((ord_less_eq_nat @ N @ zero_zero_nat) = (N = zero_zero_nat))))). % le_zero_eq
thf(fact_39_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_40_neg__equal__0__iff__equal, axiom,
    ((![A : complex]: (((uminus1204672759omplex @ A) = zero_zero_complex) = (A = zero_zero_complex))))). % neg_equal_0_iff_equal
thf(fact_41_neg__0__equal__iff__equal, axiom,
    ((![A : complex]: ((zero_zero_complex = (uminus1204672759omplex @ A)) = (zero_zero_complex = A))))). % neg_0_equal_iff_equal
thf(fact_42_add_Oinverse__neutral, axiom,
    (((uminus1204672759omplex @ zero_zero_complex) = zero_zero_complex))). % add.inverse_neutral
thf(fact_43_dvd__0__left__iff, axiom,
    ((![A : nat]: ((dvd_dvd_nat @ zero_zero_nat @ A) = (A = zero_zero_nat))))). % dvd_0_left_iff
thf(fact_44_dvd__0__left__iff, axiom,
    ((![A : complex]: ((dvd_dvd_complex @ zero_zero_complex @ A) = (A = zero_zero_complex))))). % dvd_0_left_iff
thf(fact_45_dvd__0__right, axiom,
    ((![A : nat]: (dvd_dvd_nat @ A @ zero_zero_nat)))). % dvd_0_right
thf(fact_46_dvd__0__right, axiom,
    ((![A : complex]: (dvd_dvd_complex @ A @ zero_zero_complex)))). % dvd_0_right
thf(fact_47_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_48_less__nat__zero__code, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_49_neq0__conv, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N))))). % neq0_conv
thf(fact_50_nat__zero__less__power__iff, axiom,
    ((![X : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ (power_power_nat @ X @ N)) = (((ord_less_nat @ zero_zero_nat @ X)) | ((N = zero_zero_nat))))))). % nat_zero_less_power_iff
thf(fact_51_power__inject__exp, axiom,
    ((![A : nat, M : nat, N : nat]: ((ord_less_nat @ one_one_nat @ A) => (((power_power_nat @ A @ M) = (power_power_nat @ A @ N)) = (M = N)))))). % power_inject_exp
thf(fact_52_less__one, axiom,
    ((![N : nat]: ((ord_less_nat @ N @ one_one_nat) = (N = zero_zero_nat))))). % less_one
thf(fact_53_power__increasing__iff, axiom,
    ((![B : nat, X : nat, Y : nat]: ((ord_less_nat @ one_one_nat @ B) => ((ord_less_eq_nat @ (power_power_nat @ B @ X) @ (power_power_nat @ B @ Y)) = (ord_less_eq_nat @ X @ Y)))))). % power_increasing_iff
thf(fact_54_power__strict__increasing__iff, axiom,
    ((![B : nat, X : nat, Y : nat]: ((ord_less_nat @ one_one_nat @ B) => ((ord_less_nat @ (power_power_nat @ B @ X) @ (power_power_nat @ B @ Y)) = (ord_less_nat @ X @ Y)))))). % power_strict_increasing_iff
thf(fact_55_power__decreasing__iff, axiom,
    ((![B : nat, M : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ B) => ((ord_less_nat @ B @ one_one_nat) => ((ord_less_eq_nat @ (power_power_nat @ B @ M) @ (power_power_nat @ B @ N)) = (ord_less_eq_nat @ N @ M))))))). % power_decreasing_iff
thf(fact_56_power__strict__decreasing__iff, axiom,
    ((![B : nat, M : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ B) => ((ord_less_nat @ B @ one_one_nat) => ((ord_less_nat @ (power_power_nat @ B @ M) @ (power_power_nat @ B @ N)) = (ord_less_nat @ N @ M))))))). % power_strict_decreasing_iff
thf(fact_57_power__mono__iff, axiom,
    ((![A : nat, B : nat, N : nat]: ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_eq_nat @ zero_zero_nat @ B) => ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_eq_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ B @ N)) = (ord_less_eq_nat @ A @ B)))))))). % power_mono_iff
thf(fact_58_dvd__imp__le, axiom,
    ((![K : nat, N : nat]: ((dvd_dvd_nat @ K @ N) => ((ord_less_nat @ zero_zero_nat @ N) => (ord_less_eq_nat @ K @ N)))))). % dvd_imp_le
thf(fact_59_nat__dvd__not__less, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ M) => ((ord_less_nat @ M @ N) => (~ ((dvd_dvd_nat @ N @ M)))))))). % nat_dvd_not_less
thf(fact_60_power__dvd__imp__le, axiom,
    ((![I : nat, M : nat, N : nat]: ((dvd_dvd_nat @ (power_power_nat @ I @ M) @ (power_power_nat @ I @ N)) => ((ord_less_nat @ one_one_nat @ I) => (ord_less_eq_nat @ M @ N)))))). % power_dvd_imp_le
thf(fact_61_dvd__power__le, axiom,
    ((![X : complex, Y : complex, N : nat, M : nat]: ((dvd_dvd_complex @ X @ Y) => ((ord_less_eq_nat @ N @ M) => (dvd_dvd_complex @ (power_power_complex @ X @ N) @ (power_power_complex @ Y @ M))))))). % dvd_power_le
thf(fact_62_dvd__power__le, axiom,
    ((![X : nat, Y : nat, N : nat, M : nat]: ((dvd_dvd_nat @ X @ Y) => ((ord_less_eq_nat @ N @ M) => (dvd_dvd_nat @ (power_power_nat @ X @ N) @ (power_power_nat @ Y @ M))))))). % dvd_power_le
thf(fact_63_power__le__dvd, axiom,
    ((![A : complex, N : nat, B : complex, M : nat]: ((dvd_dvd_complex @ (power_power_complex @ A @ N) @ B) => ((ord_less_eq_nat @ M @ N) => (dvd_dvd_complex @ (power_power_complex @ A @ M) @ B)))))). % power_le_dvd
thf(fact_64_power__le__dvd, axiom,
    ((![A : nat, N : nat, B : nat, M : nat]: ((dvd_dvd_nat @ (power_power_nat @ A @ N) @ B) => ((ord_less_eq_nat @ M @ N) => (dvd_dvd_nat @ (power_power_nat @ A @ M) @ B)))))). % power_le_dvd
thf(fact_65_le__imp__power__dvd, axiom,
    ((![M : nat, N : nat, A : complex]: ((ord_less_eq_nat @ M @ N) => (dvd_dvd_complex @ (power_power_complex @ A @ M) @ (power_power_complex @ A @ N)))))). % le_imp_power_dvd
thf(fact_66_le__imp__power__dvd, axiom,
    ((![M : nat, N : nat, A : nat]: ((ord_less_eq_nat @ M @ N) => (dvd_dvd_nat @ (power_power_nat @ A @ M) @ (power_power_nat @ A @ N)))))). % le_imp_power_dvd
thf(fact_67_power__le__imp__le__exp, axiom,
    ((![A : nat, M : nat, N : nat]: ((ord_less_nat @ one_one_nat @ A) => ((ord_less_eq_nat @ (power_power_nat @ A @ M) @ (power_power_nat @ A @ N)) => (ord_less_eq_nat @ M @ N)))))). % power_le_imp_le_exp
thf(fact_68_ex__least__nat__le, axiom,
    ((![P : nat > $o, N : nat]: ((P @ N) => ((~ ((P @ zero_zero_nat))) => (?[K2 : nat]: ((ord_less_eq_nat @ K2 @ N) & ((![I2 : nat]: ((ord_less_nat @ I2 @ K2) => (~ ((P @ I2))))) & (P @ K2))))))))). % ex_least_nat_le
thf(fact_69_dvd__trans, axiom,
    ((![A : nat, B : nat, C : nat]: ((dvd_dvd_nat @ A @ B) => ((dvd_dvd_nat @ B @ C) => (dvd_dvd_nat @ A @ C)))))). % dvd_trans
thf(fact_70_dvd__refl, axiom,
    ((![A : nat]: (dvd_dvd_nat @ A @ A)))). % dvd_refl
thf(fact_71_less__mono__imp__le__mono, axiom,
    ((![F : nat > nat, I : nat, J : nat]: ((![I3 : nat, J2 : nat]: ((ord_less_nat @ I3 @ J2) => (ord_less_nat @ (F @ I3) @ (F @ J2)))) => ((ord_less_eq_nat @ I @ J) => (ord_less_eq_nat @ (F @ I) @ (F @ J))))))). % less_mono_imp_le_mono
thf(fact_72_le__neq__implies__less, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ M @ N) => ((~ ((M = N))) => (ord_less_nat @ M @ N)))))). % le_neq_implies_less
thf(fact_73_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_74_less__or__eq__imp__le, axiom,
    ((![M : nat, N : nat]: (((ord_less_nat @ M @ N) | (M = N)) => (ord_less_eq_nat @ M @ N))))). % less_or_eq_imp_le
thf(fact_75_le__eq__less__or__eq, axiom,
    ((ord_less_eq_nat = (^[M2 : nat]: (^[N2 : nat]: (((ord_less_nat @ M2 @ N2)) | ((M2 = N2)))))))). % le_eq_less_or_eq
thf(fact_76_infinite__descent, axiom,
    ((![P : nat > $o, N : nat]: ((![N3 : nat]: ((~ ((P @ N3))) => (?[M3 : nat]: ((ord_less_nat @ M3 @ N3) & (~ ((P @ M3))))))) => (P @ N))))). % infinite_descent
thf(fact_77_nat__less__induct, axiom,
    ((![P : nat > $o, N : nat]: ((![N3 : nat]: ((![M3 : nat]: ((ord_less_nat @ M3 @ N3) => (P @ M3))) => (P @ N3))) => (P @ N))))). % nat_less_induct
thf(fact_78_less__irrefl__nat, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_irrefl_nat
thf(fact_79_less__imp__le__nat, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (ord_less_eq_nat @ M @ N))))). % less_imp_le_nat
thf(fact_80_less__not__refl3, axiom,
    ((![S : nat, T : nat]: ((ord_less_nat @ S @ T) => (~ ((S = T))))))). % less_not_refl3
thf(fact_81_less__not__refl2, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ N @ M) => (~ ((M = N))))))). % less_not_refl2
thf(fact_82_less__not__refl, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_not_refl
thf(fact_83_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_84_nat__less__le, axiom,
    ((ord_less_nat = (^[M2 : nat]: (^[N2 : nat]: (((ord_less_eq_nat @ M2 @ N2)) & ((~ ((M2 = N2)))))))))). % nat_less_le
thf(fact_85_power__strict__mono, axiom,
    ((![A : nat, B : nat, N : nat]: ((ord_less_nat @ A @ B) => ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_nat @ zero_zero_nat @ N) => (ord_less_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ B @ N)))))))). % power_strict_mono
thf(fact_86_power__less__imp__less__base, axiom,
    ((![A : nat, N : nat, B : nat]: ((ord_less_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ B @ N)) => ((ord_less_eq_nat @ zero_zero_nat @ B) => (ord_less_nat @ A @ B)))))). % power_less_imp_less_base
thf(fact_87_power__less__imp__less__exp, axiom,
    ((![A : nat, M : nat, N : nat]: ((ord_less_nat @ one_one_nat @ A) => ((ord_less_nat @ (power_power_nat @ A @ M) @ (power_power_nat @ A @ N)) => (ord_less_nat @ M @ N)))))). % power_less_imp_less_exp
thf(fact_88_power__strict__increasing, axiom,
    ((![N : nat, N4 : nat, A : nat]: ((ord_less_nat @ N @ N4) => ((ord_less_nat @ one_one_nat @ A) => (ord_less_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ A @ N4))))))). % power_strict_increasing
thf(fact_89_power__increasing, axiom,
    ((![N : nat, N4 : nat, A : nat]: ((ord_less_eq_nat @ N @ N4) => ((ord_less_eq_nat @ one_one_nat @ A) => (ord_less_eq_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ A @ N4))))))). % power_increasing
thf(fact_90_dvd__0__left, axiom,
    ((![A : nat]: ((dvd_dvd_nat @ zero_zero_nat @ A) => (A = zero_zero_nat))))). % dvd_0_left
thf(fact_91_dvd__0__left, axiom,
    ((![A : complex]: ((dvd_dvd_complex @ zero_zero_complex @ A) => (A = zero_zero_complex))))). % dvd_0_left
thf(fact_92_dvd__unit__imp__unit, axiom,
    ((![A : nat, B : nat]: ((dvd_dvd_nat @ A @ B) => ((dvd_dvd_nat @ B @ one_one_nat) => (dvd_dvd_nat @ A @ one_one_nat)))))). % dvd_unit_imp_unit
thf(fact_93_unit__imp__dvd, axiom,
    ((![B : nat, A : nat]: ((dvd_dvd_nat @ B @ one_one_nat) => (dvd_dvd_nat @ B @ A))))). % unit_imp_dvd
thf(fact_94_one__dvd, axiom,
    ((![A : complex]: (dvd_dvd_complex @ one_one_complex @ A)))). % one_dvd
thf(fact_95_one__dvd, axiom,
    ((![A : nat]: (dvd_dvd_nat @ one_one_nat @ A)))). % one_dvd
thf(fact_96_dvd__power__same, axiom,
    ((![X : complex, Y : complex, N : nat]: ((dvd_dvd_complex @ X @ Y) => (dvd_dvd_complex @ (power_power_complex @ X @ N) @ (power_power_complex @ Y @ N)))))). % dvd_power_same
thf(fact_97_dvd__power__same, axiom,
    ((![X : nat, Y : nat, N : nat]: ((dvd_dvd_nat @ X @ Y) => (dvd_dvd_nat @ (power_power_nat @ X @ N) @ (power_power_nat @ Y @ N)))))). % dvd_power_same
thf(fact_98_one__neq__neg__one, axiom,
    ((~ ((one_one_complex = (uminus1204672759omplex @ one_one_complex)))))). % one_neq_neg_one
thf(fact_99_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_100_gr__implies__not__zero, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not_zero
thf(fact_101_not__less__zero, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less_zero
thf(fact_102_gr__zeroI, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr_zeroI
thf(fact_103_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_nat @ zero_zero_nat @ zero_zero_nat))))). % less_numeral_extra(3)
thf(fact_104_zero__le, axiom,
    ((![X : nat]: (ord_less_eq_nat @ zero_zero_nat @ X)))). % zero_le
thf(fact_105_le__numeral__extra_I3_J, axiom,
    ((ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat))). % le_numeral_extra(3)
thf(fact_106_less__numeral__extra_I4_J, axiom,
    ((~ ((ord_less_nat @ one_one_nat @ one_one_nat))))). % less_numeral_extra(4)
thf(fact_107_le__numeral__extra_I4_J, axiom,
    ((ord_less_eq_nat @ one_one_nat @ one_one_nat))). % le_numeral_extra(4)
thf(fact_108_bot__nat__0_Oextremum__strict, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ zero_zero_nat)))))). % bot_nat_0.extremum_strict
thf(fact_109_infinite__descent0, axiom,
    ((![P : nat > $o, N : nat]: ((P @ zero_zero_nat) => ((![N3 : nat]: ((ord_less_nat @ zero_zero_nat @ N3) => ((~ ((P @ N3))) => (?[M3 : nat]: ((ord_less_nat @ M3 @ N3) & (~ ((P @ M3)))))))) => (P @ N)))))). % infinite_descent0
thf(fact_110_gr__implies__not0, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not0
thf(fact_111_less__zeroE, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_zeroE
thf(fact_112_not__less0, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less0
thf(fact_113_not__gr0, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr0
thf(fact_114_gr0I, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr0I
thf(fact_115_nat__power__less__imp__less, axiom,
    ((![I : nat, M : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ I) => ((ord_less_nat @ (power_power_nat @ I @ M) @ (power_power_nat @ I @ N)) => (ord_less_nat @ M @ N)))))). % nat_power_less_imp_less
thf(fact_116_bot__nat__0_Oextremum__uniqueI, axiom,
    ((![A : nat]: ((ord_less_eq_nat @ A @ zero_zero_nat) => (A = zero_zero_nat))))). % bot_nat_0.extremum_uniqueI
thf(fact_117_bot__nat__0_Oextremum__unique, axiom,
    ((![A : nat]: ((ord_less_eq_nat @ A @ zero_zero_nat) = (A = zero_zero_nat))))). % bot_nat_0.extremum_unique

% Conjectures (1)
thf(conj_0, conjecture,
    (((fFT_Mirabelle_root @ zero_zero_nat) = one_one_complex))).
