% 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_414__3226678_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:11:09.037

% Could-be-implicit typings (3)
thf(ty_n_t__Set__Oset_It__Nat__Onat_J, type,
    set_nat : $tType).
thf(ty_n_t__Complex__Ocomplex, type,
    complex : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).

% Explicit typings (21)
thf(sy_c_FFT__Mirabelle__ulikgskiun_Oroot, type,
    fFT_Mirabelle_root : nat > complex).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat, type,
    minus_minus_nat : nat > nat > nat).
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_Otimes__class_Otimes_001t__Complex__Ocomplex, type,
    times_times_complex : complex > complex > complex).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat, type,
    times_times_nat : nat > nat > nat).
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_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Complex__Ocomplex, type,
    groups59700922omplex : (nat > complex) > set_nat > complex).
thf(sy_c_If_001t__Complex__Ocomplex, type,
    if_complex : $o > complex > complex > complex).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Complex__Ocomplex, type,
    semiri356525583omplex : nat > complex).
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__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_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_Odivide__class_Odivide_001t__Complex__Ocomplex, type,
    divide1210191872omplex : complex > complex > complex).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat, type,
    set_or562006527an_nat : nat > nat > set_nat).
thf(sy_v_a, type,
    a : nat > complex).
thf(sy_v_i, type,
    i : nat).
thf(sy_v_n, type,
    n : nat).

% Relevant facts (134)
thf(fact_0_root0, axiom,
    (((fFT_Mirabelle_root @ zero_zero_nat) = one_one_complex))). % root0
thf(fact_1_i__less, axiom,
    ((ord_less_nat @ i @ n))). % i_less
thf(fact_2_root__unity, axiom,
    ((![N : nat]: ((power_power_complex @ (fFT_Mirabelle_root @ N) @ N) = one_one_complex)))). % root_unity
thf(fact_3_power__diff__rev__if, axiom,
    ((![A : complex, N : nat, M : nat]: ((~ ((A = zero_zero_complex))) => (((ord_less_eq_nat @ N @ M) => ((divide1210191872omplex @ (power_power_complex @ A @ M) @ (power_power_complex @ A @ N)) = (power_power_complex @ A @ (minus_minus_nat @ M @ N)))) & ((~ ((ord_less_eq_nat @ N @ M))) => ((divide1210191872omplex @ (power_power_complex @ A @ M) @ (power_power_complex @ A @ N)) = (power_power_complex @ (divide1210191872omplex @ one_one_complex @ A) @ (minus_minus_nat @ N @ M))))))))). % power_diff_rev_if
thf(fact_4_of__nat__le__of__nat__power__cancel__iff, axiom,
    ((![B : nat, W : nat, X : nat]: ((ord_less_eq_nat @ (power_power_nat @ (semiri1382578993at_nat @ B) @ W) @ (semiri1382578993at_nat @ X)) = (ord_less_eq_nat @ (power_power_nat @ B @ W) @ X))))). % of_nat_le_of_nat_power_cancel_iff
thf(fact_5_of__nat__power__le__of__nat__cancel__iff, axiom,
    ((![X : nat, B : nat, W : nat]: ((ord_less_eq_nat @ (semiri1382578993at_nat @ X) @ (power_power_nat @ (semiri1382578993at_nat @ B) @ W)) = (ord_less_eq_nat @ X @ (power_power_nat @ B @ W)))))). % of_nat_power_le_of_nat_cancel_iff
thf(fact_6_of__nat__le__0__iff, axiom,
    ((![M : nat]: ((ord_less_eq_nat @ (semiri1382578993at_nat @ M) @ zero_zero_nat) = (M = zero_zero_nat))))). % of_nat_le_0_iff
thf(fact_7_nonzero__divide__mult__cancel__left, axiom,
    ((![A : complex, B : complex]: ((~ ((A = zero_zero_complex))) => ((divide1210191872omplex @ A @ (times_times_complex @ A @ B)) = (divide1210191872omplex @ one_one_complex @ B)))))). % nonzero_divide_mult_cancel_left
thf(fact_8_nonzero__divide__mult__cancel__right, axiom,
    ((![B : complex, A : complex]: ((~ ((B = zero_zero_complex))) => ((divide1210191872omplex @ B @ (times_times_complex @ A @ B)) = (divide1210191872omplex @ one_one_complex @ A)))))). % nonzero_divide_mult_cancel_right
thf(fact_9_diff__is__0__eq, axiom,
    ((![M : nat, N : nat]: (((minus_minus_nat @ M @ N) = zero_zero_nat) = (ord_less_eq_nat @ M @ N))))). % diff_is_0_eq
thf(fact_10_diff__is__0__eq_H, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ M @ N) => ((minus_minus_nat @ M @ N) = zero_zero_nat))))). % diff_is_0_eq'
thf(fact_11_of__nat__le__iff, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ (semiri1382578993at_nat @ M) @ (semiri1382578993at_nat @ N)) = (ord_less_eq_nat @ M @ N))))). % of_nat_le_iff
thf(fact_12_of__nat__0, axiom,
    (((semiri356525583omplex @ zero_zero_nat) = zero_zero_complex))). % of_nat_0
thf(fact_13_of__nat__0, axiom,
    (((semiri1382578993at_nat @ zero_zero_nat) = zero_zero_nat))). % of_nat_0
thf(fact_14_of__nat__0__eq__iff, axiom,
    ((![N : nat]: ((zero_zero_complex = (semiri356525583omplex @ N)) = (zero_zero_nat = N))))). % of_nat_0_eq_iff
thf(fact_15_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_16_power__one__right, axiom,
    ((![A : complex]: ((power_power_complex @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_17_power__one__right, axiom,
    ((![A : nat]: ((power_power_nat @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_18_mult__cancel2, axiom,
    ((![M : nat, K : nat, N : nat]: (((times_times_nat @ M @ K) = (times_times_nat @ N @ K)) = (((M = N)) | ((K = zero_zero_nat))))))). % mult_cancel2
thf(fact_19_mult__cancel1, axiom,
    ((![K : nat, M : nat, N : nat]: (((times_times_nat @ K @ M) = (times_times_nat @ K @ N)) = (((M = N)) | ((K = zero_zero_nat))))))). % mult_cancel1
thf(fact_20_mult__0__right, axiom,
    ((![M : nat]: ((times_times_nat @ M @ zero_zero_nat) = zero_zero_nat)))). % mult_0_right
thf(fact_21_mult__is__0, axiom,
    ((![M : nat, N : nat]: (((times_times_nat @ M @ N) = zero_zero_nat) = (((M = zero_zero_nat)) | ((N = zero_zero_nat))))))). % mult_is_0
thf(fact_22_of__nat__eq__iff, axiom,
    ((![M : nat, N : nat]: (((semiri356525583omplex @ M) = (semiri356525583omplex @ N)) = (M = N))))). % of_nat_eq_iff
thf(fact_23_of__nat__eq__iff, axiom,
    ((![M : nat, N : nat]: (((semiri1382578993at_nat @ M) = (semiri1382578993at_nat @ N)) = (M = N))))). % of_nat_eq_iff
thf(fact_24_division__ring__divide__zero, axiom,
    ((![A : complex]: ((divide1210191872omplex @ A @ zero_zero_complex) = zero_zero_complex)))). % division_ring_divide_zero
thf(fact_25_divide__cancel__right, axiom,
    ((![A : complex, C : complex, B : complex]: (((divide1210191872omplex @ A @ C) = (divide1210191872omplex @ B @ C)) = (((C = zero_zero_complex)) | ((A = B))))))). % divide_cancel_right
thf(fact_26_divide__cancel__left, axiom,
    ((![C : complex, A : complex, B : complex]: (((divide1210191872omplex @ C @ A) = (divide1210191872omplex @ C @ B)) = (((C = zero_zero_complex)) | ((A = B))))))). % divide_cancel_left
thf(fact_27_divide__eq__0__iff, axiom,
    ((![A : complex, B : complex]: (((divide1210191872omplex @ A @ B) = zero_zero_complex) = (((A = zero_zero_complex)) | ((B = zero_zero_complex))))))). % divide_eq_0_iff
thf(fact_28_times__divide__eq__right, axiom,
    ((![A : complex, B : complex, C : complex]: ((times_times_complex @ A @ (divide1210191872omplex @ B @ C)) = (divide1210191872omplex @ (times_times_complex @ A @ B) @ C))))). % times_divide_eq_right
thf(fact_29_times__divide__eq__left, axiom,
    ((![B : complex, C : complex, A : complex]: ((times_times_complex @ (divide1210191872omplex @ B @ C) @ A) = (divide1210191872omplex @ (times_times_complex @ B @ A) @ C))))). % times_divide_eq_left
thf(fact_30_power__one, axiom,
    ((![N : nat]: ((power_power_complex @ one_one_complex @ N) = one_one_complex)))). % power_one
thf(fact_31_power__one, axiom,
    ((![N : nat]: ((power_power_nat @ one_one_nat @ N) = one_one_nat)))). % power_one
thf(fact_32_of__nat__mult, axiom,
    ((![M : nat, N : nat]: ((semiri356525583omplex @ (times_times_nat @ M @ N)) = (times_times_complex @ (semiri356525583omplex @ M) @ (semiri356525583omplex @ N)))))). % of_nat_mult
thf(fact_33_of__nat__mult, axiom,
    ((![M : nat, N : nat]: ((semiri1382578993at_nat @ (times_times_nat @ M @ N)) = (times_times_nat @ (semiri1382578993at_nat @ M) @ (semiri1382578993at_nat @ N)))))). % of_nat_mult
thf(fact_34_of__nat__eq__1__iff, axiom,
    ((![N : nat]: (((semiri356525583omplex @ N) = one_one_complex) = (N = one_one_nat))))). % of_nat_eq_1_iff
thf(fact_35_of__nat__eq__1__iff, axiom,
    ((![N : nat]: (((semiri1382578993at_nat @ N) = one_one_nat) = (N = one_one_nat))))). % of_nat_eq_1_iff
thf(fact_36_of__nat__1__eq__iff, axiom,
    ((![N : nat]: ((one_one_complex = (semiri356525583omplex @ N)) = (N = one_one_nat))))). % of_nat_1_eq_iff
thf(fact_37_of__nat__1__eq__iff, axiom,
    ((![N : nat]: ((one_one_nat = (semiri1382578993at_nat @ N)) = (N = one_one_nat))))). % of_nat_1_eq_iff
thf(fact_38_of__nat__1, axiom,
    (((semiri356525583omplex @ one_one_nat) = one_one_complex))). % of_nat_1
thf(fact_39_of__nat__1, axiom,
    (((semiri1382578993at_nat @ one_one_nat) = one_one_nat))). % of_nat_1
thf(fact_40_less__one, axiom,
    ((![N : nat]: ((ord_less_nat @ N @ one_one_nat) = (N = zero_zero_nat))))). % less_one
thf(fact_41_mult__less__cancel2, axiom,
    ((![M : nat, K : nat, N : nat]: ((ord_less_nat @ (times_times_nat @ M @ K) @ (times_times_nat @ N @ K)) = (((ord_less_nat @ zero_zero_nat @ K)) & ((ord_less_nat @ M @ N))))))). % mult_less_cancel2
thf(fact_42_nat__0__less__mult__iff, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ (times_times_nat @ M @ N)) = (((ord_less_nat @ zero_zero_nat @ M)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % nat_0_less_mult_iff
thf(fact_43_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_44_less__nat__zero__code, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_45_neq0__conv, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N))))). % neq0_conv
thf(fact_46_bot__nat__0_Oextremum, axiom,
    ((![A : nat]: (ord_less_eq_nat @ zero_zero_nat @ A)))). % bot_nat_0.extremum
thf(fact_47_le0, axiom,
    ((![N : nat]: (ord_less_eq_nat @ zero_zero_nat @ N)))). % le0
thf(fact_48_diff__self__eq__0, axiom,
    ((![M : nat]: ((minus_minus_nat @ M @ M) = zero_zero_nat)))). % diff_self_eq_0
thf(fact_49_diff__0__eq__0, axiom,
    ((![N : nat]: ((minus_minus_nat @ zero_zero_nat @ N) = zero_zero_nat)))). % diff_0_eq_0
thf(fact_50_diff__diff__cancel, axiom,
    ((![I : nat, N : nat]: ((ord_less_eq_nat @ I @ N) => ((minus_minus_nat @ N @ (minus_minus_nat @ N @ I)) = I))))). % diff_diff_cancel
thf(fact_51_root1, axiom,
    (((fFT_Mirabelle_root @ one_one_nat) = one_one_complex))). % root1
thf(fact_52_nonzero__mult__divide__mult__cancel__right2, axiom,
    ((![C : complex, A : complex, B : complex]: ((~ ((C = zero_zero_complex))) => ((divide1210191872omplex @ (times_times_complex @ A @ C) @ (times_times_complex @ C @ B)) = (divide1210191872omplex @ A @ B)))))). % nonzero_mult_divide_mult_cancel_right2
thf(fact_53_nonzero__mult__divide__mult__cancel__right, axiom,
    ((![C : complex, A : complex, B : complex]: ((~ ((C = zero_zero_complex))) => ((divide1210191872omplex @ (times_times_complex @ A @ C) @ (times_times_complex @ B @ C)) = (divide1210191872omplex @ A @ B)))))). % nonzero_mult_divide_mult_cancel_right
thf(fact_54_nonzero__mult__divide__mult__cancel__left2, axiom,
    ((![C : complex, A : complex, B : complex]: ((~ ((C = zero_zero_complex))) => ((divide1210191872omplex @ (times_times_complex @ C @ A) @ (times_times_complex @ B @ C)) = (divide1210191872omplex @ A @ B)))))). % nonzero_mult_divide_mult_cancel_left2
thf(fact_55_nonzero__mult__divide__mult__cancel__left, axiom,
    ((![C : complex, A : complex, B : complex]: ((~ ((C = zero_zero_complex))) => ((divide1210191872omplex @ (times_times_complex @ C @ A) @ (times_times_complex @ C @ B)) = (divide1210191872omplex @ A @ B)))))). % nonzero_mult_divide_mult_cancel_left
thf(fact_56_mult__divide__mult__cancel__left__if, axiom,
    ((![C : complex, A : complex, B : complex]: (((C = zero_zero_complex) => ((divide1210191872omplex @ (times_times_complex @ C @ A) @ (times_times_complex @ C @ B)) = zero_zero_complex)) & ((~ ((C = zero_zero_complex))) => ((divide1210191872omplex @ (times_times_complex @ C @ A) @ (times_times_complex @ C @ B)) = (divide1210191872omplex @ A @ B))))))). % mult_divide_mult_cancel_left_if
thf(fact_57_divide__self__if, axiom,
    ((![A : complex]: (((A = zero_zero_complex) => ((divide1210191872omplex @ A @ A) = zero_zero_complex)) & ((~ ((A = zero_zero_complex))) => ((divide1210191872omplex @ A @ A) = one_one_complex)))))). % divide_self_if
thf(fact_58_divide__self, axiom,
    ((![A : complex]: ((~ ((A = zero_zero_complex))) => ((divide1210191872omplex @ A @ A) = one_one_complex))))). % divide_self
thf(fact_59_one__eq__divide__iff, axiom,
    ((![A : complex, B : complex]: ((one_one_complex = (divide1210191872omplex @ A @ B)) = (((~ ((B = zero_zero_complex)))) & ((A = B))))))). % one_eq_divide_iff
thf(fact_60_divide__eq__1__iff, axiom,
    ((![A : complex, B : complex]: (((divide1210191872omplex @ A @ B) = one_one_complex) = (((~ ((B = zero_zero_complex)))) & ((A = B))))))). % divide_eq_1_iff
thf(fact_61_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_62_of__nat__eq__0__iff, axiom,
    ((![M : nat]: (((semiri356525583omplex @ M) = zero_zero_complex) = (M = zero_zero_nat))))). % of_nat_eq_0_iff
thf(fact_63_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_64_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_65_mult__le__cancel2, axiom,
    ((![M : nat, K : nat, N : nat]: ((ord_less_eq_nat @ (times_times_nat @ M @ K) @ (times_times_nat @ N @ K)) = (((ord_less_nat @ zero_zero_nat @ K)) => ((ord_less_eq_nat @ M @ N))))))). % mult_le_cancel2
thf(fact_66_zero__less__diff, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ zero_zero_nat @ (minus_minus_nat @ N @ M)) = (ord_less_nat @ M @ N))))). % zero_less_diff
thf(fact_67_of__nat__power__eq__of__nat__cancel__iff, axiom,
    ((![X : nat, B : nat, W : nat]: (((semiri356525583omplex @ X) = (power_power_complex @ (semiri356525583omplex @ B) @ W)) = (X = (power_power_nat @ B @ W)))))). % of_nat_power_eq_of_nat_cancel_iff
thf(fact_68_of__nat__power__eq__of__nat__cancel__iff, axiom,
    ((![X : nat, B : nat, W : nat]: (((semiri1382578993at_nat @ X) = (power_power_nat @ (semiri1382578993at_nat @ B) @ W)) = (X = (power_power_nat @ B @ W)))))). % of_nat_power_eq_of_nat_cancel_iff
thf(fact_69_of__nat__eq__of__nat__power__cancel__iff, axiom,
    ((![B : nat, W : nat, X : nat]: (((power_power_complex @ (semiri356525583omplex @ B) @ W) = (semiri356525583omplex @ X)) = ((power_power_nat @ B @ W) = X))))). % of_nat_eq_of_nat_power_cancel_iff
thf(fact_70_of__nat__eq__of__nat__power__cancel__iff, axiom,
    ((![B : nat, W : nat, X : nat]: (((power_power_nat @ (semiri1382578993at_nat @ B) @ W) = (semiri1382578993at_nat @ X)) = ((power_power_nat @ B @ W) = X))))). % of_nat_eq_of_nat_power_cancel_iff
thf(fact_71_of__nat__power, axiom,
    ((![M : nat, N : nat]: ((semiri356525583omplex @ (power_power_nat @ M @ N)) = (power_power_complex @ (semiri356525583omplex @ M) @ N))))). % of_nat_power
thf(fact_72_of__nat__power, axiom,
    ((![M : nat, N : nat]: ((semiri1382578993at_nat @ (power_power_nat @ M @ N)) = (power_power_nat @ (semiri1382578993at_nat @ M) @ N))))). % of_nat_power
thf(fact_73_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_74_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_75_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_76_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_77_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_78_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_79_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_80_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_81_of__nat__power__less__of__nat__cancel__iff, axiom,
    ((![X : nat, B : nat, W : nat]: ((ord_less_nat @ (semiri1382578993at_nat @ X) @ (power_power_nat @ (semiri1382578993at_nat @ B) @ W)) = (ord_less_nat @ X @ (power_power_nat @ B @ W)))))). % of_nat_power_less_of_nat_cancel_iff
thf(fact_82_of__nat__less__of__nat__power__cancel__iff, axiom,
    ((![B : nat, W : nat, X : nat]: ((ord_less_nat @ (power_power_nat @ (semiri1382578993at_nat @ B) @ W) @ (semiri1382578993at_nat @ X)) = (ord_less_nat @ (power_power_nat @ B @ W) @ X))))). % of_nat_less_of_nat_power_cancel_iff
thf(fact_83_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_84_of__nat__zero__less__power__iff, axiom,
    ((![X : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ (power_power_nat @ (semiri1382578993at_nat @ X) @ N)) = (((ord_less_nat @ zero_zero_nat @ X)) | ((N = zero_zero_nat))))))). % of_nat_zero_less_power_iff
thf(fact_85_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_86_less__not__refl, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_not_refl
thf(fact_87_less__not__refl2, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ N @ M) => (~ ((M = N))))))). % less_not_refl2
thf(fact_88_less__not__refl3, axiom,
    ((![S : nat, T : nat]: ((ord_less_nat @ S @ T) => (~ ((S = T))))))). % less_not_refl3
thf(fact_89_less__irrefl__nat, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_irrefl_nat
thf(fact_90_mult__less__mono1, axiom,
    ((![I : nat, J : nat, K : nat]: ((ord_less_nat @ I @ J) => ((ord_less_nat @ zero_zero_nat @ K) => (ord_less_nat @ (times_times_nat @ I @ K) @ (times_times_nat @ J @ K))))))). % mult_less_mono1
thf(fact_91_mult__less__mono2, axiom,
    ((![I : nat, J : nat, K : nat]: ((ord_less_nat @ I @ J) => ((ord_less_nat @ zero_zero_nat @ K) => (ord_less_nat @ (times_times_nat @ K @ I) @ (times_times_nat @ K @ J))))))). % mult_less_mono2
thf(fact_92_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_93_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_94_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_95_mult__eq__self__implies__10, axiom,
    ((![M : nat, N : nat]: ((M = (times_times_nat @ M @ N)) => ((N = one_one_nat) | (M = zero_zero_nat)))))). % mult_eq_self_implies_10
thf(fact_96_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_97_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_98_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_99_power__mult, axiom,
    ((![A : complex, M : nat, N : nat]: ((power_power_complex @ A @ (times_times_nat @ M @ N)) = (power_power_complex @ (power_power_complex @ A @ M) @ N))))). % power_mult
thf(fact_100_power__mult, axiom,
    ((![A : nat, M : nat, N : nat]: ((power_power_nat @ A @ (times_times_nat @ M @ N)) = (power_power_nat @ (power_power_nat @ A @ M) @ N))))). % power_mult
thf(fact_101_mult__0, axiom,
    ((![N : nat]: ((times_times_nat @ zero_zero_nat @ N) = zero_zero_nat)))). % mult_0
thf(fact_102_bot__nat__0_Oextremum__strict, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ zero_zero_nat)))))). % bot_nat_0.extremum_strict
thf(fact_103_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_104_gr__implies__not0, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not0
thf(fact_105_less__zeroE, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_zeroE
thf(fact_106_not__less0, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less0
thf(fact_107_not__gr0, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr0
thf(fact_108_gr0I, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr0I
thf(fact_109_mult__le__mono2, axiom,
    ((![I : nat, J : nat, K : nat]: ((ord_less_eq_nat @ I @ J) => (ord_less_eq_nat @ (times_times_nat @ K @ I) @ (times_times_nat @ K @ J)))))). % mult_le_mono2
thf(fact_110_mult__le__mono1, axiom,
    ((![I : nat, J : nat, K : nat]: ((ord_less_eq_nat @ I @ J) => (ord_less_eq_nat @ (times_times_nat @ I @ K) @ (times_times_nat @ J @ K)))))). % mult_le_mono1
thf(fact_111_mult__le__mono, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: ((ord_less_eq_nat @ I @ J) => ((ord_less_eq_nat @ K @ L) => (ord_less_eq_nat @ (times_times_nat @ I @ K) @ (times_times_nat @ J @ L))))))). % mult_le_mono
thf(fact_112_le__square, axiom,
    ((![M : nat]: (ord_less_eq_nat @ M @ (times_times_nat @ M @ M))))). % le_square
thf(fact_113_le__cube, axiom,
    ((![M : nat]: (ord_less_eq_nat @ M @ (times_times_nat @ M @ (times_times_nat @ M @ M)))))). % le_cube
thf(fact_114_less__mono__imp__le__mono, axiom,
    ((![F : nat > nat, I : nat, J : nat]: ((![I2 : nat, J2 : nat]: ((ord_less_nat @ I2 @ J2) => (ord_less_nat @ (F @ I2) @ (F @ J2)))) => ((ord_less_eq_nat @ I @ J) => (ord_less_eq_nat @ (F @ I) @ (F @ J))))))). % less_mono_imp_le_mono
thf(fact_115_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_116_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_117_le__eq__less__or__eq, axiom,
    ((ord_less_eq_nat = (^[M3 : nat]: (^[N3 : nat]: (((ord_less_nat @ M3 @ N3)) | ((M3 = N3)))))))). % le_eq_less_or_eq
thf(fact_118_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_119_nat__less__le, axiom,
    ((ord_less_nat = (^[M3 : nat]: (^[N3 : nat]: (((ord_less_eq_nat @ M3 @ N3)) & ((~ ((M3 = N3)))))))))). % nat_less_le
thf(fact_120_diff__mult__distrib2, axiom,
    ((![K : nat, M : nat, N : nat]: ((times_times_nat @ K @ (minus_minus_nat @ M @ N)) = (minus_minus_nat @ (times_times_nat @ K @ M) @ (times_times_nat @ K @ N)))))). % diff_mult_distrib2
thf(fact_121_diff__mult__distrib, axiom,
    ((![M : nat, N : nat, K : nat]: ((times_times_nat @ (minus_minus_nat @ M @ N) @ K) = (minus_minus_nat @ (times_times_nat @ M @ K) @ (times_times_nat @ N @ K)))))). % diff_mult_distrib
thf(fact_122_less__imp__diff__less, axiom,
    ((![J : nat, K : nat, N : nat]: ((ord_less_nat @ J @ K) => (ord_less_nat @ (minus_minus_nat @ J @ N) @ K))))). % less_imp_diff_less
thf(fact_123_diff__less__mono2, axiom,
    ((![M : nat, N : nat, L : nat]: ((ord_less_nat @ M @ N) => ((ord_less_nat @ M @ L) => (ord_less_nat @ (minus_minus_nat @ L @ N) @ (minus_minus_nat @ L @ M))))))). % diff_less_mono2
thf(fact_124_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_125_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_126_power__strict__decreasing, axiom,
    ((![N : nat, N4 : nat, A : nat]: ((ord_less_nat @ N @ N4) => ((ord_less_nat @ zero_zero_nat @ A) => ((ord_less_nat @ A @ one_one_nat) => (ord_less_nat @ (power_power_nat @ A @ N4) @ (power_power_nat @ A @ N)))))))). % power_strict_decreasing
thf(fact_127_one__less__power, axiom,
    ((![A : nat, N : nat]: ((ord_less_nat @ one_one_nat @ A) => ((ord_less_nat @ zero_zero_nat @ N) => (ord_less_nat @ one_one_nat @ (power_power_nat @ A @ N))))))). % one_less_power
thf(fact_128_zero__less__power, axiom,
    ((![A : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ A) => (ord_less_nat @ zero_zero_nat @ (power_power_nat @ A @ N)))))). % zero_less_power
thf(fact_129_of__nat__less__0__iff, axiom,
    ((![M : nat]: (~ ((ord_less_nat @ (semiri1382578993at_nat @ M) @ zero_zero_nat)))))). % of_nat_less_0_iff
thf(fact_130_ex__least__nat__le, axiom,
    ((![P : nat > $o, N : nat]: ((P @ N) => ((~ ((P @ zero_zero_nat))) => (?[K2 : nat]: ((ord_less_eq_nat @ K2 @ N) & ((![I3 : nat]: ((ord_less_nat @ I3 @ K2) => (~ ((P @ I3))))) & (P @ K2))))))))). % ex_least_nat_le
thf(fact_131_diff__less, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_nat @ zero_zero_nat @ M) => (ord_less_nat @ (minus_minus_nat @ M @ N) @ M)))))). % diff_less
thf(fact_132_diff__less__mono, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => ((ord_less_eq_nat @ C @ A) => (ord_less_nat @ (minus_minus_nat @ A @ C) @ (minus_minus_nat @ B @ C))))))). % diff_less_mono
thf(fact_133_less__diff__iff, axiom,
    ((![K : nat, M : nat, N : nat]: ((ord_less_eq_nat @ K @ M) => ((ord_less_eq_nat @ K @ N) => ((ord_less_nat @ (minus_minus_nat @ M @ K) @ (minus_minus_nat @ N @ K)) = (ord_less_nat @ M @ N))))))). % less_diff_iff

% Helper facts (3)
thf(help_If_3_1_If_001t__Complex__Ocomplex_T, axiom,
    ((![P : $o]: ((P = $true) | (P = $false))))).
thf(help_If_2_1_If_001t__Complex__Ocomplex_T, axiom,
    ((![X : complex, Y : complex]: ((if_complex @ $false @ X @ Y) = Y)))).
thf(help_If_1_1_If_001t__Complex__Ocomplex_T, axiom,
    ((![X : complex, Y : complex]: ((if_complex @ $true @ X @ Y) = X)))).

% Conjectures (1)
thf(conj_0, conjecture,
    (((groups59700922omplex @ (^[J3 : nat]: (groups59700922omplex @ (^[I4 : nat]: (times_times_complex @ (power_power_complex @ (if_complex @ (ord_less_eq_nat @ i @ J3) @ (power_power_complex @ (fFT_Mirabelle_root @ n) @ (minus_minus_nat @ J3 @ i)) @ (power_power_complex @ (divide1210191872omplex @ one_one_complex @ (fFT_Mirabelle_root @ n)) @ (minus_minus_nat @ i @ J3))) @ I4) @ (a @ J3))) @ (set_or562006527an_nat @ zero_zero_nat @ n))) @ (set_or562006527an_nat @ zero_zero_nat @ n)) = (times_times_complex @ (semiri356525583omplex @ n) @ (a @ i))))).
