% 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_303__3225642_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:10:19.207

% Could-be-implicit typings (4)
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__Num__Onum, type,
    num : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).

% Explicit typings (33)
thf(sy_c_FFT__Mirabelle__ulikgskiun_ODFT, type,
    fFT_Mirabelle_DFT : nat > (nat > complex) > nat > complex).
thf(sy_c_FFT__Mirabelle__ulikgskiun_Oroot, type,
    fFT_Mirabelle_root : nat > complex).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Complex__Ocomplex, type,
    minus_minus_complex : complex > complex > 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_Oplus__class_Oplus_001t__Complex__Ocomplex, type,
    plus_plus_complex : complex > complex > complex).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat, type,
    plus_plus_nat : nat > nat > nat).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Num__Onum, type,
    plus_plus_num : num > num > num).
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_Otimes__class_Otimes_001t__Num__Onum, type,
    times_times_num : num > num > num).
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_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Nat__Onat, type,
    groups1842438620at_nat : (nat > nat) > set_nat > nat).
thf(sy_c_If_001t__Complex__Ocomplex, type,
    if_complex : $o > complex > complex > complex).
thf(sy_c_If_001t__Nat__Onat, type,
    if_nat : $o > nat > nat > nat).
thf(sy_c_Num_Onum_OBit0, type,
    bit0 : num > num).
thf(sy_c_Num_Onum_OOne, type,
    one : num).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Complex__Ocomplex, type,
    numera632737353omplex : num > complex).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat, type,
    numeral_numeral_nat : num > 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_001t__Num__Onum, type,
    ord_less_num : num > num > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat, type,
    ord_less_eq_nat : nat > nat > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum, type,
    ord_less_eq_num : num > num > $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_Set_OCollect_001t__Nat__Onat, type,
    collect_nat : (nat > $o) > set_nat).
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_m, type,
    m : nat).

% Relevant facts (182)
thf(fact_0_mbound, axiom,
    ((ord_less_nat @ zero_zero_nat @ m))). % mbound
thf(fact_1_ibound, axiom,
    ((ord_less_eq_nat @ m @ i))). % ibound
thf(fact_2_calculation, axiom,
    (((groups59700922omplex @ (^[J : nat]: (times_times_complex @ (power_power_complex @ (fFT_Mirabelle_root @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ m)) @ (times_times_nat @ i @ J)) @ (a @ J))) @ (set_or562006527an_nat @ zero_zero_nat @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ m))) = (plus_plus_complex @ (groups59700922omplex @ (^[J : nat]: (times_times_complex @ (power_power_complex @ (fFT_Mirabelle_root @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ m)) @ (times_times_nat @ i @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ J))) @ (a @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ J)))) @ (set_or562006527an_nat @ zero_zero_nat @ m)) @ (groups59700922omplex @ (^[J : nat]: (times_times_complex @ (power_power_complex @ (fFT_Mirabelle_root @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ m)) @ (times_times_nat @ i @ (plus_plus_nat @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ J) @ one_one_nat))) @ (a @ (plus_plus_nat @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ J) @ one_one_nat)))) @ (set_or562006527an_nat @ zero_zero_nat @ m)))))). % calculation
thf(fact_3_sum__splice, axiom,
    ((![F : nat > nat, N : nat]: ((groups1842438620at_nat @ F @ (set_or562006527an_nat @ zero_zero_nat @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N))) = (plus_plus_nat @ (groups1842438620at_nat @ (^[I : nat]: (F @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ I))) @ (set_or562006527an_nat @ zero_zero_nat @ N)) @ (groups1842438620at_nat @ (^[I : nat]: (F @ (plus_plus_nat @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ I) @ one_one_nat))) @ (set_or562006527an_nat @ zero_zero_nat @ N))))))). % sum_splice
thf(fact_4_sum__splice, axiom,
    ((![F : nat > complex, N : nat]: ((groups59700922omplex @ F @ (set_or562006527an_nat @ zero_zero_nat @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N))) = (plus_plus_complex @ (groups59700922omplex @ (^[I : nat]: (F @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ I))) @ (set_or562006527an_nat @ zero_zero_nat @ N)) @ (groups59700922omplex @ (^[I : nat]: (F @ (plus_plus_nat @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ I) @ one_one_nat))) @ (set_or562006527an_nat @ zero_zero_nat @ N))))))). % sum_splice
thf(fact_5_root__cancel1, axiom,
    ((![M : nat, I2 : nat, J2 : nat]: ((power_power_complex @ (fFT_Mirabelle_root @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ M)) @ (times_times_nat @ I2 @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ J2))) = (power_power_complex @ (fFT_Mirabelle_root @ M) @ (times_times_nat @ I2 @ J2)))))). % root_cancel1
thf(fact_6_zero__eq__power2, axiom,
    ((![A : complex]: (((power_power_complex @ A @ (numeral_numeral_nat @ (bit0 @ one))) = zero_zero_complex) = (A = zero_zero_complex))))). % zero_eq_power2
thf(fact_7_zero__eq__power2, axiom,
    ((![A : nat]: (((power_power_nat @ A @ (numeral_numeral_nat @ (bit0 @ one))) = zero_zero_nat) = (A = zero_zero_nat))))). % zero_eq_power2
thf(fact_8_one__add__one, axiom,
    (((plus_plus_complex @ one_one_complex @ one_one_complex) = (numera632737353omplex @ (bit0 @ one))))). % one_add_one
thf(fact_9_one__add__one, axiom,
    (((plus_plus_nat @ one_one_nat @ one_one_nat) = (numeral_numeral_nat @ (bit0 @ one))))). % one_add_one
thf(fact_10_numeral__plus__one, axiom,
    ((![N : num]: ((plus_plus_complex @ (numera632737353omplex @ N) @ one_one_complex) = (numera632737353omplex @ (plus_plus_num @ N @ one)))))). % numeral_plus_one
thf(fact_11_numeral__plus__one, axiom,
    ((![N : num]: ((plus_plus_nat @ (numeral_numeral_nat @ N) @ one_one_nat) = (numeral_numeral_nat @ (plus_plus_num @ N @ one)))))). % numeral_plus_one
thf(fact_12_one__plus__numeral, axiom,
    ((![N : num]: ((plus_plus_complex @ one_one_complex @ (numera632737353omplex @ N)) = (numera632737353omplex @ (plus_plus_num @ one @ N)))))). % one_plus_numeral
thf(fact_13_one__plus__numeral, axiom,
    ((![N : num]: ((plus_plus_nat @ one_one_nat @ (numeral_numeral_nat @ N)) = (numeral_numeral_nat @ (plus_plus_num @ one @ N)))))). % one_plus_numeral
thf(fact_14_DFT__lower, axiom,
    ((![M : nat, A : nat > complex, I2 : nat]: ((fFT_Mirabelle_DFT @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ M) @ A @ I2) = (plus_plus_complex @ (fFT_Mirabelle_DFT @ M @ (^[I : nat]: (A @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ I))) @ I2) @ (times_times_complex @ (power_power_complex @ (fFT_Mirabelle_root @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ M)) @ I2) @ (fFT_Mirabelle_DFT @ M @ (^[I : nat]: (A @ (plus_plus_nat @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ I) @ one_one_nat))) @ I2))))))). % DFT_lower
thf(fact_15_power2__diff, axiom,
    ((![X : complex, Y : complex]: ((power_power_complex @ (minus_minus_complex @ X @ Y) @ (numeral_numeral_nat @ (bit0 @ one))) = (minus_minus_complex @ (plus_plus_complex @ (power_power_complex @ X @ (numeral_numeral_nat @ (bit0 @ one))) @ (power_power_complex @ Y @ (numeral_numeral_nat @ (bit0 @ one)))) @ (times_times_complex @ (times_times_complex @ (numera632737353omplex @ (bit0 @ one)) @ X) @ Y)))))). % power2_diff
thf(fact_16_power__zero__numeral, axiom,
    ((![K : num]: ((power_power_complex @ zero_zero_complex @ (numeral_numeral_nat @ K)) = zero_zero_complex)))). % power_zero_numeral
thf(fact_17_power__zero__numeral, axiom,
    ((![K : num]: ((power_power_nat @ zero_zero_nat @ (numeral_numeral_nat @ K)) = zero_zero_nat)))). % power_zero_numeral
thf(fact_18_numeral__eq__one__iff, axiom,
    ((![N : num]: (((numeral_numeral_nat @ N) = one_one_nat) = (N = one))))). % numeral_eq_one_iff
thf(fact_19_numeral__eq__one__iff, axiom,
    ((![N : num]: (((numera632737353omplex @ N) = one_one_complex) = (N = one))))). % numeral_eq_one_iff
thf(fact_20_one__eq__numeral__iff, axiom,
    ((![N : num]: ((one_one_nat = (numeral_numeral_nat @ N)) = (one = N))))). % one_eq_numeral_iff
thf(fact_21_one__eq__numeral__iff, axiom,
    ((![N : num]: ((one_one_complex = (numera632737353omplex @ N)) = (one = N))))). % one_eq_numeral_iff
thf(fact_22_left__diff__distrib__numeral, axiom,
    ((![A : complex, B : complex, V : num]: ((times_times_complex @ (minus_minus_complex @ A @ B) @ (numera632737353omplex @ V)) = (minus_minus_complex @ (times_times_complex @ A @ (numera632737353omplex @ V)) @ (times_times_complex @ B @ (numera632737353omplex @ V))))))). % left_diff_distrib_numeral
thf(fact_23_right__diff__distrib__numeral, axiom,
    ((![V : num, B : complex, C : complex]: ((times_times_complex @ (numera632737353omplex @ V) @ (minus_minus_complex @ B @ C)) = (minus_minus_complex @ (times_times_complex @ (numera632737353omplex @ V) @ B) @ (times_times_complex @ (numera632737353omplex @ V) @ C)))))). % right_diff_distrib_numeral
thf(fact_24_numeral__eq__iff, axiom,
    ((![M : num, N : num]: (((numeral_numeral_nat @ M) = (numeral_numeral_nat @ N)) = (M = N))))). % numeral_eq_iff
thf(fact_25_numeral__eq__iff, axiom,
    ((![M : num, N : num]: (((numera632737353omplex @ M) = (numera632737353omplex @ N)) = (M = N))))). % numeral_eq_iff
thf(fact_26_numeral__le__iff, axiom,
    ((![M : num, N : num]: ((ord_less_eq_nat @ (numeral_numeral_nat @ M) @ (numeral_numeral_nat @ N)) = (ord_less_eq_num @ M @ N))))). % numeral_le_iff
thf(fact_27_numeral__less__iff, axiom,
    ((![M : num, N : num]: ((ord_less_nat @ (numeral_numeral_nat @ M) @ (numeral_numeral_nat @ N)) = (ord_less_num @ M @ N))))). % numeral_less_iff
thf(fact_28_numeral__times__numeral, axiom,
    ((![M : num, N : num]: ((times_times_nat @ (numeral_numeral_nat @ M) @ (numeral_numeral_nat @ N)) = (numeral_numeral_nat @ (times_times_num @ M @ N)))))). % numeral_times_numeral
thf(fact_29_numeral__times__numeral, axiom,
    ((![M : num, N : num]: ((times_times_complex @ (numera632737353omplex @ M) @ (numera632737353omplex @ N)) = (numera632737353omplex @ (times_times_num @ M @ N)))))). % numeral_times_numeral
thf(fact_30_mult__numeral__left__semiring__numeral, axiom,
    ((![V : num, W : num, Z : nat]: ((times_times_nat @ (numeral_numeral_nat @ V) @ (times_times_nat @ (numeral_numeral_nat @ W) @ Z)) = (times_times_nat @ (numeral_numeral_nat @ (times_times_num @ V @ W)) @ Z))))). % mult_numeral_left_semiring_numeral
thf(fact_31_mult__numeral__left__semiring__numeral, axiom,
    ((![V : num, W : num, Z : complex]: ((times_times_complex @ (numera632737353omplex @ V) @ (times_times_complex @ (numera632737353omplex @ W) @ Z)) = (times_times_complex @ (numera632737353omplex @ (times_times_num @ V @ W)) @ Z))))). % mult_numeral_left_semiring_numeral
thf(fact_32_power__one, axiom,
    ((![N : nat]: ((power_power_complex @ one_one_complex @ N) = one_one_complex)))). % power_one
thf(fact_33_power__one, axiom,
    ((![N : nat]: ((power_power_nat @ one_one_nat @ N) = one_one_nat)))). % power_one
thf(fact_34_num__double, axiom,
    ((![N : num]: ((times_times_num @ (bit0 @ one) @ N) = (bit0 @ N))))). % num_double
thf(fact_35_power__mult__numeral, axiom,
    ((![A : complex, M : num, N : num]: ((power_power_complex @ (power_power_complex @ A @ (numeral_numeral_nat @ M)) @ (numeral_numeral_nat @ N)) = (power_power_complex @ A @ (numeral_numeral_nat @ (times_times_num @ M @ N))))))). % power_mult_numeral
thf(fact_36_power__mult__numeral, axiom,
    ((![A : nat, M : num, N : num]: ((power_power_nat @ (power_power_nat @ A @ (numeral_numeral_nat @ M)) @ (numeral_numeral_nat @ N)) = (power_power_nat @ A @ (numeral_numeral_nat @ (times_times_num @ M @ N))))))). % power_mult_numeral
thf(fact_37_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_38_power__one__right, axiom,
    ((![A : complex]: ((power_power_complex @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_39_power__one__right, axiom,
    ((![A : nat]: ((power_power_nat @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_40_root0, axiom,
    (((fFT_Mirabelle_root @ zero_zero_nat) = one_one_complex))). % root0
thf(fact_41_root1, axiom,
    (((fFT_Mirabelle_root @ one_one_nat) = one_one_complex))). % root1
thf(fact_42_diff__numeral__special_I9_J, axiom,
    (((minus_minus_complex @ one_one_complex @ one_one_complex) = zero_zero_complex))). % diff_numeral_special(9)
thf(fact_43_distrib__right__numeral, axiom,
    ((![A : nat, B : nat, V : num]: ((times_times_nat @ (plus_plus_nat @ A @ B) @ (numeral_numeral_nat @ V)) = (plus_plus_nat @ (times_times_nat @ A @ (numeral_numeral_nat @ V)) @ (times_times_nat @ B @ (numeral_numeral_nat @ V))))))). % distrib_right_numeral
thf(fact_44_distrib__right__numeral, axiom,
    ((![A : complex, B : complex, V : num]: ((times_times_complex @ (plus_plus_complex @ A @ B) @ (numera632737353omplex @ V)) = (plus_plus_complex @ (times_times_complex @ A @ (numera632737353omplex @ V)) @ (times_times_complex @ B @ (numera632737353omplex @ V))))))). % distrib_right_numeral
thf(fact_45_distrib__left__numeral, axiom,
    ((![V : num, B : nat, C : nat]: ((times_times_nat @ (numeral_numeral_nat @ V) @ (plus_plus_nat @ B @ C)) = (plus_plus_nat @ (times_times_nat @ (numeral_numeral_nat @ V) @ B) @ (times_times_nat @ (numeral_numeral_nat @ V) @ C)))))). % distrib_left_numeral
thf(fact_46_distrib__left__numeral, axiom,
    ((![V : num, B : complex, C : complex]: ((times_times_complex @ (numera632737353omplex @ V) @ (plus_plus_complex @ B @ C)) = (plus_plus_complex @ (times_times_complex @ (numera632737353omplex @ V) @ B) @ (times_times_complex @ (numera632737353omplex @ V) @ C)))))). % distrib_left_numeral
thf(fact_47_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_48_numeral__plus__numeral, axiom,
    ((![M : num, N : num]: ((plus_plus_nat @ (numeral_numeral_nat @ M) @ (numeral_numeral_nat @ N)) = (numeral_numeral_nat @ (plus_plus_num @ M @ N)))))). % numeral_plus_numeral
thf(fact_49_numeral__plus__numeral, axiom,
    ((![M : num, N : num]: ((plus_plus_complex @ (numera632737353omplex @ M) @ (numera632737353omplex @ N)) = (numera632737353omplex @ (plus_plus_num @ M @ N)))))). % numeral_plus_numeral
thf(fact_50_add__numeral__left, axiom,
    ((![V : num, W : num, Z : nat]: ((plus_plus_nat @ (numeral_numeral_nat @ V) @ (plus_plus_nat @ (numeral_numeral_nat @ W) @ Z)) = (plus_plus_nat @ (numeral_numeral_nat @ (plus_plus_num @ V @ W)) @ Z))))). % add_numeral_left
thf(fact_51_add__numeral__left, axiom,
    ((![V : num, W : num, Z : complex]: ((plus_plus_complex @ (numera632737353omplex @ V) @ (plus_plus_complex @ (numera632737353omplex @ W) @ Z)) = (plus_plus_complex @ (numera632737353omplex @ (plus_plus_num @ V @ W)) @ Z))))). % add_numeral_left
thf(fact_52_numeral__le__one__iff, axiom,
    ((![N : num]: ((ord_less_eq_nat @ (numeral_numeral_nat @ N) @ one_one_nat) = (ord_less_eq_num @ N @ one))))). % numeral_le_one_iff
thf(fact_53_one__less__numeral__iff, axiom,
    ((![N : num]: ((ord_less_nat @ one_one_nat @ (numeral_numeral_nat @ N)) = (ord_less_num @ one @ N))))). % one_less_numeral_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__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_56_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_57_power__add__numeral2, axiom,
    ((![A : complex, M : num, N : num, B : complex]: ((times_times_complex @ (power_power_complex @ A @ (numeral_numeral_nat @ M)) @ (times_times_complex @ (power_power_complex @ A @ (numeral_numeral_nat @ N)) @ B)) = (times_times_complex @ (power_power_complex @ A @ (numeral_numeral_nat @ (plus_plus_num @ M @ N))) @ B))))). % power_add_numeral2
thf(fact_58_power__add__numeral2, axiom,
    ((![A : nat, M : num, N : num, B : nat]: ((times_times_nat @ (power_power_nat @ A @ (numeral_numeral_nat @ M)) @ (times_times_nat @ (power_power_nat @ A @ (numeral_numeral_nat @ N)) @ B)) = (times_times_nat @ (power_power_nat @ A @ (numeral_numeral_nat @ (plus_plus_num @ M @ N))) @ B))))). % power_add_numeral2
thf(fact_59_power__add__numeral, axiom,
    ((![A : complex, M : num, N : num]: ((times_times_complex @ (power_power_complex @ A @ (numeral_numeral_nat @ M)) @ (power_power_complex @ A @ (numeral_numeral_nat @ N))) = (power_power_complex @ A @ (numeral_numeral_nat @ (plus_plus_num @ M @ N))))))). % power_add_numeral
thf(fact_60_power__add__numeral, axiom,
    ((![A : nat, M : num, N : num]: ((times_times_nat @ (power_power_nat @ A @ (numeral_numeral_nat @ M)) @ (power_power_nat @ A @ (numeral_numeral_nat @ N))) = (power_power_nat @ A @ (numeral_numeral_nat @ (plus_plus_num @ M @ N))))))). % power_add_numeral
thf(fact_61_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_62_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_63_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_64_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_65_power2__eq__iff__nonneg, axiom,
    ((![X : nat, Y : nat]: ((ord_less_eq_nat @ zero_zero_nat @ X) => ((ord_less_eq_nat @ zero_zero_nat @ Y) => (((power_power_nat @ X @ (numeral_numeral_nat @ (bit0 @ one))) = (power_power_nat @ Y @ (numeral_numeral_nat @ (bit0 @ one)))) = (X = Y))))))). % power2_eq_iff_nonneg
thf(fact_66_nat__power__less__imp__less, axiom,
    ((![I2 : nat, M : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ I2) => ((ord_less_nat @ (power_power_nat @ I2 @ M) @ (power_power_nat @ I2 @ N)) => (ord_less_nat @ M @ N)))))). % nat_power_less_imp_less
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_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_nat @ zero_zero_nat @ zero_zero_nat))))). % less_numeral_extra(3)
thf(fact_69_le__numeral__extra_I3_J, axiom,
    ((ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat))). % le_numeral_extra(3)
thf(fact_70_less__numeral__extra_I4_J, axiom,
    ((~ ((ord_less_nat @ one_one_nat @ one_one_nat))))). % less_numeral_extra(4)
thf(fact_71_add__One__commute, axiom,
    ((![N : num]: ((plus_plus_num @ one @ N) = (plus_plus_num @ N @ one))))). % add_One_commute
thf(fact_72_le__numeral__extra_I4_J, axiom,
    ((ord_less_eq_nat @ one_one_nat @ one_one_nat))). % le_numeral_extra(4)
thf(fact_73_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_74_power__strict__increasing, axiom,
    ((![N : nat, N2 : nat, A : nat]: ((ord_less_nat @ N @ N2) => ((ord_less_nat @ one_one_nat @ A) => (ord_less_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ A @ N2))))))). % power_strict_increasing
thf(fact_75_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_76_power__increasing, axiom,
    ((![N : nat, N2 : nat, A : nat]: ((ord_less_eq_nat @ N @ N2) => ((ord_less_eq_nat @ one_one_nat @ A) => (ord_less_eq_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ A @ N2))))))). % power_increasing
thf(fact_77_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_78_power__strict__decreasing, axiom,
    ((![N : nat, N2 : nat, A : nat]: ((ord_less_nat @ N @ N2) => ((ord_less_nat @ zero_zero_nat @ A) => ((ord_less_nat @ A @ one_one_nat) => (ord_less_nat @ (power_power_nat @ A @ N2) @ (power_power_nat @ A @ N)))))))). % power_strict_decreasing
thf(fact_79_power__decreasing, axiom,
    ((![N : nat, N2 : nat, A : nat]: ((ord_less_eq_nat @ N @ N2) => ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_eq_nat @ A @ one_one_nat) => (ord_less_eq_nat @ (power_power_nat @ A @ N2) @ (power_power_nat @ A @ N)))))))). % power_decreasing
thf(fact_80_power__eq__imp__eq__base, axiom,
    ((![A : nat, N : nat, B : nat]: (((power_power_nat @ A @ N) = (power_power_nat @ B @ N)) => ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_eq_nat @ zero_zero_nat @ B) => ((ord_less_nat @ zero_zero_nat @ N) => (A = B)))))))). % power_eq_imp_eq_base
thf(fact_81_power__eq__iff__eq__base, axiom,
    ((![N : nat, A : nat, B : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_eq_nat @ zero_zero_nat @ B) => (((power_power_nat @ A @ N) = (power_power_nat @ B @ N)) = (A = B)))))))). % power_eq_iff_eq_base
thf(fact_82_power2__nat__le__imp__le, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ (power_power_nat @ M @ (numeral_numeral_nat @ (bit0 @ one))) @ N) => (ord_less_eq_nat @ M @ N))))). % power2_nat_le_imp_le
thf(fact_83_power2__nat__le__eq__le, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ (power_power_nat @ M @ (numeral_numeral_nat @ (bit0 @ one))) @ (power_power_nat @ N @ (numeral_numeral_nat @ (bit0 @ one)))) = (ord_less_eq_nat @ M @ N))))). % power2_nat_le_eq_le
thf(fact_84_self__le__ge2__pow, axiom,
    ((![K : nat, M : nat]: ((ord_less_eq_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ K) => (ord_less_eq_nat @ M @ (power_power_nat @ K @ M)))))). % self_le_ge2_pow
thf(fact_85_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_86_self__le__power, axiom,
    ((![A : nat, N : nat]: ((ord_less_eq_nat @ one_one_nat @ A) => ((ord_less_nat @ zero_zero_nat @ N) => (ord_less_eq_nat @ A @ (power_power_nat @ A @ N))))))). % self_le_power
thf(fact_87_not__numeral__less__zero, axiom,
    ((![N : num]: (~ ((ord_less_nat @ (numeral_numeral_nat @ N) @ zero_zero_nat)))))). % not_numeral_less_zero
thf(fact_88_zero__less__numeral, axiom,
    ((![N : num]: (ord_less_nat @ zero_zero_nat @ (numeral_numeral_nat @ N))))). % zero_less_numeral
thf(fact_89_less__numeral__extra_I1_J, axiom,
    ((ord_less_nat @ zero_zero_nat @ one_one_nat))). % less_numeral_extra(1)
thf(fact_90_not__numeral__le__zero, axiom,
    ((![N : num]: (~ ((ord_less_eq_nat @ (numeral_numeral_nat @ N) @ zero_zero_nat)))))). % not_numeral_le_zero
thf(fact_91_zero__le__numeral, axiom,
    ((![N : num]: (ord_less_eq_nat @ zero_zero_nat @ (numeral_numeral_nat @ N))))). % zero_le_numeral
thf(fact_92_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_93_not__numeral__less__one, axiom,
    ((![N : num]: (~ ((ord_less_nat @ (numeral_numeral_nat @ N) @ one_one_nat)))))). % not_numeral_less_one
thf(fact_94_zero__le__power, axiom,
    ((![A : nat, N : nat]: ((ord_less_eq_nat @ zero_zero_nat @ A) => (ord_less_eq_nat @ zero_zero_nat @ (power_power_nat @ A @ N)))))). % zero_le_power
thf(fact_95_power__mono, axiom,
    ((![A : nat, B : nat, N : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ zero_zero_nat @ A) => (ord_less_eq_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ B @ N))))))). % power_mono
thf(fact_96_one__le__numeral, axiom,
    ((![N : num]: (ord_less_eq_nat @ one_one_nat @ (numeral_numeral_nat @ N))))). % one_le_numeral
thf(fact_97_ex__power__ivl2, axiom,
    ((![B : nat, K : nat]: ((ord_less_eq_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ B) => ((ord_less_eq_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ K) => (?[N3 : nat]: ((ord_less_nat @ (power_power_nat @ B @ N3) @ K) & (ord_less_eq_nat @ K @ (power_power_nat @ B @ (plus_plus_nat @ N3 @ one_one_nat)))))))))). % ex_power_ivl2
thf(fact_98_ex__power__ivl1, axiom,
    ((![B : nat, K : nat]: ((ord_less_eq_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ B) => ((ord_less_eq_nat @ one_one_nat @ K) => (?[N3 : nat]: ((ord_less_eq_nat @ (power_power_nat @ B @ N3) @ K) & (ord_less_nat @ K @ (power_power_nat @ B @ (plus_plus_nat @ N3 @ one_one_nat)))))))))). % ex_power_ivl1
thf(fact_99_one__le__power, axiom,
    ((![A : nat, N : nat]: ((ord_less_eq_nat @ one_one_nat @ A) => (ord_less_eq_nat @ one_one_nat @ (power_power_nat @ A @ N)))))). % one_le_power
thf(fact_100_diff__le__diff__pow, axiom,
    ((![K : nat, M : nat, N : nat]: ((ord_less_eq_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ K) => (ord_less_eq_nat @ (minus_minus_nat @ M @ N) @ (minus_minus_nat @ (power_power_nat @ K @ M) @ (power_power_nat @ K @ N))))))). % diff_le_diff_pow
thf(fact_101_sum__add__split__nat__ivl, axiom,
    ((![M : nat, K : nat, N : nat, G : nat > complex, F : nat > complex, H : nat > complex]: ((ord_less_eq_nat @ M @ K) => ((ord_less_eq_nat @ K @ N) => ((![I3 : nat]: ((ord_less_eq_nat @ M @ I3) => ((ord_less_nat @ I3 @ K) => ((G @ I3) = (F @ I3))))) => ((![I3 : nat]: ((ord_less_eq_nat @ K @ I3) => ((ord_less_nat @ I3 @ N) => ((H @ I3) = (F @ I3))))) => ((plus_plus_complex @ (groups59700922omplex @ G @ (set_or562006527an_nat @ M @ K)) @ (groups59700922omplex @ H @ (set_or562006527an_nat @ K @ N))) = (groups59700922omplex @ F @ (set_or562006527an_nat @ M @ N)))))))))). % sum_add_split_nat_ivl
thf(fact_102_sum__add__split__nat__ivl, axiom,
    ((![M : nat, K : nat, N : nat, G : nat > nat, F : nat > nat, H : nat > nat]: ((ord_less_eq_nat @ M @ K) => ((ord_less_eq_nat @ K @ N) => ((![I3 : nat]: ((ord_less_eq_nat @ M @ I3) => ((ord_less_nat @ I3 @ K) => ((G @ I3) = (F @ I3))))) => ((![I3 : nat]: ((ord_less_eq_nat @ K @ I3) => ((ord_less_nat @ I3 @ N) => ((H @ I3) = (F @ I3))))) => ((plus_plus_nat @ (groups1842438620at_nat @ G @ (set_or562006527an_nat @ M @ K)) @ (groups1842438620at_nat @ H @ (set_or562006527an_nat @ K @ N))) = (groups1842438620at_nat @ F @ (set_or562006527an_nat @ M @ N)))))))))). % sum_add_split_nat_ivl
thf(fact_103_root__nonzero, axiom,
    ((![N : nat]: (~ (((fFT_Mirabelle_root @ N) = zero_zero_complex)))))). % root_nonzero
thf(fact_104_power2__less__imp__less, axiom,
    ((![X : nat, Y : nat]: ((ord_less_nat @ (power_power_nat @ X @ (numeral_numeral_nat @ (bit0 @ one))) @ (power_power_nat @ Y @ (numeral_numeral_nat @ (bit0 @ one)))) => ((ord_less_eq_nat @ zero_zero_nat @ Y) => (ord_less_nat @ X @ Y)))))). % power2_less_imp_less
thf(fact_105_power__less__power__Suc, axiom,
    ((![A : nat, N : nat]: ((ord_less_nat @ one_one_nat @ A) => (ord_less_nat @ (power_power_nat @ A @ N) @ (times_times_nat @ A @ (power_power_nat @ A @ N))))))). % power_less_power_Suc
thf(fact_106_power__gt1__lemma, axiom,
    ((![A : nat, N : nat]: ((ord_less_nat @ one_one_nat @ A) => (ord_less_nat @ one_one_nat @ (times_times_nat @ A @ (power_power_nat @ A @ N))))))). % power_gt1_lemma
thf(fact_107_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_108_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_109_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_110_root__unity, axiom,
    ((![N : nat]: ((power_power_complex @ (fFT_Mirabelle_root @ N) @ N) = one_one_complex)))). % root_unity
thf(fact_111_power__Suc__less, axiom,
    ((![A : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ A) => ((ord_less_nat @ A @ one_one_nat) => (ord_less_nat @ (times_times_nat @ A @ (power_power_nat @ A @ N)) @ (power_power_nat @ A @ N))))))). % power_Suc_less
thf(fact_112_is__num__normalize_I1_J, axiom,
    ((![A : complex, B : complex, C : complex]: ((plus_plus_complex @ (plus_plus_complex @ A @ B) @ C) = (plus_plus_complex @ A @ (plus_plus_complex @ B @ C)))))). % is_num_normalize(1)
thf(fact_113_root__summation, axiom,
    ((![K : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ K) => ((ord_less_nat @ K @ N) => ((groups59700922omplex @ (power_power_complex @ (power_power_complex @ (fFT_Mirabelle_root @ N) @ K)) @ (set_or562006527an_nat @ zero_zero_nat @ N)) = zero_zero_complex)))))). % root_summation
thf(fact_114_power2__le__imp__le, axiom,
    ((![X : nat, Y : nat]: ((ord_less_eq_nat @ (power_power_nat @ X @ (numeral_numeral_nat @ (bit0 @ one))) @ (power_power_nat @ Y @ (numeral_numeral_nat @ (bit0 @ one)))) => ((ord_less_eq_nat @ zero_zero_nat @ Y) => (ord_less_eq_nat @ X @ Y)))))). % power2_le_imp_le
thf(fact_115_power2__eq__imp__eq, axiom,
    ((![X : nat, Y : nat]: (((power_power_nat @ X @ (numeral_numeral_nat @ (bit0 @ one))) = (power_power_nat @ Y @ (numeral_numeral_nat @ (bit0 @ one)))) => ((ord_less_eq_nat @ zero_zero_nat @ X) => ((ord_less_eq_nat @ zero_zero_nat @ Y) => (X = Y))))))). % power2_eq_imp_eq
thf(fact_116_power__minus__mult, axiom,
    ((![N : nat, A : complex]: ((ord_less_nat @ zero_zero_nat @ N) => ((times_times_complex @ (power_power_complex @ A @ (minus_minus_nat @ N @ one_one_nat)) @ A) = (power_power_complex @ A @ N)))))). % power_minus_mult
thf(fact_117_power__minus__mult, axiom,
    ((![N : nat, A : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((times_times_nat @ (power_power_nat @ A @ (minus_minus_nat @ N @ one_one_nat)) @ A) = (power_power_nat @ A @ N)))))). % power_minus_mult
thf(fact_118_root__cancel, axiom,
    ((![D : nat, N : nat, K : nat]: ((ord_less_nat @ zero_zero_nat @ D) => ((power_power_complex @ (fFT_Mirabelle_root @ (times_times_nat @ D @ N)) @ (times_times_nat @ D @ K)) = (power_power_complex @ (fFT_Mirabelle_root @ N) @ K)))))). % root_cancel
thf(fact_119_zero__neq__numeral, axiom,
    ((![N : num]: (~ ((zero_zero_nat = (numeral_numeral_nat @ N))))))). % zero_neq_numeral
thf(fact_120_zero__neq__numeral, axiom,
    ((![N : num]: (~ ((zero_zero_complex = (numera632737353omplex @ N))))))). % zero_neq_numeral
thf(fact_121_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_122_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_123_power__commuting__commutes, axiom,
    ((![X : complex, Y : complex, N : nat]: (((times_times_complex @ X @ Y) = (times_times_complex @ Y @ X)) => ((times_times_complex @ (power_power_complex @ X @ N) @ Y) = (times_times_complex @ Y @ (power_power_complex @ X @ N))))))). % power_commuting_commutes
thf(fact_124_power__commuting__commutes, axiom,
    ((![X : nat, Y : nat, N : nat]: (((times_times_nat @ X @ Y) = (times_times_nat @ Y @ X)) => ((times_times_nat @ (power_power_nat @ X @ N) @ Y) = (times_times_nat @ Y @ (power_power_nat @ X @ N))))))). % power_commuting_commutes
thf(fact_125_power__mult__distrib, axiom,
    ((![A : complex, B : complex, N : nat]: ((power_power_complex @ (times_times_complex @ A @ B) @ N) = (times_times_complex @ (power_power_complex @ A @ N) @ (power_power_complex @ B @ N)))))). % power_mult_distrib
thf(fact_126_power__mult__distrib, axiom,
    ((![A : nat, B : nat, N : nat]: ((power_power_nat @ (times_times_nat @ A @ B) @ N) = (times_times_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ B @ N)))))). % power_mult_distrib
thf(fact_127_power__commutes, axiom,
    ((![A : complex, N : nat]: ((times_times_complex @ (power_power_complex @ A @ N) @ A) = (times_times_complex @ A @ (power_power_complex @ A @ N)))))). % power_commutes
thf(fact_128_power__commutes, axiom,
    ((![A : nat, N : nat]: ((times_times_nat @ (power_power_nat @ A @ N) @ A) = (times_times_nat @ A @ (power_power_nat @ A @ N)))))). % power_commutes
thf(fact_129_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_130_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_131_one__plus__numeral__commute, axiom,
    ((![X : num]: ((plus_plus_nat @ one_one_nat @ (numeral_numeral_nat @ X)) = (plus_plus_nat @ (numeral_numeral_nat @ X) @ one_one_nat))))). % one_plus_numeral_commute
thf(fact_132_one__plus__numeral__commute, axiom,
    ((![X : num]: ((plus_plus_complex @ one_one_complex @ (numera632737353omplex @ X)) = (plus_plus_complex @ (numera632737353omplex @ X) @ one_one_complex))))). % one_plus_numeral_commute
thf(fact_133_mult__numeral__1__right, axiom,
    ((![A : nat]: ((times_times_nat @ A @ (numeral_numeral_nat @ one)) = A)))). % mult_numeral_1_right
thf(fact_134_mult__numeral__1__right, axiom,
    ((![A : complex]: ((times_times_complex @ A @ (numera632737353omplex @ one)) = A)))). % mult_numeral_1_right
thf(fact_135_mult__numeral__1, axiom,
    ((![A : nat]: ((times_times_nat @ (numeral_numeral_nat @ one) @ A) = A)))). % mult_numeral_1
thf(fact_136_mult__numeral__1, axiom,
    ((![A : complex]: ((times_times_complex @ (numera632737353omplex @ one) @ A) = A)))). % mult_numeral_1
thf(fact_137_numeral__Bit0, axiom,
    ((![N : num]: ((numeral_numeral_nat @ (bit0 @ N)) = (plus_plus_nat @ (numeral_numeral_nat @ N) @ (numeral_numeral_nat @ N)))))). % numeral_Bit0
thf(fact_138_numeral__Bit0, axiom,
    ((![N : num]: ((numera632737353omplex @ (bit0 @ N)) = (plus_plus_complex @ (numera632737353omplex @ N) @ (numera632737353omplex @ N)))))). % numeral_Bit0
thf(fact_139_numeral__One, axiom,
    (((numeral_numeral_nat @ one) = one_one_nat))). % numeral_One
thf(fact_140_numeral__One, axiom,
    (((numera632737353omplex @ one) = one_one_complex))). % numeral_One
thf(fact_141_left__right__inverse__power, axiom,
    ((![X : complex, Y : complex, N : nat]: (((times_times_complex @ X @ Y) = one_one_complex) => ((times_times_complex @ (power_power_complex @ X @ N) @ (power_power_complex @ Y @ N)) = one_one_complex))))). % left_right_inverse_power
thf(fact_142_left__right__inverse__power, axiom,
    ((![X : nat, Y : nat, N : nat]: (((times_times_nat @ X @ Y) = one_one_nat) => ((times_times_nat @ (power_power_nat @ X @ N) @ (power_power_nat @ Y @ N)) = one_one_nat))))). % left_right_inverse_power
thf(fact_143_power__0, axiom,
    ((![A : complex]: ((power_power_complex @ A @ zero_zero_nat) = one_one_complex)))). % power_0
thf(fact_144_power__0, axiom,
    ((![A : nat]: ((power_power_nat @ A @ zero_zero_nat) = one_one_nat)))). % power_0
thf(fact_145_power__add, axiom,
    ((![A : complex, M : nat, N : nat]: ((power_power_complex @ A @ (plus_plus_nat @ M @ N)) = (times_times_complex @ (power_power_complex @ A @ M) @ (power_power_complex @ A @ N)))))). % power_add
thf(fact_146_power__add, axiom,
    ((![A : nat, M : nat, N : nat]: ((power_power_nat @ A @ (plus_plus_nat @ M @ N)) = (times_times_nat @ (power_power_nat @ A @ M) @ (power_power_nat @ A @ N)))))). % power_add
thf(fact_147_numerals_I1_J, axiom,
    (((numeral_numeral_nat @ one) = one_one_nat))). % numerals(1)
thf(fact_148_numeral__code_I2_J, axiom,
    ((![N : num]: ((numeral_numeral_nat @ (bit0 @ N)) = (plus_plus_nat @ (numeral_numeral_nat @ N) @ (numeral_numeral_nat @ N)))))). % numeral_code(2)
thf(fact_149_numeral__code_I2_J, axiom,
    ((![N : num]: ((numera632737353omplex @ (bit0 @ N)) = (plus_plus_complex @ (numera632737353omplex @ N) @ (numera632737353omplex @ N)))))). % numeral_code(2)
thf(fact_150_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_151_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_152_power__numeral__even, axiom,
    ((![Z : complex, W : num]: ((power_power_complex @ Z @ (numeral_numeral_nat @ (bit0 @ W))) = (times_times_complex @ (power_power_complex @ Z @ (numeral_numeral_nat @ W)) @ (power_power_complex @ Z @ (numeral_numeral_nat @ W))))))). % power_numeral_even
thf(fact_153_power__numeral__even, axiom,
    ((![Z : nat, W : num]: ((power_power_nat @ Z @ (numeral_numeral_nat @ (bit0 @ W))) = (times_times_nat @ (power_power_nat @ Z @ (numeral_numeral_nat @ W)) @ (power_power_nat @ Z @ (numeral_numeral_nat @ W))))))). % power_numeral_even
thf(fact_154_left__add__twice, axiom,
    ((![A : nat, B : nat]: ((plus_plus_nat @ A @ (plus_plus_nat @ A @ B)) = (plus_plus_nat @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ A) @ B))))). % left_add_twice
thf(fact_155_left__add__twice, axiom,
    ((![A : complex, B : complex]: ((plus_plus_complex @ A @ (plus_plus_complex @ A @ B)) = (plus_plus_complex @ (times_times_complex @ (numera632737353omplex @ (bit0 @ one)) @ A) @ B))))). % left_add_twice
thf(fact_156_mult__2__right, axiom,
    ((![Z : nat]: ((times_times_nat @ Z @ (numeral_numeral_nat @ (bit0 @ one))) = (plus_plus_nat @ Z @ Z))))). % mult_2_right
thf(fact_157_mult__2__right, axiom,
    ((![Z : complex]: ((times_times_complex @ Z @ (numera632737353omplex @ (bit0 @ one))) = (plus_plus_complex @ Z @ Z))))). % mult_2_right
thf(fact_158_mult__2, axiom,
    ((![Z : nat]: ((times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ Z) = (plus_plus_nat @ Z @ Z))))). % mult_2
thf(fact_159_mult__2, axiom,
    ((![Z : complex]: ((times_times_complex @ (numera632737353omplex @ (bit0 @ one)) @ Z) = (plus_plus_complex @ Z @ Z))))). % mult_2
thf(fact_160_zero__power2, axiom,
    (((power_power_complex @ zero_zero_complex @ (numeral_numeral_nat @ (bit0 @ one))) = zero_zero_complex))). % zero_power2
thf(fact_161_zero__power2, axiom,
    (((power_power_nat @ zero_zero_nat @ (numeral_numeral_nat @ (bit0 @ one))) = zero_zero_nat))). % zero_power2
thf(fact_162_power2__eq__square, axiom,
    ((![A : complex]: ((power_power_complex @ A @ (numeral_numeral_nat @ (bit0 @ one))) = (times_times_complex @ A @ A))))). % power2_eq_square
thf(fact_163_power2__eq__square, axiom,
    ((![A : nat]: ((power_power_nat @ A @ (numeral_numeral_nat @ (bit0 @ one))) = (times_times_nat @ A @ A))))). % power2_eq_square
thf(fact_164_power4__eq__xxxx, axiom,
    ((![X : complex]: ((power_power_complex @ X @ (numeral_numeral_nat @ (bit0 @ (bit0 @ one)))) = (times_times_complex @ (times_times_complex @ (times_times_complex @ X @ X) @ X) @ X))))). % power4_eq_xxxx
thf(fact_165_power4__eq__xxxx, axiom,
    ((![X : nat]: ((power_power_nat @ X @ (numeral_numeral_nat @ (bit0 @ (bit0 @ one)))) = (times_times_nat @ (times_times_nat @ (times_times_nat @ X @ X) @ X) @ X))))). % power4_eq_xxxx
thf(fact_166_one__power2, axiom,
    (((power_power_complex @ one_one_complex @ (numeral_numeral_nat @ (bit0 @ one))) = one_one_complex))). % one_power2
thf(fact_167_one__power2, axiom,
    (((power_power_nat @ one_one_nat @ (numeral_numeral_nat @ (bit0 @ one))) = one_one_nat))). % one_power2
thf(fact_168_power2__commute, axiom,
    ((![X : complex, Y : complex]: ((power_power_complex @ (minus_minus_complex @ X @ Y) @ (numeral_numeral_nat @ (bit0 @ one))) = (power_power_complex @ (minus_minus_complex @ Y @ X) @ (numeral_numeral_nat @ (bit0 @ one))))))). % power2_commute
thf(fact_169_power__even__eq, axiom,
    ((![A : complex, N : nat]: ((power_power_complex @ A @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N)) = (power_power_complex @ (power_power_complex @ A @ N) @ (numeral_numeral_nat @ (bit0 @ one))))))). % power_even_eq
thf(fact_170_power__even__eq, axiom,
    ((![A : nat, N : nat]: ((power_power_nat @ A @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N)) = (power_power_nat @ (power_power_nat @ A @ N) @ (numeral_numeral_nat @ (bit0 @ one))))))). % power_even_eq
thf(fact_171_nat__1__add__1, axiom,
    (((plus_plus_nat @ one_one_nat @ one_one_nat) = (numeral_numeral_nat @ (bit0 @ one))))). % nat_1_add_1
thf(fact_172_power__eq__if, axiom,
    ((power_power_complex = (^[P : complex]: (^[M2 : nat]: (if_complex @ (M2 = zero_zero_nat) @ one_one_complex @ (times_times_complex @ P @ (power_power_complex @ P @ (minus_minus_nat @ M2 @ one_one_nat))))))))). % power_eq_if
thf(fact_173_power__eq__if, axiom,
    ((power_power_nat = (^[P : nat]: (^[M2 : nat]: (if_nat @ (M2 = zero_zero_nat) @ one_one_nat @ (times_times_nat @ P @ (power_power_nat @ P @ (minus_minus_nat @ M2 @ one_one_nat))))))))). % power_eq_if
thf(fact_174_DFT__def, axiom,
    ((fFT_Mirabelle_DFT = (^[N4 : nat]: (^[A2 : nat > complex]: (^[I : nat]: (groups59700922omplex @ (^[J : nat]: (times_times_complex @ (power_power_complex @ (fFT_Mirabelle_root @ N4) @ (times_times_nat @ I @ J)) @ (A2 @ J))) @ (set_or562006527an_nat @ zero_zero_nat @ N4)))))))). % DFT_def
thf(fact_175_power2__sum, axiom,
    ((![X : nat, Y : nat]: ((power_power_nat @ (plus_plus_nat @ X @ Y) @ (numeral_numeral_nat @ (bit0 @ one))) = (plus_plus_nat @ (plus_plus_nat @ (power_power_nat @ X @ (numeral_numeral_nat @ (bit0 @ one))) @ (power_power_nat @ Y @ (numeral_numeral_nat @ (bit0 @ one)))) @ (times_times_nat @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ X) @ Y)))))). % power2_sum
thf(fact_176_power2__sum, axiom,
    ((![X : complex, Y : complex]: ((power_power_complex @ (plus_plus_complex @ X @ Y) @ (numeral_numeral_nat @ (bit0 @ one))) = (plus_plus_complex @ (plus_plus_complex @ (power_power_complex @ X @ (numeral_numeral_nat @ (bit0 @ one))) @ (power_power_complex @ Y @ (numeral_numeral_nat @ (bit0 @ one)))) @ (times_times_complex @ (times_times_complex @ (numera632737353omplex @ (bit0 @ one)) @ X) @ Y)))))). % power2_sum
thf(fact_177_nat__mult__le__cancel__disj, axiom,
    ((![K : nat, M : nat, N : nat]: ((ord_less_eq_nat @ (times_times_nat @ K @ M) @ (times_times_nat @ K @ N)) = (((ord_less_nat @ zero_zero_nat @ K)) => ((ord_less_eq_nat @ M @ N))))))). % nat_mult_le_cancel_disj
thf(fact_178_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_179_mask__eq__sum__exp, axiom,
    ((![N : nat]: ((minus_minus_nat @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N) @ one_one_nat) = (groups1842438620at_nat @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one))) @ (collect_nat @ (^[Q : nat]: (ord_less_nat @ Q @ N)))))))). % mask_eq_sum_exp
thf(fact_180_semiring__norm_I2_J, axiom,
    (((plus_plus_num @ one @ one) = (bit0 @ one)))). % semiring_norm(2)
thf(fact_181_Nat_Odiff__diff__right, axiom,
    ((![K : nat, J2 : nat, I2 : nat]: ((ord_less_eq_nat @ K @ J2) => ((minus_minus_nat @ I2 @ (minus_minus_nat @ J2 @ K)) = (minus_minus_nat @ (plus_plus_nat @ I2 @ K) @ J2)))))). % Nat.diff_diff_right

% Helper facts (5)
thf(help_If_2_1_If_001t__Nat__Onat_T, axiom,
    ((![X : nat, Y : nat]: ((if_nat @ $false @ X @ Y) = Y)))).
thf(help_If_1_1_If_001t__Nat__Onat_T, axiom,
    ((![X : nat, Y : nat]: ((if_nat @ $true @ X @ Y) = X)))).
thf(help_If_3_1_If_001t__Complex__Ocomplex_T, axiom,
    ((![P2 : $o]: ((P2 = $true) | (P2 = $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,
    (((plus_plus_complex @ (groups59700922omplex @ (^[J : nat]: (times_times_complex @ (power_power_complex @ (fFT_Mirabelle_root @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ m)) @ (times_times_nat @ i @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ J))) @ (a @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ J)))) @ (set_or562006527an_nat @ zero_zero_nat @ m)) @ (groups59700922omplex @ (^[J : nat]: (times_times_complex @ (power_power_complex @ (fFT_Mirabelle_root @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ m)) @ (times_times_nat @ i @ (plus_plus_nat @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ J) @ one_one_nat))) @ (a @ (plus_plus_nat @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ J) @ one_one_nat)))) @ (set_or562006527an_nat @ zero_zero_nat @ m))) = (minus_minus_complex @ (groups59700922omplex @ (^[J : nat]: (times_times_complex @ (power_power_complex @ (fFT_Mirabelle_root @ m) @ (times_times_nat @ (minus_minus_nat @ i @ m) @ J)) @ (a @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ J)))) @ (set_or562006527an_nat @ zero_zero_nat @ m)) @ (times_times_complex @ (power_power_complex @ (fFT_Mirabelle_root @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ m)) @ (minus_minus_nat @ i @ m)) @ (groups59700922omplex @ (^[J : nat]: (times_times_complex @ (power_power_complex @ (fFT_Mirabelle_root @ m) @ (times_times_nat @ (minus_minus_nat @ i @ m) @ J)) @ (a @ (plus_plus_nat @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ J) @ one_one_nat)))) @ (set_or562006527an_nat @ zero_zero_nat @ m))))))).
