% 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_320__3225778_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:10:28.948

% Could-be-implicit typings (5)
thf(ty_n_t__Complex__Ocomplex, type,
    complex : $tType).
thf(ty_n_t__Real__Oreal, type,
    real : $tType).
thf(ty_n_t__Num__Onum, type,
    num : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).
thf(ty_n_t__Int__Oint, type,
    int : $tType).

% Explicit typings (36)
thf(sy_c_FFT__Mirabelle__ulikgskiun_ODFT, type,
    fFT_Mirabelle_DFT : nat > (nat > complex) > nat > complex).
thf(sy_c_FFT__Mirabelle__ulikgskiun_OIDFT, type,
    fFT_Mirabelle_IDFT : nat > (nat > complex) > nat > complex).
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__Int__Oint, type,
    one_one_int : int).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat, type,
    one_one_nat : nat).
thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal, type,
    one_one_real : real).
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__Int__Oint, type,
    plus_plus_int : int > int > int).
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_Oplus__class_Oplus_001t__Real__Oreal, type,
    plus_plus_real : real > real > real).
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__Int__Oint, type,
    times_times_int : int > int > int).
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_Otimes__class_Otimes_001t__Real__Oreal, type,
    times_times_real : real > real > real).
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__Int__Oint, type,
    numeral_numeral_int : num > int).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat, type,
    numeral_numeral_nat : num > nat).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal, type,
    numeral_numeral_real : num > real).
thf(sy_c_Num_Opow, type,
    pow : num > num > num).
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__Int__Oint, type,
    power_power_int : int > nat > int).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat, type,
    power_power_nat : nat > nat > nat).
thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal, type,
    power_power_real : real > nat > real).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex, type,
    divide1210191872omplex : complex > complex > complex).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Int__Oint, type,
    divide_divide_int : int > int > int).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat, type,
    divide_divide_nat : nat > nat > nat).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal, type,
    divide_divide_real : real > real > real).
thf(sy_c_Rings_Odvd__class_Odvd_001t__Nat__Onat, type,
    dvd_dvd_nat : nat > nat > $o).
thf(sy_v_a, type,
    a : nat > complex).
thf(sy_v_i, type,
    i : nat).
thf(sy_v_m, type,
    m : nat).

% Relevant facts (245)
thf(fact_0_root1, axiom,
    (((fFT_Mirabelle_root @ one_one_nat) = one_one_complex))). % root1
thf(fact_1_root__unity, axiom,
    ((![N : nat]: ((power_power_complex @ (fFT_Mirabelle_root @ N) @ N) = one_one_complex)))). % root_unity
thf(fact_2_root__cancel1, axiom,
    ((![M : nat, I : nat, J : nat]: ((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))) = (power_power_complex @ (fFT_Mirabelle_root @ M) @ (times_times_nat @ I @ J)))))). % root_cancel1
thf(fact_3_one__add__one, axiom,
    (((plus_plus_complex @ one_one_complex @ one_one_complex) = (numera632737353omplex @ (bit0 @ one))))). % one_add_one
thf(fact_4_one__add__one, axiom,
    (((plus_plus_nat @ one_one_nat @ one_one_nat) = (numeral_numeral_nat @ (bit0 @ one))))). % one_add_one
thf(fact_5_one__add__one, axiom,
    (((plus_plus_real @ one_one_real @ one_one_real) = (numeral_numeral_real @ (bit0 @ one))))). % one_add_one
thf(fact_6_one__add__one, axiom,
    (((plus_plus_int @ one_one_int @ one_one_int) = (numeral_numeral_int @ (bit0 @ one))))). % one_add_one
thf(fact_7_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_8_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_9_numeral__plus__one, axiom,
    ((![N : num]: ((plus_plus_real @ (numeral_numeral_real @ N) @ one_one_real) = (numeral_numeral_real @ (plus_plus_num @ N @ one)))))). % numeral_plus_one
thf(fact_10_numeral__plus__one, axiom,
    ((![N : num]: ((plus_plus_int @ (numeral_numeral_int @ N) @ one_one_int) = (numeral_numeral_int @ (plus_plus_num @ N @ one)))))). % numeral_plus_one
thf(fact_11_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_12_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_13_one__plus__numeral, axiom,
    ((![N : num]: ((plus_plus_real @ one_one_real @ (numeral_numeral_real @ N)) = (numeral_numeral_real @ (plus_plus_num @ one @ N)))))). % one_plus_numeral
thf(fact_14_one__plus__numeral, axiom,
    ((![N : num]: ((plus_plus_int @ one_one_int @ (numeral_numeral_int @ N)) = (numeral_numeral_int @ (plus_plus_num @ one @ N)))))). % one_plus_numeral
thf(fact_15_numeral__eq__one__iff, axiom,
    ((![N : num]: (((numera632737353omplex @ N) = one_one_complex) = (N = one))))). % numeral_eq_one_iff
thf(fact_16_numeral__eq__one__iff, axiom,
    ((![N : num]: (((numeral_numeral_nat @ N) = one_one_nat) = (N = one))))). % numeral_eq_one_iff
thf(fact_17_numeral__eq__one__iff, axiom,
    ((![N : num]: (((numeral_numeral_real @ N) = one_one_real) = (N = one))))). % numeral_eq_one_iff
thf(fact_18_numeral__eq__one__iff, axiom,
    ((![N : num]: (((numeral_numeral_int @ N) = one_one_int) = (N = one))))). % numeral_eq_one_iff
thf(fact_19_one__eq__numeral__iff, axiom,
    ((![N : num]: ((one_one_complex = (numera632737353omplex @ N)) = (one = N))))). % one_eq_numeral_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_real = (numeral_numeral_real @ N)) = (one = N))))). % one_eq_numeral_iff
thf(fact_22_one__eq__numeral__iff, axiom,
    ((![N : num]: ((one_one_int = (numeral_numeral_int @ N)) = (one = N))))). % one_eq_numeral_iff
thf(fact_23_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_24_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_25_distrib__left__numeral, axiom,
    ((![V : num, B : real, C : real]: ((times_times_real @ (numeral_numeral_real @ V) @ (plus_plus_real @ B @ C)) = (plus_plus_real @ (times_times_real @ (numeral_numeral_real @ V) @ B) @ (times_times_real @ (numeral_numeral_real @ V) @ C)))))). % distrib_left_numeral
thf(fact_26_distrib__left__numeral, axiom,
    ((![V : num, B : int, C : int]: ((times_times_int @ (numeral_numeral_int @ V) @ (plus_plus_int @ B @ C)) = (plus_plus_int @ (times_times_int @ (numeral_numeral_int @ V) @ B) @ (times_times_int @ (numeral_numeral_int @ V) @ C)))))). % distrib_left_numeral
thf(fact_27_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_28_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_29_distrib__right__numeral, axiom,
    ((![A : real, B : real, V : num]: ((times_times_real @ (plus_plus_real @ A @ B) @ (numeral_numeral_real @ V)) = (plus_plus_real @ (times_times_real @ A @ (numeral_numeral_real @ V)) @ (times_times_real @ B @ (numeral_numeral_real @ V))))))). % distrib_right_numeral
thf(fact_30_distrib__right__numeral, axiom,
    ((![A : int, B : int, V : num]: ((times_times_int @ (plus_plus_int @ A @ B) @ (numeral_numeral_int @ V)) = (plus_plus_int @ (times_times_int @ A @ (numeral_numeral_int @ V)) @ (times_times_int @ B @ (numeral_numeral_int @ V))))))). % distrib_right_numeral
thf(fact_31_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_32_power2__sum, axiom,
    ((![X : real, Y : real]: ((power_power_real @ (plus_plus_real @ X @ Y) @ (numeral_numeral_nat @ (bit0 @ one))) = (plus_plus_real @ (plus_plus_real @ (power_power_real @ X @ (numeral_numeral_nat @ (bit0 @ one))) @ (power_power_real @ Y @ (numeral_numeral_nat @ (bit0 @ one)))) @ (times_times_real @ (times_times_real @ (numeral_numeral_real @ (bit0 @ one)) @ X) @ Y)))))). % power2_sum
thf(fact_33_power2__sum, axiom,
    ((![X : int, Y : int]: ((power_power_int @ (plus_plus_int @ X @ Y) @ (numeral_numeral_nat @ (bit0 @ one))) = (plus_plus_int @ (plus_plus_int @ (power_power_int @ X @ (numeral_numeral_nat @ (bit0 @ one))) @ (power_power_int @ Y @ (numeral_numeral_nat @ (bit0 @ one)))) @ (times_times_int @ (times_times_int @ (numeral_numeral_int @ (bit0 @ one)) @ X) @ Y)))))). % power2_sum
thf(fact_34_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_35_DFT__lower, axiom,
    ((![M : nat, A : nat > complex, I : nat]: ((fFT_Mirabelle_DFT @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ M) @ A @ I) = (plus_plus_complex @ (fFT_Mirabelle_DFT @ M @ (^[I2 : nat]: (A @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ I2))) @ I) @ (times_times_complex @ (power_power_complex @ (fFT_Mirabelle_root @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ M)) @ I) @ (fFT_Mirabelle_DFT @ M @ (^[I2 : nat]: (A @ (plus_plus_nat @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ I2) @ one_one_nat))) @ I))))))). % DFT_lower
thf(fact_36_div__exp__eq, axiom,
    ((![A : nat, M : nat, N : nat]: ((divide_divide_nat @ (divide_divide_nat @ A @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ M)) @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N)) = (divide_divide_nat @ A @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (plus_plus_nat @ M @ N))))))). % div_exp_eq
thf(fact_37_div__exp__eq, axiom,
    ((![A : int, M : nat, N : nat]: ((divide_divide_int @ (divide_divide_int @ A @ (power_power_int @ (numeral_numeral_int @ (bit0 @ one)) @ M)) @ (power_power_int @ (numeral_numeral_int @ (bit0 @ one)) @ N)) = (divide_divide_int @ A @ (power_power_int @ (numeral_numeral_int @ (bit0 @ one)) @ (plus_plus_nat @ M @ N))))))). % div_exp_eq
thf(fact_38_nat__1__eq__mult__iff, axiom,
    ((![M : nat, N : nat]: ((one_one_nat = (times_times_nat @ M @ N)) = (((M = one_one_nat)) & ((N = one_one_nat))))))). % nat_1_eq_mult_iff
thf(fact_39_nat__mult__eq__1__iff, axiom,
    ((![M : nat, N : nat]: (((times_times_nat @ M @ N) = one_one_nat) = (((M = one_one_nat)) & ((N = one_one_nat))))))). % nat_mult_eq_1_iff
thf(fact_40_numeral__eq__iff, axiom,
    ((![M : num, N : num]: (((numeral_numeral_nat @ M) = (numeral_numeral_nat @ N)) = (M = N))))). % numeral_eq_iff
thf(fact_41_numeral__eq__iff, axiom,
    ((![M : num, N : num]: (((numeral_numeral_real @ M) = (numeral_numeral_real @ N)) = (M = N))))). % numeral_eq_iff
thf(fact_42_numeral__eq__iff, axiom,
    ((![M : num, N : num]: (((numeral_numeral_int @ M) = (numeral_numeral_int @ N)) = (M = N))))). % numeral_eq_iff
thf(fact_43_numeral__eq__iff, axiom,
    ((![M : num, N : num]: (((numera632737353omplex @ M) = (numera632737353omplex @ N)) = (M = N))))). % numeral_eq_iff
thf(fact_44_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_45_numeral__times__numeral, axiom,
    ((![M : num, N : num]: ((times_times_real @ (numeral_numeral_real @ M) @ (numeral_numeral_real @ N)) = (numeral_numeral_real @ (times_times_num @ M @ N)))))). % numeral_times_numeral
thf(fact_46_numeral__times__numeral, axiom,
    ((![M : num, N : num]: ((times_times_int @ (numeral_numeral_int @ M) @ (numeral_numeral_int @ N)) = (numeral_numeral_int @ (times_times_num @ M @ N)))))). % numeral_times_numeral
thf(fact_47_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_48_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_49_mult__numeral__left__semiring__numeral, axiom,
    ((![V : num, W : num, Z : real]: ((times_times_real @ (numeral_numeral_real @ V) @ (times_times_real @ (numeral_numeral_real @ W) @ Z)) = (times_times_real @ (numeral_numeral_real @ (times_times_num @ V @ W)) @ Z))))). % mult_numeral_left_semiring_numeral
thf(fact_50_mult__numeral__left__semiring__numeral, axiom,
    ((![V : num, W : num, Z : int]: ((times_times_int @ (numeral_numeral_int @ V) @ (times_times_int @ (numeral_numeral_int @ W) @ Z)) = (times_times_int @ (numeral_numeral_int @ (times_times_num @ V @ W)) @ Z))))). % mult_numeral_left_semiring_numeral
thf(fact_51_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_52_bits__div__by__1, axiom,
    ((![A : nat]: ((divide_divide_nat @ A @ one_one_nat) = A)))). % bits_div_by_1
thf(fact_53_bits__div__by__1, axiom,
    ((![A : int]: ((divide_divide_int @ A @ one_one_int) = A)))). % bits_div_by_1
thf(fact_54_power__one, axiom,
    ((![N : nat]: ((power_power_nat @ one_one_nat @ N) = one_one_nat)))). % power_one
thf(fact_55_power__one, axiom,
    ((![N : nat]: ((power_power_int @ one_one_int @ N) = one_one_int)))). % power_one
thf(fact_56_power__one, axiom,
    ((![N : nat]: ((power_power_complex @ one_one_complex @ N) = one_one_complex)))). % power_one
thf(fact_57_power__one, axiom,
    ((![N : nat]: ((power_power_real @ one_one_real @ N) = one_one_real)))). % power_one
thf(fact_58_num__double, axiom,
    ((![N : num]: ((times_times_num @ (bit0 @ one) @ N) = (bit0 @ N))))). % num_double
thf(fact_59_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_60_power__mult__numeral, axiom,
    ((![A : real, M : num, N : num]: ((power_power_real @ (power_power_real @ A @ (numeral_numeral_nat @ M)) @ (numeral_numeral_nat @ N)) = (power_power_real @ A @ (numeral_numeral_nat @ (times_times_num @ M @ N))))))). % power_mult_numeral
thf(fact_61_power__one__right, axiom,
    ((![A : complex]: ((power_power_complex @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_62_power__one__right, axiom,
    ((![A : real]: ((power_power_real @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_63_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_64_numeral__plus__numeral, axiom,
    ((![M : num, N : num]: ((plus_plus_real @ (numeral_numeral_real @ M) @ (numeral_numeral_real @ N)) = (numeral_numeral_real @ (plus_plus_num @ M @ N)))))). % numeral_plus_numeral
thf(fact_65_numeral__plus__numeral, axiom,
    ((![M : num, N : num]: ((plus_plus_int @ (numeral_numeral_int @ M) @ (numeral_numeral_int @ N)) = (numeral_numeral_int @ (plus_plus_num @ M @ N)))))). % numeral_plus_numeral
thf(fact_66_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_67_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_68_add__numeral__left, axiom,
    ((![V : num, W : num, Z : real]: ((plus_plus_real @ (numeral_numeral_real @ V) @ (plus_plus_real @ (numeral_numeral_real @ W) @ Z)) = (plus_plus_real @ (numeral_numeral_real @ (plus_plus_num @ V @ W)) @ Z))))). % add_numeral_left
thf(fact_69_add__numeral__left, axiom,
    ((![V : num, W : num, Z : int]: ((plus_plus_int @ (numeral_numeral_int @ V) @ (plus_plus_int @ (numeral_numeral_int @ W) @ Z)) = (plus_plus_int @ (numeral_numeral_int @ (plus_plus_num @ V @ W)) @ Z))))). % add_numeral_left
thf(fact_70_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_71_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_72_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_73_power__add__numeral2, axiom,
    ((![A : real, M : num, N : num, B : real]: ((times_times_real @ (power_power_real @ A @ (numeral_numeral_nat @ M)) @ (times_times_real @ (power_power_real @ A @ (numeral_numeral_nat @ N)) @ B)) = (times_times_real @ (power_power_real @ A @ (numeral_numeral_nat @ (plus_plus_num @ M @ N))) @ B))))). % power_add_numeral2
thf(fact_74_power__add__numeral2, axiom,
    ((![A : int, M : num, N : num, B : int]: ((times_times_int @ (power_power_int @ A @ (numeral_numeral_nat @ M)) @ (times_times_int @ (power_power_int @ A @ (numeral_numeral_nat @ N)) @ B)) = (times_times_int @ (power_power_int @ A @ (numeral_numeral_nat @ (plus_plus_num @ M @ N))) @ B))))). % power_add_numeral2
thf(fact_75_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_76_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_77_power__add__numeral, axiom,
    ((![A : real, M : num, N : num]: ((times_times_real @ (power_power_real @ A @ (numeral_numeral_nat @ M)) @ (power_power_real @ A @ (numeral_numeral_nat @ N))) = (power_power_real @ A @ (numeral_numeral_nat @ (plus_plus_num @ M @ N))))))). % power_add_numeral
thf(fact_78_power__add__numeral, axiom,
    ((![A : int, M : num, N : num]: ((times_times_int @ (power_power_int @ A @ (numeral_numeral_nat @ M)) @ (power_power_int @ A @ (numeral_numeral_nat @ N))) = (power_power_int @ A @ (numeral_numeral_nat @ (plus_plus_num @ M @ N))))))). % power_add_numeral
thf(fact_79_add__One__commute, axiom,
    ((![N : num]: ((plus_plus_num @ one @ N) = (plus_plus_num @ N @ one))))). % add_One_commute
thf(fact_80_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_81_is__num__normalize_I1_J, axiom,
    ((![A : int, B : int, C : int]: ((plus_plus_int @ (plus_plus_int @ A @ B) @ C) = (plus_plus_int @ A @ (plus_plus_int @ B @ C)))))). % is_num_normalize(1)
thf(fact_82_is__num__normalize_I1_J, axiom,
    ((![A : real, B : real, C : real]: ((plus_plus_real @ (plus_plus_real @ A @ B) @ C) = (plus_plus_real @ A @ (plus_plus_real @ B @ C)))))). % is_num_normalize(1)
thf(fact_83_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_84_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_85_power__commuting__commutes, axiom,
    ((![X : real, Y : real, N : nat]: (((times_times_real @ X @ Y) = (times_times_real @ Y @ X)) => ((times_times_real @ (power_power_real @ X @ N) @ Y) = (times_times_real @ Y @ (power_power_real @ X @ N))))))). % power_commuting_commutes
thf(fact_86_power__commuting__commutes, axiom,
    ((![X : int, Y : int, N : nat]: (((times_times_int @ X @ Y) = (times_times_int @ Y @ X)) => ((times_times_int @ (power_power_int @ X @ N) @ Y) = (times_times_int @ Y @ (power_power_int @ X @ N))))))). % power_commuting_commutes
thf(fact_87_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_88_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_89_power__mult__distrib, axiom,
    ((![A : real, B : real, N : nat]: ((power_power_real @ (times_times_real @ A @ B) @ N) = (times_times_real @ (power_power_real @ A @ N) @ (power_power_real @ B @ N)))))). % power_mult_distrib
thf(fact_90_power__mult__distrib, axiom,
    ((![A : int, B : int, N : nat]: ((power_power_int @ (times_times_int @ A @ B) @ N) = (times_times_int @ (power_power_int @ A @ N) @ (power_power_int @ B @ N)))))). % power_mult_distrib
thf(fact_91_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_92_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_93_power__commutes, axiom,
    ((![A : real, N : nat]: ((times_times_real @ (power_power_real @ A @ N) @ A) = (times_times_real @ A @ (power_power_real @ A @ N)))))). % power_commutes
thf(fact_94_power__commutes, axiom,
    ((![A : int, N : nat]: ((times_times_int @ (power_power_int @ A @ N) @ A) = (times_times_int @ A @ (power_power_int @ A @ N)))))). % power_commutes
thf(fact_95_power__divide, axiom,
    ((![A : complex, B : complex, N : nat]: ((power_power_complex @ (divide1210191872omplex @ A @ B) @ N) = (divide1210191872omplex @ (power_power_complex @ A @ N) @ (power_power_complex @ B @ N)))))). % power_divide
thf(fact_96_power__divide, axiom,
    ((![A : real, B : real, N : nat]: ((power_power_real @ (divide_divide_real @ A @ B) @ N) = (divide_divide_real @ (power_power_real @ A @ N) @ (power_power_real @ B @ N)))))). % power_divide
thf(fact_97_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_98_power__mult, axiom,
    ((![A : real, M : nat, N : nat]: ((power_power_real @ A @ (times_times_nat @ M @ N)) = (power_power_real @ (power_power_real @ A @ M) @ N))))). % power_mult
thf(fact_99_add__mult__distrib2, axiom,
    ((![K : nat, M : nat, N : nat]: ((times_times_nat @ K @ (plus_plus_nat @ M @ N)) = (plus_plus_nat @ (times_times_nat @ K @ M) @ (times_times_nat @ K @ N)))))). % add_mult_distrib2
thf(fact_100_add__mult__distrib, axiom,
    ((![M : nat, N : nat, K : nat]: ((times_times_nat @ (plus_plus_nat @ M @ N) @ K) = (plus_plus_nat @ (times_times_nat @ M @ K) @ (times_times_nat @ N @ K)))))). % add_mult_distrib
thf(fact_101_nat__mult__1__right, axiom,
    ((![N : nat]: ((times_times_nat @ N @ one_one_nat) = N)))). % nat_mult_1_right
thf(fact_102_nat__mult__1, axiom,
    ((![N : nat]: ((times_times_nat @ one_one_nat @ N) = N)))). % nat_mult_1
thf(fact_103_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_104_one__plus__numeral__commute, axiom,
    ((![X : num]: ((plus_plus_real @ one_one_real @ (numeral_numeral_real @ X)) = (plus_plus_real @ (numeral_numeral_real @ X) @ one_one_real))))). % one_plus_numeral_commute
thf(fact_105_one__plus__numeral__commute, axiom,
    ((![X : num]: ((plus_plus_int @ one_one_int @ (numeral_numeral_int @ X)) = (plus_plus_int @ (numeral_numeral_int @ X) @ one_one_int))))). % one_plus_numeral_commute
thf(fact_106_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_107_mult__numeral__1__right, axiom,
    ((![A : nat]: ((times_times_nat @ A @ (numeral_numeral_nat @ one)) = A)))). % mult_numeral_1_right
thf(fact_108_mult__numeral__1__right, axiom,
    ((![A : real]: ((times_times_real @ A @ (numeral_numeral_real @ one)) = A)))). % mult_numeral_1_right
thf(fact_109_mult__numeral__1__right, axiom,
    ((![A : int]: ((times_times_int @ A @ (numeral_numeral_int @ one)) = A)))). % mult_numeral_1_right
thf(fact_110_mult__numeral__1__right, axiom,
    ((![A : complex]: ((times_times_complex @ A @ (numera632737353omplex @ one)) = A)))). % mult_numeral_1_right
thf(fact_111_mult__numeral__1, axiom,
    ((![A : nat]: ((times_times_nat @ (numeral_numeral_nat @ one) @ A) = A)))). % mult_numeral_1
thf(fact_112_mult__numeral__1, axiom,
    ((![A : real]: ((times_times_real @ (numeral_numeral_real @ one) @ A) = A)))). % mult_numeral_1
thf(fact_113_mult__numeral__1, axiom,
    ((![A : int]: ((times_times_int @ (numeral_numeral_int @ one) @ A) = A)))). % mult_numeral_1
thf(fact_114_mult__numeral__1, axiom,
    ((![A : complex]: ((times_times_complex @ (numera632737353omplex @ one) @ A) = A)))). % mult_numeral_1
thf(fact_115_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_116_numeral__Bit0, axiom,
    ((![N : num]: ((numeral_numeral_real @ (bit0 @ N)) = (plus_plus_real @ (numeral_numeral_real @ N) @ (numeral_numeral_real @ N)))))). % numeral_Bit0
thf(fact_117_numeral__Bit0, axiom,
    ((![N : num]: ((numeral_numeral_int @ (bit0 @ N)) = (plus_plus_int @ (numeral_numeral_int @ N) @ (numeral_numeral_int @ N)))))). % numeral_Bit0
thf(fact_118_numeral__Bit0, axiom,
    ((![N : num]: ((numera632737353omplex @ (bit0 @ N)) = (plus_plus_complex @ (numera632737353omplex @ N) @ (numera632737353omplex @ N)))))). % numeral_Bit0
thf(fact_119_numeral__One, axiom,
    (((numeral_numeral_nat @ one) = one_one_nat))). % numeral_One
thf(fact_120_numeral__One, axiom,
    (((numeral_numeral_real @ one) = one_one_real))). % numeral_One
thf(fact_121_numeral__One, axiom,
    (((numeral_numeral_int @ one) = one_one_int))). % numeral_One
thf(fact_122_numeral__One, axiom,
    (((numera632737353omplex @ one) = one_one_complex))). % numeral_One
thf(fact_123_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_124_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_125_left__right__inverse__power, axiom,
    ((![X : real, Y : real, N : nat]: (((times_times_real @ X @ Y) = one_one_real) => ((times_times_real @ (power_power_real @ X @ N) @ (power_power_real @ Y @ N)) = one_one_real))))). % left_right_inverse_power
thf(fact_126_left__right__inverse__power, axiom,
    ((![X : int, Y : int, N : nat]: (((times_times_int @ X @ Y) = one_one_int) => ((times_times_int @ (power_power_int @ X @ N) @ (power_power_int @ Y @ N)) = one_one_int))))). % left_right_inverse_power
thf(fact_127_divide__numeral__1, axiom,
    ((![A : complex]: ((divide1210191872omplex @ A @ (numera632737353omplex @ one)) = A)))). % divide_numeral_1
thf(fact_128_divide__numeral__1, axiom,
    ((![A : real]: ((divide_divide_real @ A @ (numeral_numeral_real @ one)) = A)))). % divide_numeral_1
thf(fact_129_power__one__over, axiom,
    ((![A : complex, N : nat]: ((power_power_complex @ (divide1210191872omplex @ one_one_complex @ A) @ N) = (divide1210191872omplex @ one_one_complex @ (power_power_complex @ A @ N)))))). % power_one_over
thf(fact_130_power__one__over, axiom,
    ((![A : real, N : nat]: ((power_power_real @ (divide_divide_real @ one_one_real @ A) @ N) = (divide_divide_real @ one_one_real @ (power_power_real @ A @ N)))))). % power_one_over
thf(fact_131_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_132_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_133_power__add, axiom,
    ((![A : real, M : nat, N : nat]: ((power_power_real @ A @ (plus_plus_nat @ M @ N)) = (times_times_real @ (power_power_real @ A @ M) @ (power_power_real @ A @ N)))))). % power_add
thf(fact_134_power__add, axiom,
    ((![A : int, M : nat, N : nat]: ((power_power_int @ A @ (plus_plus_nat @ M @ N)) = (times_times_int @ (power_power_int @ A @ M) @ (power_power_int @ A @ N)))))). % power_add
thf(fact_135_numerals_I1_J, axiom,
    (((numeral_numeral_nat @ one) = one_one_nat))). % numerals(1)
thf(fact_136_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_137_numeral__code_I2_J, axiom,
    ((![N : num]: ((numeral_numeral_real @ (bit0 @ N)) = (plus_plus_real @ (numeral_numeral_real @ N) @ (numeral_numeral_real @ N)))))). % numeral_code(2)
thf(fact_138_numeral__code_I2_J, axiom,
    ((![N : num]: ((numeral_numeral_int @ (bit0 @ N)) = (plus_plus_int @ (numeral_numeral_int @ N) @ (numeral_numeral_int @ N)))))). % numeral_code(2)
thf(fact_139_numeral__code_I2_J, axiom,
    ((![N : num]: ((numera632737353omplex @ (bit0 @ N)) = (plus_plus_complex @ (numera632737353omplex @ N) @ (numera632737353omplex @ N)))))). % numeral_code(2)
thf(fact_140_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_141_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_142_power__numeral__even, axiom,
    ((![Z : real, W : num]: ((power_power_real @ Z @ (numeral_numeral_nat @ (bit0 @ W))) = (times_times_real @ (power_power_real @ Z @ (numeral_numeral_nat @ W)) @ (power_power_real @ Z @ (numeral_numeral_nat @ W))))))). % power_numeral_even
thf(fact_143_power__numeral__even, axiom,
    ((![Z : int, W : num]: ((power_power_int @ Z @ (numeral_numeral_nat @ (bit0 @ W))) = (times_times_int @ (power_power_int @ Z @ (numeral_numeral_nat @ W)) @ (power_power_int @ Z @ (numeral_numeral_nat @ W))))))). % power_numeral_even
thf(fact_144_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_145_left__add__twice, axiom,
    ((![A : real, B : real]: ((plus_plus_real @ A @ (plus_plus_real @ A @ B)) = (plus_plus_real @ (times_times_real @ (numeral_numeral_real @ (bit0 @ one)) @ A) @ B))))). % left_add_twice
thf(fact_146_left__add__twice, axiom,
    ((![A : int, B : int]: ((plus_plus_int @ A @ (plus_plus_int @ A @ B)) = (plus_plus_int @ (times_times_int @ (numeral_numeral_int @ (bit0 @ one)) @ A) @ B))))). % left_add_twice
thf(fact_147_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_148_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_149_mult__2__right, axiom,
    ((![Z : real]: ((times_times_real @ Z @ (numeral_numeral_real @ (bit0 @ one))) = (plus_plus_real @ Z @ Z))))). % mult_2_right
thf(fact_150_mult__2__right, axiom,
    ((![Z : int]: ((times_times_int @ Z @ (numeral_numeral_int @ (bit0 @ one))) = (plus_plus_int @ Z @ Z))))). % mult_2_right
thf(fact_151_mult__2__right, axiom,
    ((![Z : complex]: ((times_times_complex @ Z @ (numera632737353omplex @ (bit0 @ one))) = (plus_plus_complex @ Z @ Z))))). % mult_2_right
thf(fact_152_mult__2, axiom,
    ((![Z : nat]: ((times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ Z) = (plus_plus_nat @ Z @ Z))))). % mult_2
thf(fact_153_mult__2, axiom,
    ((![Z : real]: ((times_times_real @ (numeral_numeral_real @ (bit0 @ one)) @ Z) = (plus_plus_real @ Z @ Z))))). % mult_2
thf(fact_154_mult__2, axiom,
    ((![Z : int]: ((times_times_int @ (numeral_numeral_int @ (bit0 @ one)) @ Z) = (plus_plus_int @ Z @ Z))))). % mult_2
thf(fact_155_mult__2, axiom,
    ((![Z : complex]: ((times_times_complex @ (numera632737353omplex @ (bit0 @ one)) @ Z) = (plus_plus_complex @ Z @ Z))))). % mult_2
thf(fact_156_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_157_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_158_power2__eq__square, axiom,
    ((![A : real]: ((power_power_real @ A @ (numeral_numeral_nat @ (bit0 @ one))) = (times_times_real @ A @ A))))). % power2_eq_square
thf(fact_159_power2__eq__square, axiom,
    ((![A : int]: ((power_power_int @ A @ (numeral_numeral_nat @ (bit0 @ one))) = (times_times_int @ A @ A))))). % power2_eq_square
thf(fact_160_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_161_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_162_power4__eq__xxxx, axiom,
    ((![X : real]: ((power_power_real @ X @ (numeral_numeral_nat @ (bit0 @ (bit0 @ one)))) = (times_times_real @ (times_times_real @ (times_times_real @ X @ X) @ X) @ X))))). % power4_eq_xxxx
thf(fact_163_power4__eq__xxxx, axiom,
    ((![X : int]: ((power_power_int @ X @ (numeral_numeral_nat @ (bit0 @ (bit0 @ one)))) = (times_times_int @ (times_times_int @ (times_times_int @ X @ X) @ X) @ X))))). % power4_eq_xxxx
thf(fact_164_one__power2, axiom,
    (((power_power_nat @ one_one_nat @ (numeral_numeral_nat @ (bit0 @ one))) = one_one_nat))). % one_power2
thf(fact_165_one__power2, axiom,
    (((power_power_int @ one_one_int @ (numeral_numeral_nat @ (bit0 @ one))) = one_one_int))). % one_power2
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_real @ one_one_real @ (numeral_numeral_nat @ (bit0 @ one))) = one_one_real))). % one_power2
thf(fact_168_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_169_power__even__eq, axiom,
    ((![A : real, N : nat]: ((power_power_real @ A @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N)) = (power_power_real @ (power_power_real @ A @ N) @ (numeral_numeral_nat @ (bit0 @ one))))))). % power_even_eq
thf(fact_170_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_171_add__self__div__2, axiom,
    ((![M : nat]: ((divide_divide_nat @ (plus_plus_nat @ M @ M) @ (numeral_numeral_nat @ (bit0 @ one))) = M)))). % add_self_div_2
thf(fact_172_semiring__norm_I2_J, axiom,
    (((plus_plus_num @ one @ one) = (bit0 @ one)))). % semiring_norm(2)
thf(fact_173_semiring__norm_I6_J, axiom,
    ((![M : num, N : num]: ((plus_plus_num @ (bit0 @ M) @ (bit0 @ N)) = (bit0 @ (plus_plus_num @ M @ N)))))). % semiring_norm(6)
thf(fact_174_semiring__norm_I83_J, axiom,
    ((![N : num]: (~ ((one = (bit0 @ N))))))). % semiring_norm(83)
thf(fact_175_semiring__norm_I85_J, axiom,
    ((![M : num]: (~ (((bit0 @ M) = one)))))). % semiring_norm(85)
thf(fact_176_div__by__1, axiom,
    ((![A : complex]: ((divide1210191872omplex @ A @ one_one_complex) = A)))). % div_by_1
thf(fact_177_div__by__1, axiom,
    ((![A : nat]: ((divide_divide_nat @ A @ one_one_nat) = A)))). % div_by_1
thf(fact_178_div__by__1, axiom,
    ((![A : real]: ((divide_divide_real @ A @ one_one_real) = A)))). % div_by_1
thf(fact_179_div__by__1, axiom,
    ((![A : int]: ((divide_divide_int @ A @ one_one_int) = A)))). % div_by_1
thf(fact_180_field__sum__of__halves, axiom,
    ((![X : real]: ((plus_plus_real @ (divide_divide_real @ X @ (numeral_numeral_real @ (bit0 @ one))) @ (divide_divide_real @ X @ (numeral_numeral_real @ (bit0 @ one)))) = X)))). % field_sum_of_halves
thf(fact_181_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_182_times__divide__eq__right, axiom,
    ((![A : real, B : real, C : real]: ((times_times_real @ A @ (divide_divide_real @ B @ C)) = (divide_divide_real @ (times_times_real @ A @ B) @ C))))). % times_divide_eq_right
thf(fact_183_divide__divide__eq__right, axiom,
    ((![A : complex, B : complex, C : complex]: ((divide1210191872omplex @ A @ (divide1210191872omplex @ B @ C)) = (divide1210191872omplex @ (times_times_complex @ A @ C) @ B))))). % divide_divide_eq_right
thf(fact_184_divide__divide__eq__right, axiom,
    ((![A : real, B : real, C : real]: ((divide_divide_real @ A @ (divide_divide_real @ B @ C)) = (divide_divide_real @ (times_times_real @ A @ C) @ B))))). % divide_divide_eq_right
thf(fact_185_divide__divide__eq__left, axiom,
    ((![A : complex, B : complex, C : complex]: ((divide1210191872omplex @ (divide1210191872omplex @ A @ B) @ C) = (divide1210191872omplex @ A @ (times_times_complex @ B @ C)))))). % divide_divide_eq_left
thf(fact_186_divide__divide__eq__left, axiom,
    ((![A : real, B : real, C : real]: ((divide_divide_real @ (divide_divide_real @ A @ B) @ C) = (divide_divide_real @ A @ (times_times_real @ B @ C)))))). % divide_divide_eq_left
thf(fact_187_semiring__norm_I87_J, axiom,
    ((![M : num, N : num]: (((bit0 @ M) = (bit0 @ N)) = (M = N))))). % semiring_norm(87)
thf(fact_188_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_189_times__divide__eq__left, axiom,
    ((![B : real, C : real, A : real]: ((times_times_real @ (divide_divide_real @ B @ C) @ A) = (divide_divide_real @ (times_times_real @ B @ A) @ C))))). % times_divide_eq_left
thf(fact_190_semiring__norm_I13_J, axiom,
    ((![M : num, N : num]: ((times_times_num @ (bit0 @ M) @ (bit0 @ N)) = (bit0 @ (bit0 @ (times_times_num @ M @ N))))))). % semiring_norm(13)
thf(fact_191_semiring__norm_I12_J, axiom,
    ((![N : num]: ((times_times_num @ one @ N) = N)))). % semiring_norm(12)
thf(fact_192_semiring__norm_I11_J, axiom,
    ((![M : num]: ((times_times_num @ M @ one) = M)))). % semiring_norm(11)
thf(fact_193_div__mult2__eq, axiom,
    ((![M : nat, N : nat, Q : nat]: ((divide_divide_nat @ M @ (times_times_nat @ N @ Q)) = (divide_divide_nat @ (divide_divide_nat @ M @ N) @ Q))))). % div_mult2_eq
thf(fact_194_ring__class_Oring__distribs_I2_J, axiom,
    ((![A : complex, B : complex, C : complex]: ((times_times_complex @ (plus_plus_complex @ A @ B) @ C) = (plus_plus_complex @ (times_times_complex @ A @ C) @ (times_times_complex @ B @ C)))))). % ring_class.ring_distribs(2)
thf(fact_195_ring__class_Oring__distribs_I2_J, axiom,
    ((![A : real, B : real, C : real]: ((times_times_real @ (plus_plus_real @ A @ B) @ C) = (plus_plus_real @ (times_times_real @ A @ C) @ (times_times_real @ B @ C)))))). % ring_class.ring_distribs(2)
thf(fact_196_ring__class_Oring__distribs_I2_J, axiom,
    ((![A : int, B : int, C : int]: ((times_times_int @ (plus_plus_int @ A @ B) @ C) = (plus_plus_int @ (times_times_int @ A @ C) @ (times_times_int @ B @ C)))))). % ring_class.ring_distribs(2)
thf(fact_197_ring__class_Oring__distribs_I1_J, axiom,
    ((![A : complex, B : complex, C : complex]: ((times_times_complex @ A @ (plus_plus_complex @ B @ C)) = (plus_plus_complex @ (times_times_complex @ A @ B) @ (times_times_complex @ A @ C)))))). % ring_class.ring_distribs(1)
thf(fact_198_ring__class_Oring__distribs_I1_J, axiom,
    ((![A : real, B : real, C : real]: ((times_times_real @ A @ (plus_plus_real @ B @ C)) = (plus_plus_real @ (times_times_real @ A @ B) @ (times_times_real @ A @ C)))))). % ring_class.ring_distribs(1)
thf(fact_199_ring__class_Oring__distribs_I1_J, axiom,
    ((![A : int, B : int, C : int]: ((times_times_int @ A @ (plus_plus_int @ B @ C)) = (plus_plus_int @ (times_times_int @ A @ B) @ (times_times_int @ A @ C)))))). % ring_class.ring_distribs(1)
thf(fact_200_comm__semiring__class_Odistrib, axiom,
    ((![A : nat, B : nat, C : nat]: ((times_times_nat @ (plus_plus_nat @ A @ B) @ C) = (plus_plus_nat @ (times_times_nat @ A @ C) @ (times_times_nat @ B @ C)))))). % comm_semiring_class.distrib
thf(fact_201_comm__semiring__class_Odistrib, axiom,
    ((![A : complex, B : complex, C : complex]: ((times_times_complex @ (plus_plus_complex @ A @ B) @ C) = (plus_plus_complex @ (times_times_complex @ A @ C) @ (times_times_complex @ B @ C)))))). % comm_semiring_class.distrib
thf(fact_202_comm__semiring__class_Odistrib, axiom,
    ((![A : real, B : real, C : real]: ((times_times_real @ (plus_plus_real @ A @ B) @ C) = (plus_plus_real @ (times_times_real @ A @ C) @ (times_times_real @ B @ C)))))). % comm_semiring_class.distrib
thf(fact_203_comm__semiring__class_Odistrib, axiom,
    ((![A : int, B : int, C : int]: ((times_times_int @ (plus_plus_int @ A @ B) @ C) = (plus_plus_int @ (times_times_int @ A @ C) @ (times_times_int @ B @ C)))))). % comm_semiring_class.distrib
thf(fact_204_distrib__left, axiom,
    ((![A : nat, B : nat, C : nat]: ((times_times_nat @ A @ (plus_plus_nat @ B @ C)) = (plus_plus_nat @ (times_times_nat @ A @ B) @ (times_times_nat @ A @ C)))))). % distrib_left
thf(fact_205_distrib__left, axiom,
    ((![A : complex, B : complex, C : complex]: ((times_times_complex @ A @ (plus_plus_complex @ B @ C)) = (plus_plus_complex @ (times_times_complex @ A @ B) @ (times_times_complex @ A @ C)))))). % distrib_left
thf(fact_206_distrib__left, axiom,
    ((![A : real, B : real, C : real]: ((times_times_real @ A @ (plus_plus_real @ B @ C)) = (plus_plus_real @ (times_times_real @ A @ B) @ (times_times_real @ A @ C)))))). % distrib_left
thf(fact_207_distrib__left, axiom,
    ((![A : int, B : int, C : int]: ((times_times_int @ A @ (plus_plus_int @ B @ C)) = (plus_plus_int @ (times_times_int @ A @ B) @ (times_times_int @ A @ C)))))). % distrib_left
thf(fact_208_distrib__right, axiom,
    ((![A : nat, B : nat, C : nat]: ((times_times_nat @ (plus_plus_nat @ A @ B) @ C) = (plus_plus_nat @ (times_times_nat @ A @ C) @ (times_times_nat @ B @ C)))))). % distrib_right
thf(fact_209_distrib__right, axiom,
    ((![A : complex, B : complex, C : complex]: ((times_times_complex @ (plus_plus_complex @ A @ B) @ C) = (plus_plus_complex @ (times_times_complex @ A @ C) @ (times_times_complex @ B @ C)))))). % distrib_right
thf(fact_210_distrib__right, axiom,
    ((![A : real, B : real, C : real]: ((times_times_real @ (plus_plus_real @ A @ B) @ C) = (plus_plus_real @ (times_times_real @ A @ C) @ (times_times_real @ B @ C)))))). % distrib_right
thf(fact_211_distrib__right, axiom,
    ((![A : int, B : int, C : int]: ((times_times_int @ (plus_plus_int @ A @ B) @ C) = (plus_plus_int @ (times_times_int @ A @ C) @ (times_times_int @ B @ C)))))). % distrib_right
thf(fact_212_combine__common__factor, axiom,
    ((![A : nat, E : nat, B : nat, C : nat]: ((plus_plus_nat @ (times_times_nat @ A @ E) @ (plus_plus_nat @ (times_times_nat @ B @ E) @ C)) = (plus_plus_nat @ (times_times_nat @ (plus_plus_nat @ A @ B) @ E) @ C))))). % combine_common_factor
thf(fact_213_combine__common__factor, axiom,
    ((![A : complex, E : complex, B : complex, C : complex]: ((plus_plus_complex @ (times_times_complex @ A @ E) @ (plus_plus_complex @ (times_times_complex @ B @ E) @ C)) = (plus_plus_complex @ (times_times_complex @ (plus_plus_complex @ A @ B) @ E) @ C))))). % combine_common_factor
thf(fact_214_combine__common__factor, axiom,
    ((![A : real, E : real, B : real, C : real]: ((plus_plus_real @ (times_times_real @ A @ E) @ (plus_plus_real @ (times_times_real @ B @ E) @ C)) = (plus_plus_real @ (times_times_real @ (plus_plus_real @ A @ B) @ E) @ C))))). % combine_common_factor
thf(fact_215_combine__common__factor, axiom,
    ((![A : int, E : int, B : int, C : int]: ((plus_plus_int @ (times_times_int @ A @ E) @ (plus_plus_int @ (times_times_int @ B @ E) @ C)) = (plus_plus_int @ (times_times_int @ (plus_plus_int @ A @ B) @ E) @ C))))). % combine_common_factor
thf(fact_216_times__divide__times__eq, axiom,
    ((![X : complex, Y : complex, Z : complex, W : complex]: ((times_times_complex @ (divide1210191872omplex @ X @ Y) @ (divide1210191872omplex @ Z @ W)) = (divide1210191872omplex @ (times_times_complex @ X @ Z) @ (times_times_complex @ Y @ W)))))). % times_divide_times_eq
thf(fact_217_times__divide__times__eq, axiom,
    ((![X : real, Y : real, Z : real, W : real]: ((times_times_real @ (divide_divide_real @ X @ Y) @ (divide_divide_real @ Z @ W)) = (divide_divide_real @ (times_times_real @ X @ Z) @ (times_times_real @ Y @ W)))))). % times_divide_times_eq
thf(fact_218_divide__divide__times__eq, axiom,
    ((![X : complex, Y : complex, Z : complex, W : complex]: ((divide1210191872omplex @ (divide1210191872omplex @ X @ Y) @ (divide1210191872omplex @ Z @ W)) = (divide1210191872omplex @ (times_times_complex @ X @ W) @ (times_times_complex @ Y @ Z)))))). % divide_divide_times_eq
thf(fact_219_divide__divide__times__eq, axiom,
    ((![X : real, Y : real, Z : real, W : real]: ((divide_divide_real @ (divide_divide_real @ X @ Y) @ (divide_divide_real @ Z @ W)) = (divide_divide_real @ (times_times_real @ X @ W) @ (times_times_real @ Y @ Z)))))). % divide_divide_times_eq
thf(fact_220_divide__divide__eq__left_H, axiom,
    ((![A : complex, B : complex, C : complex]: ((divide1210191872omplex @ (divide1210191872omplex @ A @ B) @ C) = (divide1210191872omplex @ A @ (times_times_complex @ C @ B)))))). % divide_divide_eq_left'
thf(fact_221_divide__divide__eq__left_H, axiom,
    ((![A : real, B : real, C : real]: ((divide_divide_real @ (divide_divide_real @ A @ B) @ C) = (divide_divide_real @ A @ (times_times_real @ C @ B)))))). % divide_divide_eq_left'
thf(fact_222_div__mult2__numeral__eq, axiom,
    ((![A : nat, K : num, L : num]: ((divide_divide_nat @ (divide_divide_nat @ A @ (numeral_numeral_nat @ K)) @ (numeral_numeral_nat @ L)) = (divide_divide_nat @ A @ (numeral_numeral_nat @ (times_times_num @ K @ L))))))). % div_mult2_numeral_eq
thf(fact_223_div__mult2__numeral__eq, axiom,
    ((![A : int, K : num, L : num]: ((divide_divide_int @ (divide_divide_int @ A @ (numeral_numeral_int @ K)) @ (numeral_numeral_int @ L)) = (divide_divide_int @ A @ (numeral_numeral_int @ (times_times_num @ K @ L))))))). % div_mult2_numeral_eq
thf(fact_224_add__divide__distrib, axiom,
    ((![A : complex, B : complex, C : complex]: ((divide1210191872omplex @ (plus_plus_complex @ A @ B) @ C) = (plus_plus_complex @ (divide1210191872omplex @ A @ C) @ (divide1210191872omplex @ B @ C)))))). % add_divide_distrib
thf(fact_225_add__divide__distrib, axiom,
    ((![A : real, B : real, C : real]: ((divide_divide_real @ (plus_plus_real @ A @ B) @ C) = (plus_plus_real @ (divide_divide_real @ A @ C) @ (divide_divide_real @ B @ C)))))). % add_divide_distrib
thf(fact_226_left__add__mult__distrib, axiom,
    ((![I : nat, U : nat, J : nat, K : nat]: ((plus_plus_nat @ (times_times_nat @ I @ U) @ (plus_plus_nat @ (times_times_nat @ J @ U) @ K)) = (plus_plus_nat @ (times_times_nat @ (plus_plus_nat @ I @ J) @ U) @ K))))). % left_add_mult_distrib
thf(fact_227_lambda__one, axiom,
    (((^[X2 : nat]: X2) = (times_times_nat @ one_one_nat)))). % lambda_one
thf(fact_228_lambda__one, axiom,
    (((^[X2 : complex]: X2) = (times_times_complex @ one_one_complex)))). % lambda_one
thf(fact_229_lambda__one, axiom,
    (((^[X2 : real]: X2) = (times_times_real @ one_one_real)))). % lambda_one
thf(fact_230_lambda__one, axiom,
    (((^[X2 : int]: X2) = (times_times_int @ one_one_int)))). % lambda_one
thf(fact_231_numeral__Bit0__div__2, axiom,
    ((![N : num]: ((divide_divide_nat @ (numeral_numeral_nat @ (bit0 @ N)) @ (numeral_numeral_nat @ (bit0 @ one))) = (numeral_numeral_nat @ N))))). % numeral_Bit0_div_2
thf(fact_232_numeral__Bit0__div__2, axiom,
    ((![N : num]: ((divide_divide_int @ (numeral_numeral_int @ (bit0 @ N)) @ (numeral_numeral_int @ (bit0 @ one))) = (numeral_numeral_int @ N))))). % numeral_Bit0_div_2
thf(fact_233_mult_Oleft__neutral, axiom,
    ((![A : nat]: ((times_times_nat @ one_one_nat @ A) = A)))). % mult.left_neutral
thf(fact_234_mult_Oleft__neutral, axiom,
    ((![A : complex]: ((times_times_complex @ one_one_complex @ A) = A)))). % mult.left_neutral
thf(fact_235_mult_Oleft__neutral, axiom,
    ((![A : real]: ((times_times_real @ one_one_real @ A) = A)))). % mult.left_neutral
thf(fact_236_mult_Oleft__neutral, axiom,
    ((![A : int]: ((times_times_int @ one_one_int @ A) = A)))). % mult.left_neutral
thf(fact_237_mult_Oright__neutral, axiom,
    ((![A : int]: ((times_times_int @ A @ one_one_int) = A)))). % mult.right_neutral
thf(fact_238_four__x__squared, axiom,
    ((![X : real]: ((times_times_real @ (numeral_numeral_real @ (bit0 @ (bit0 @ one))) @ (power_power_real @ X @ (numeral_numeral_nat @ (bit0 @ one)))) = (power_power_real @ (times_times_real @ (numeral_numeral_real @ (bit0 @ one)) @ X) @ (numeral_numeral_nat @ (bit0 @ one))))))). % four_x_squared
thf(fact_239_real__divide__square__eq, axiom,
    ((![R : real, A : real]: ((divide_divide_real @ (times_times_real @ R @ A) @ (times_times_real @ R @ R)) = (divide_divide_real @ A @ R))))). % real_divide_square_eq
thf(fact_240_zdiv__numeral__Bit0, axiom,
    ((![V : num, W : num]: ((divide_divide_int @ (numeral_numeral_int @ (bit0 @ V)) @ (numeral_numeral_int @ (bit0 @ W))) = (divide_divide_int @ (numeral_numeral_int @ V) @ (numeral_numeral_int @ W)))))). % zdiv_numeral_Bit0
thf(fact_241_verit__eq__simplify_I8_J, axiom,
    ((![X22 : num, Y2 : num]: (((bit0 @ X22) = (bit0 @ Y2)) = (X22 = Y2))))). % verit_eq_simplify(8)
thf(fact_242_pow_Osimps_I1_J, axiom,
    ((![X : num]: ((pow @ X @ one) = X)))). % pow.simps(1)
thf(fact_243_verit__eq__simplify_I10_J, axiom,
    ((![X22 : num]: (~ ((one = (bit0 @ X22))))))). % verit_eq_simplify(10)
thf(fact_244_nat__dvd__1__iff__1, axiom,
    ((![M : nat]: ((dvd_dvd_nat @ M @ one_one_nat) = (M = one_one_nat))))). % nat_dvd_1_iff_1

% Conjectures (1)
thf(conj_0, conjecture,
    (((fFT_Mirabelle_IDFT @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ m) @ a @ i) = (plus_plus_complex @ (fFT_Mirabelle_IDFT @ m @ (^[I2 : nat]: (a @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ I2))) @ i) @ (times_times_complex @ (power_power_complex @ (divide1210191872omplex @ one_one_complex @ (fFT_Mirabelle_root @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ m))) @ i) @ (fFT_Mirabelle_IDFT @ m @ (^[I2 : nat]: (a @ (plus_plus_nat @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ I2) @ one_one_nat))) @ i)))))).
