% 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_331__3225846_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:10:31.439

% 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 (27)
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__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_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_Power_Opower__class_Opower_001t__Complex__Ocomplex, type,
    power_power_complex : complex > nat > complex).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat, type,
    power_power_nat : nat > nat > nat).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex, type,
    divide1210191872omplex : complex > complex > complex).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat, type,
    divide_divide_nat : nat > nat > 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 (195)
thf(fact_0_root0, axiom,
    (((fFT_Mirabelle_root @ zero_zero_nat) = one_one_complex))). % root0
thf(fact_1_root1, axiom,
    (((fFT_Mirabelle_root @ one_one_nat) = one_one_complex))). % root1
thf(fact_2_calculation, axiom,
    (((groups59700922omplex @ (^[J : nat]: (divide1210191872omplex @ (a @ J) @ (power_power_complex @ (fFT_Mirabelle_root @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ m)) @ (times_times_nat @ i @ J)))) @ (set_or562006527an_nat @ zero_zero_nat @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ m))) = (plus_plus_complex @ (groups59700922omplex @ (^[J : nat]: (divide1210191872omplex @ (a @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ J)) @ (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))))) @ (set_or562006527an_nat @ zero_zero_nat @ m)) @ (groups59700922omplex @ (^[J : nat]: (divide1210191872omplex @ (a @ (plus_plus_nat @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ J) @ one_one_nat)) @ (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))))) @ (set_or562006527an_nat @ zero_zero_nat @ m)))))). % calculation
thf(fact_3_root__unity, axiom,
    ((![N : nat]: ((power_power_complex @ (fFT_Mirabelle_root @ N) @ N) = one_one_complex)))). % root_unity
thf(fact_4_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_5_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_6_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_7_bits__1__div__2, axiom,
    (((divide_divide_nat @ one_one_nat @ (numeral_numeral_nat @ (bit0 @ one))) = zero_zero_nat))). % bits_1_div_2
thf(fact_8_one__div__two__eq__zero, axiom,
    (((divide_divide_nat @ one_one_nat @ (numeral_numeral_nat @ (bit0 @ one))) = zero_zero_nat))). % one_div_two_eq_zero
thf(fact_9_IDFT__def, axiom,
    ((fFT_Mirabelle_IDFT = (^[N2 : nat]: (^[A : nat > complex]: (^[I : nat]: (groups59700922omplex @ (^[K : nat]: (divide1210191872omplex @ (A @ K) @ (power_power_complex @ (fFT_Mirabelle_root @ N2) @ (times_times_nat @ I @ K)))) @ (set_or562006527an_nat @ zero_zero_nat @ N2)))))))). % IDFT_def
thf(fact_10_zero__eq__power2, axiom,
    ((![A2 : nat]: (((power_power_nat @ A2 @ (numeral_numeral_nat @ (bit0 @ one))) = zero_zero_nat) = (A2 = zero_zero_nat))))). % zero_eq_power2
thf(fact_11_zero__eq__power2, axiom,
    ((![A2 : complex]: (((power_power_complex @ A2 @ (numeral_numeral_nat @ (bit0 @ one))) = zero_zero_complex) = (A2 = zero_zero_complex))))). % zero_eq_power2
thf(fact_12_one__add__one, axiom,
    (((plus_plus_complex @ one_one_complex @ one_one_complex) = (numera632737353omplex @ (bit0 @ one))))). % one_add_one
thf(fact_13_one__add__one, axiom,
    (((plus_plus_nat @ one_one_nat @ one_one_nat) = (numeral_numeral_nat @ (bit0 @ one))))). % one_add_one
thf(fact_14_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_15_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_16_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_17_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_18_nonzero__divide__mult__cancel__left, axiom,
    ((![A2 : complex, B : complex]: ((~ ((A2 = zero_zero_complex))) => ((divide1210191872omplex @ A2 @ (times_times_complex @ A2 @ B)) = (divide1210191872omplex @ one_one_complex @ B)))))). % nonzero_divide_mult_cancel_left
thf(fact_19_nonzero__divide__mult__cancel__right, axiom,
    ((![B : complex, A2 : complex]: ((~ ((B = zero_zero_complex))) => ((divide1210191872omplex @ B @ (times_times_complex @ A2 @ B)) = (divide1210191872omplex @ one_one_complex @ A2)))))). % nonzero_divide_mult_cancel_right
thf(fact_20_div__mult__self1, axiom,
    ((![B : nat, A2 : nat, C : nat]: ((~ ((B = zero_zero_nat))) => ((divide_divide_nat @ (plus_plus_nat @ A2 @ (times_times_nat @ C @ B)) @ B) = (plus_plus_nat @ C @ (divide_divide_nat @ A2 @ B))))))). % div_mult_self1
thf(fact_21_div__mult__self2, axiom,
    ((![B : nat, A2 : nat, C : nat]: ((~ ((B = zero_zero_nat))) => ((divide_divide_nat @ (plus_plus_nat @ A2 @ (times_times_nat @ B @ C)) @ B) = (plus_plus_nat @ C @ (divide_divide_nat @ A2 @ B))))))). % div_mult_self2
thf(fact_22_numeral__eq__iff, axiom,
    ((![M : num, N : num]: (((numeral_numeral_nat @ M) = (numeral_numeral_nat @ N)) = (M = N))))). % numeral_eq_iff
thf(fact_23_division__ring__divide__zero, axiom,
    ((![A2 : complex]: ((divide1210191872omplex @ A2 @ zero_zero_complex) = zero_zero_complex)))). % division_ring_divide_zero
thf(fact_24_bits__div__by__0, axiom,
    ((![A2 : nat]: ((divide_divide_nat @ A2 @ zero_zero_nat) = zero_zero_nat)))). % bits_div_by_0
thf(fact_25_divide__cancel__right, axiom,
    ((![A2 : complex, C : complex, B : complex]: (((divide1210191872omplex @ A2 @ C) = (divide1210191872omplex @ B @ C)) = (((C = zero_zero_complex)) | ((A2 = B))))))). % divide_cancel_right
thf(fact_26_bits__div__0, axiom,
    ((![A2 : nat]: ((divide_divide_nat @ zero_zero_nat @ A2) = zero_zero_nat)))). % bits_div_0
thf(fact_27_divide__cancel__left, axiom,
    ((![C : complex, A2 : complex, B : complex]: (((divide1210191872omplex @ C @ A2) = (divide1210191872omplex @ C @ B)) = (((C = zero_zero_complex)) | ((A2 = B))))))). % divide_cancel_left
thf(fact_28_divide__eq__0__iff, axiom,
    ((![A2 : complex, B : complex]: (((divide1210191872omplex @ A2 @ B) = zero_zero_complex) = (((A2 = zero_zero_complex)) | ((B = zero_zero_complex))))))). % divide_eq_0_iff
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_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_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_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_33_times__divide__eq__right, axiom,
    ((![A2 : complex, B : complex, C : complex]: ((times_times_complex @ A2 @ (divide1210191872omplex @ B @ C)) = (divide1210191872omplex @ (times_times_complex @ A2 @ B) @ C))))). % times_divide_eq_right
thf(fact_34_divide__divide__eq__right, axiom,
    ((![A2 : complex, B : complex, C : complex]: ((divide1210191872omplex @ A2 @ (divide1210191872omplex @ B @ C)) = (divide1210191872omplex @ (times_times_complex @ A2 @ C) @ B))))). % divide_divide_eq_right
thf(fact_35_divide__divide__eq__left, axiom,
    ((![A2 : complex, B : complex, C : complex]: ((divide1210191872omplex @ (divide1210191872omplex @ A2 @ B) @ C) = (divide1210191872omplex @ A2 @ (times_times_complex @ B @ C)))))). % divide_divide_eq_left
thf(fact_36_times__divide__eq__left, axiom,
    ((![B : complex, C : complex, A2 : complex]: ((times_times_complex @ (divide1210191872omplex @ B @ C) @ A2) = (divide1210191872omplex @ (times_times_complex @ B @ A2) @ C))))). % times_divide_eq_left
thf(fact_37_bits__div__by__1, axiom,
    ((![A2 : nat]: ((divide_divide_nat @ A2 @ one_one_nat) = A2)))). % bits_div_by_1
thf(fact_38_power__one, axiom,
    ((![N : nat]: ((power_power_nat @ one_one_nat @ N) = one_one_nat)))). % power_one
thf(fact_39_power__one, axiom,
    ((![N : nat]: ((power_power_complex @ one_one_complex @ N) = one_one_complex)))). % power_one
thf(fact_40_num__double, axiom,
    ((![N : num]: ((times_times_num @ (bit0 @ one) @ N) = (bit0 @ N))))). % num_double
thf(fact_41_power__mult__numeral, axiom,
    ((![A2 : complex, M : num, N : num]: ((power_power_complex @ (power_power_complex @ A2 @ (numeral_numeral_nat @ M)) @ (numeral_numeral_nat @ N)) = (power_power_complex @ A2 @ (numeral_numeral_nat @ (times_times_num @ M @ N))))))). % power_mult_numeral
thf(fact_42_power__one__right, axiom,
    ((![A2 : complex]: ((power_power_complex @ A2 @ one_one_nat) = A2)))). % power_one_right
thf(fact_43_div__mult__mult1__if, axiom,
    ((![C : nat, A2 : nat, B : nat]: (((C = zero_zero_nat) => ((divide_divide_nat @ (times_times_nat @ C @ A2) @ (times_times_nat @ C @ B)) = zero_zero_nat)) & ((~ ((C = zero_zero_nat))) => ((divide_divide_nat @ (times_times_nat @ C @ A2) @ (times_times_nat @ C @ B)) = (divide_divide_nat @ A2 @ B))))))). % div_mult_mult1_if
thf(fact_44_div__mult__mult2, axiom,
    ((![C : nat, A2 : nat, B : nat]: ((~ ((C = zero_zero_nat))) => ((divide_divide_nat @ (times_times_nat @ A2 @ C) @ (times_times_nat @ B @ C)) = (divide_divide_nat @ A2 @ B)))))). % div_mult_mult2
thf(fact_45_div__mult__mult1, axiom,
    ((![C : nat, A2 : nat, B : nat]: ((~ ((C = zero_zero_nat))) => ((divide_divide_nat @ (times_times_nat @ C @ A2) @ (times_times_nat @ C @ B)) = (divide_divide_nat @ A2 @ B)))))). % div_mult_mult1
thf(fact_46_nonzero__mult__divide__mult__cancel__right2, axiom,
    ((![C : complex, A2 : complex, B : complex]: ((~ ((C = zero_zero_complex))) => ((divide1210191872omplex @ (times_times_complex @ A2 @ C) @ (times_times_complex @ C @ B)) = (divide1210191872omplex @ A2 @ B)))))). % nonzero_mult_divide_mult_cancel_right2
thf(fact_47_nonzero__mult__divide__mult__cancel__right, axiom,
    ((![C : complex, A2 : complex, B : complex]: ((~ ((C = zero_zero_complex))) => ((divide1210191872omplex @ (times_times_complex @ A2 @ C) @ (times_times_complex @ B @ C)) = (divide1210191872omplex @ A2 @ B)))))). % nonzero_mult_divide_mult_cancel_right
thf(fact_48_nonzero__mult__divide__mult__cancel__left2, axiom,
    ((![C : complex, A2 : complex, B : complex]: ((~ ((C = zero_zero_complex))) => ((divide1210191872omplex @ (times_times_complex @ C @ A2) @ (times_times_complex @ B @ C)) = (divide1210191872omplex @ A2 @ B)))))). % nonzero_mult_divide_mult_cancel_left2
thf(fact_49_nonzero__mult__divide__mult__cancel__left, axiom,
    ((![C : complex, A2 : complex, B : complex]: ((~ ((C = zero_zero_complex))) => ((divide1210191872omplex @ (times_times_complex @ C @ A2) @ (times_times_complex @ C @ B)) = (divide1210191872omplex @ A2 @ B)))))). % nonzero_mult_divide_mult_cancel_left
thf(fact_50_mult__divide__mult__cancel__left__if, axiom,
    ((![C : complex, A2 : complex, B : complex]: (((C = zero_zero_complex) => ((divide1210191872omplex @ (times_times_complex @ C @ A2) @ (times_times_complex @ C @ B)) = zero_zero_complex)) & ((~ ((C = zero_zero_complex))) => ((divide1210191872omplex @ (times_times_complex @ C @ A2) @ (times_times_complex @ C @ B)) = (divide1210191872omplex @ A2 @ B))))))). % mult_divide_mult_cancel_left_if
thf(fact_51_distrib__right__numeral, axiom,
    ((![A2 : complex, B : complex, V : num]: ((times_times_complex @ (plus_plus_complex @ A2 @ B) @ (numera632737353omplex @ V)) = (plus_plus_complex @ (times_times_complex @ A2 @ (numera632737353omplex @ V)) @ (times_times_complex @ B @ (numera632737353omplex @ V))))))). % distrib_right_numeral
thf(fact_52_distrib__right__numeral, axiom,
    ((![A2 : nat, B : nat, V : num]: ((times_times_nat @ (plus_plus_nat @ A2 @ B) @ (numeral_numeral_nat @ V)) = (plus_plus_nat @ (times_times_nat @ A2 @ (numeral_numeral_nat @ V)) @ (times_times_nat @ B @ (numeral_numeral_nat @ V))))))). % distrib_right_numeral
thf(fact_53_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_54_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_55_divide__self__if, axiom,
    ((![A2 : complex]: (((A2 = zero_zero_complex) => ((divide1210191872omplex @ A2 @ A2) = zero_zero_complex)) & ((~ ((A2 = zero_zero_complex))) => ((divide1210191872omplex @ A2 @ A2) = one_one_complex)))))). % divide_self_if
thf(fact_56_divide__self, axiom,
    ((![A2 : complex]: ((~ ((A2 = zero_zero_complex))) => ((divide1210191872omplex @ A2 @ A2) = one_one_complex))))). % divide_self
thf(fact_57_one__eq__divide__iff, axiom,
    ((![A2 : complex, B : complex]: ((one_one_complex = (divide1210191872omplex @ A2 @ B)) = (((~ ((B = zero_zero_complex)))) & ((A2 = B))))))). % one_eq_divide_iff
thf(fact_58_divide__eq__1__iff, axiom,
    ((![A2 : complex, B : complex]: (((divide1210191872omplex @ A2 @ B) = one_one_complex) = (((~ ((B = zero_zero_complex)))) & ((A2 = B))))))). % divide_eq_1_iff
thf(fact_59_one__eq__numeral__iff, axiom,
    ((![N : num]: ((one_one_complex = (numera632737353omplex @ N)) = (one = N))))). % one_eq_numeral_iff
thf(fact_60_one__eq__numeral__iff, axiom,
    ((![N : num]: ((one_one_nat = (numeral_numeral_nat @ N)) = (one = N))))). % one_eq_numeral_iff
thf(fact_61_numeral__eq__one__iff, axiom,
    ((![N : num]: (((numera632737353omplex @ N) = one_one_complex) = (N = one))))). % numeral_eq_one_iff
thf(fact_62_numeral__eq__one__iff, axiom,
    ((![N : num]: (((numeral_numeral_nat @ N) = one_one_nat) = (N = one))))). % numeral_eq_one_iff
thf(fact_63_power__zero__numeral, axiom,
    ((![K2 : num]: ((power_power_nat @ zero_zero_nat @ (numeral_numeral_nat @ K2)) = zero_zero_nat)))). % power_zero_numeral
thf(fact_64_power__zero__numeral, axiom,
    ((![K2 : num]: ((power_power_complex @ zero_zero_complex @ (numeral_numeral_nat @ K2)) = zero_zero_complex)))). % power_zero_numeral
thf(fact_65_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_66_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_67_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_68_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_69_eq__divide__eq__numeral1_I1_J, axiom,
    ((![A2 : complex, B : complex, W : num]: ((A2 = (divide1210191872omplex @ B @ (numera632737353omplex @ W))) = (((((~ (((numera632737353omplex @ W) = zero_zero_complex)))) => (((times_times_complex @ A2 @ (numera632737353omplex @ W)) = B)))) & (((((numera632737353omplex @ W) = zero_zero_complex)) => ((A2 = zero_zero_complex))))))))). % eq_divide_eq_numeral1(1)
thf(fact_70_divide__eq__eq__numeral1_I1_J, axiom,
    ((![B : complex, W : num, A2 : complex]: (((divide1210191872omplex @ B @ (numera632737353omplex @ W)) = A2) = (((((~ (((numera632737353omplex @ W) = zero_zero_complex)))) => ((B = (times_times_complex @ A2 @ (numera632737353omplex @ W)))))) & (((((numera632737353omplex @ W) = zero_zero_complex)) => ((A2 = zero_zero_complex))))))))). % divide_eq_eq_numeral1(1)
thf(fact_71_div__mult__self4, axiom,
    ((![B : nat, C : nat, A2 : nat]: ((~ ((B = zero_zero_nat))) => ((divide_divide_nat @ (plus_plus_nat @ (times_times_nat @ B @ C) @ A2) @ B) = (plus_plus_nat @ C @ (divide_divide_nat @ A2 @ B))))))). % div_mult_self4
thf(fact_72_div__mult__self3, axiom,
    ((![B : nat, C : nat, A2 : nat]: ((~ ((B = zero_zero_nat))) => ((divide_divide_nat @ (plus_plus_nat @ (times_times_nat @ C @ B) @ A2) @ B) = (plus_plus_nat @ C @ (divide_divide_nat @ A2 @ B))))))). % div_mult_self3
thf(fact_73_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_74_power__add__numeral2, axiom,
    ((![A2 : nat, M : num, N : num, B : nat]: ((times_times_nat @ (power_power_nat @ A2 @ (numeral_numeral_nat @ M)) @ (times_times_nat @ (power_power_nat @ A2 @ (numeral_numeral_nat @ N)) @ B)) = (times_times_nat @ (power_power_nat @ A2 @ (numeral_numeral_nat @ (plus_plus_num @ M @ N))) @ B))))). % power_add_numeral2
thf(fact_75_power__add__numeral2, axiom,
    ((![A2 : complex, M : num, N : num, B : complex]: ((times_times_complex @ (power_power_complex @ A2 @ (numeral_numeral_nat @ M)) @ (times_times_complex @ (power_power_complex @ A2 @ (numeral_numeral_nat @ N)) @ B)) = (times_times_complex @ (power_power_complex @ A2 @ (numeral_numeral_nat @ (plus_plus_num @ M @ N))) @ B))))). % power_add_numeral2
thf(fact_76_power__add__numeral, axiom,
    ((![A2 : nat, M : num, N : num]: ((times_times_nat @ (power_power_nat @ A2 @ (numeral_numeral_nat @ M)) @ (power_power_nat @ A2 @ (numeral_numeral_nat @ N))) = (power_power_nat @ A2 @ (numeral_numeral_nat @ (plus_plus_num @ M @ N))))))). % power_add_numeral
thf(fact_77_power__add__numeral, axiom,
    ((![A2 : complex, M : num, N : num]: ((times_times_complex @ (power_power_complex @ A2 @ (numeral_numeral_nat @ M)) @ (power_power_complex @ A2 @ (numeral_numeral_nat @ N))) = (power_power_complex @ A2 @ (numeral_numeral_nat @ (plus_plus_num @ M @ N))))))). % power_add_numeral
thf(fact_78_add__One__commute, axiom,
    ((![N : num]: ((plus_plus_num @ one @ N) = (plus_plus_num @ N @ one))))). % add_One_commute
thf(fact_79_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_80_div__mult2__numeral__eq, axiom,
    ((![A2 : nat, K2 : num, L : num]: ((divide_divide_nat @ (divide_divide_nat @ A2 @ (numeral_numeral_nat @ K2)) @ (numeral_numeral_nat @ L)) = (divide_divide_nat @ A2 @ (numeral_numeral_nat @ (times_times_num @ K2 @ L))))))). % div_mult2_numeral_eq
thf(fact_81_root__nonzero, axiom,
    ((![N : nat]: (~ (((fFT_Mirabelle_root @ N) = zero_zero_complex)))))). % root_nonzero
thf(fact_82_is__num__normalize_I1_J, axiom,
    ((![A2 : complex, B : complex, C : complex]: ((plus_plus_complex @ (plus_plus_complex @ A2 @ B) @ C) = (plus_plus_complex @ A2 @ (plus_plus_complex @ B @ C)))))). % is_num_normalize(1)
thf(fact_83_zero__neq__numeral, axiom,
    ((![N : num]: (~ ((zero_zero_complex = (numera632737353omplex @ N))))))). % zero_neq_numeral
thf(fact_84_zero__neq__numeral, axiom,
    ((![N : num]: (~ ((zero_zero_nat = (numeral_numeral_nat @ N))))))). % zero_neq_numeral
thf(fact_85_power__not__zero, axiom,
    ((![A2 : nat, N : nat]: ((~ ((A2 = zero_zero_nat))) => (~ (((power_power_nat @ A2 @ N) = zero_zero_nat))))))). % power_not_zero
thf(fact_86_power__not__zero, axiom,
    ((![A2 : complex, N : nat]: ((~ ((A2 = zero_zero_complex))) => (~ (((power_power_complex @ A2 @ N) = zero_zero_complex))))))). % power_not_zero
thf(fact_87_divide__divide__eq__left_H, axiom,
    ((![A2 : complex, B : complex, C : complex]: ((divide1210191872omplex @ (divide1210191872omplex @ A2 @ B) @ C) = (divide1210191872omplex @ A2 @ (times_times_complex @ C @ B)))))). % divide_divide_eq_left'
thf(fact_88_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_89_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_90_add__divide__distrib, axiom,
    ((![A2 : complex, B : complex, C : complex]: ((divide1210191872omplex @ (plus_plus_complex @ A2 @ B) @ C) = (plus_plus_complex @ (divide1210191872omplex @ A2 @ C) @ (divide1210191872omplex @ B @ C)))))). % add_divide_distrib
thf(fact_91_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_92_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_93_power__mult__distrib, axiom,
    ((![A2 : nat, B : nat, N : nat]: ((power_power_nat @ (times_times_nat @ A2 @ B) @ N) = (times_times_nat @ (power_power_nat @ A2 @ N) @ (power_power_nat @ B @ N)))))). % power_mult_distrib
thf(fact_94_power__mult__distrib, axiom,
    ((![A2 : complex, B : complex, N : nat]: ((power_power_complex @ (times_times_complex @ A2 @ B) @ N) = (times_times_complex @ (power_power_complex @ A2 @ N) @ (power_power_complex @ B @ N)))))). % power_mult_distrib
thf(fact_95_power__commutes, axiom,
    ((![A2 : nat, N : nat]: ((times_times_nat @ (power_power_nat @ A2 @ N) @ A2) = (times_times_nat @ A2 @ (power_power_nat @ A2 @ N)))))). % power_commutes
thf(fact_96_power__commutes, axiom,
    ((![A2 : complex, N : nat]: ((times_times_complex @ (power_power_complex @ A2 @ N) @ A2) = (times_times_complex @ A2 @ (power_power_complex @ A2 @ N)))))). % power_commutes
thf(fact_97_power__divide, axiom,
    ((![A2 : complex, B : complex, N : nat]: ((power_power_complex @ (divide1210191872omplex @ A2 @ B) @ N) = (divide1210191872omplex @ (power_power_complex @ A2 @ N) @ (power_power_complex @ B @ N)))))). % power_divide
thf(fact_98_power__mult, axiom,
    ((![A2 : complex, M : nat, N : nat]: ((power_power_complex @ A2 @ (times_times_nat @ M @ N)) = (power_power_complex @ (power_power_complex @ A2 @ M) @ N))))). % power_mult
thf(fact_99_nonzero__eq__divide__eq, axiom,
    ((![C : complex, A2 : complex, B : complex]: ((~ ((C = zero_zero_complex))) => ((A2 = (divide1210191872omplex @ B @ C)) = ((times_times_complex @ A2 @ C) = B)))))). % nonzero_eq_divide_eq
thf(fact_100_nonzero__divide__eq__eq, axiom,
    ((![C : complex, B : complex, A2 : complex]: ((~ ((C = zero_zero_complex))) => (((divide1210191872omplex @ B @ C) = A2) = (B = (times_times_complex @ A2 @ C))))))). % nonzero_divide_eq_eq
thf(fact_101_eq__divide__imp, axiom,
    ((![C : complex, A2 : complex, B : complex]: ((~ ((C = zero_zero_complex))) => (((times_times_complex @ A2 @ C) = B) => (A2 = (divide1210191872omplex @ B @ C))))))). % eq_divide_imp
thf(fact_102_divide__eq__imp, axiom,
    ((![C : complex, B : complex, A2 : complex]: ((~ ((C = zero_zero_complex))) => ((B = (times_times_complex @ A2 @ C)) => ((divide1210191872omplex @ B @ C) = A2)))))). % divide_eq_imp
thf(fact_103_eq__divide__eq, axiom,
    ((![A2 : complex, B : complex, C : complex]: ((A2 = (divide1210191872omplex @ B @ C)) = (((((~ ((C = zero_zero_complex)))) => (((times_times_complex @ A2 @ C) = B)))) & ((((C = zero_zero_complex)) => ((A2 = zero_zero_complex))))))))). % eq_divide_eq
thf(fact_104_divide__eq__eq, axiom,
    ((![B : complex, C : complex, A2 : complex]: (((divide1210191872omplex @ B @ C) = A2) = (((((~ ((C = zero_zero_complex)))) => ((B = (times_times_complex @ A2 @ C))))) & ((((C = zero_zero_complex)) => ((A2 = zero_zero_complex))))))))). % divide_eq_eq
thf(fact_105_frac__eq__eq, axiom,
    ((![Y : complex, Z : complex, X : complex, W : complex]: ((~ ((Y = zero_zero_complex))) => ((~ ((Z = zero_zero_complex))) => (((divide1210191872omplex @ X @ Y) = (divide1210191872omplex @ W @ Z)) = ((times_times_complex @ X @ Z) = (times_times_complex @ W @ Y)))))))). % frac_eq_eq
thf(fact_106_right__inverse__eq, axiom,
    ((![B : complex, A2 : complex]: ((~ ((B = zero_zero_complex))) => (((divide1210191872omplex @ A2 @ B) = one_one_complex) = (A2 = B)))))). % right_inverse_eq
thf(fact_107_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_108_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_109_mult__numeral__1__right, axiom,
    ((![A2 : complex]: ((times_times_complex @ A2 @ (numera632737353omplex @ one)) = A2)))). % mult_numeral_1_right
thf(fact_110_mult__numeral__1__right, axiom,
    ((![A2 : nat]: ((times_times_nat @ A2 @ (numeral_numeral_nat @ one)) = A2)))). % mult_numeral_1_right
thf(fact_111_mult__numeral__1, axiom,
    ((![A2 : complex]: ((times_times_complex @ (numera632737353omplex @ one) @ A2) = A2)))). % mult_numeral_1
thf(fact_112_mult__numeral__1, axiom,
    ((![A2 : nat]: ((times_times_nat @ (numeral_numeral_nat @ one) @ A2) = A2)))). % mult_numeral_1
thf(fact_113_numeral__Bit0, axiom,
    ((![N : num]: ((numera632737353omplex @ (bit0 @ N)) = (plus_plus_complex @ (numera632737353omplex @ N) @ (numera632737353omplex @ N)))))). % numeral_Bit0
thf(fact_114_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_115_numeral__One, axiom,
    (((numera632737353omplex @ one) = one_one_complex))). % numeral_One
thf(fact_116_numeral__One, axiom,
    (((numeral_numeral_nat @ one) = one_one_nat))). % numeral_One
thf(fact_117_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_118_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_119_divide__numeral__1, axiom,
    ((![A2 : complex]: ((divide1210191872omplex @ A2 @ (numera632737353omplex @ one)) = A2)))). % divide_numeral_1
thf(fact_120_power__one__over, axiom,
    ((![A2 : complex, N : nat]: ((power_power_complex @ (divide1210191872omplex @ one_one_complex @ A2) @ N) = (divide1210191872omplex @ one_one_complex @ (power_power_complex @ A2 @ N)))))). % power_one_over
thf(fact_121_power__0, axiom,
    ((![A2 : nat]: ((power_power_nat @ A2 @ zero_zero_nat) = one_one_nat)))). % power_0
thf(fact_122_power__0, axiom,
    ((![A2 : complex]: ((power_power_complex @ A2 @ zero_zero_nat) = one_one_complex)))). % power_0
thf(fact_123_power__add, axiom,
    ((![A2 : nat, M : nat, N : nat]: ((power_power_nat @ A2 @ (plus_plus_nat @ M @ N)) = (times_times_nat @ (power_power_nat @ A2 @ M) @ (power_power_nat @ A2 @ N)))))). % power_add
thf(fact_124_power__add, axiom,
    ((![A2 : complex, M : nat, N : nat]: ((power_power_complex @ A2 @ (plus_plus_nat @ M @ N)) = (times_times_complex @ (power_power_complex @ A2 @ M) @ (power_power_complex @ A2 @ N)))))). % power_add
thf(fact_125_numerals_I1_J, axiom,
    (((numeral_numeral_nat @ one) = one_one_nat))). % numerals(1)
thf(fact_126_numeral__code_I2_J, axiom,
    ((![N : num]: ((numera632737353omplex @ (bit0 @ N)) = (plus_plus_complex @ (numera632737353omplex @ N) @ (numera632737353omplex @ N)))))). % numeral_code(2)
thf(fact_127_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_128_eq__divide__eq__numeral_I1_J, axiom,
    ((![W : num, B : complex, C : complex]: (((numera632737353omplex @ W) = (divide1210191872omplex @ B @ C)) = (((((~ ((C = zero_zero_complex)))) => (((times_times_complex @ (numera632737353omplex @ W) @ C) = B)))) & ((((C = zero_zero_complex)) => (((numera632737353omplex @ W) = zero_zero_complex))))))))). % eq_divide_eq_numeral(1)
thf(fact_129_divide__eq__eq__numeral_I1_J, axiom,
    ((![B : complex, C : complex, W : num]: (((divide1210191872omplex @ B @ C) = (numera632737353omplex @ W)) = (((((~ ((C = zero_zero_complex)))) => ((B = (times_times_complex @ (numera632737353omplex @ W) @ C))))) & ((((C = zero_zero_complex)) => (((numera632737353omplex @ W) = zero_zero_complex))))))))). % divide_eq_eq_numeral(1)
thf(fact_130_divide__add__eq__iff, axiom,
    ((![Z : complex, X : complex, Y : complex]: ((~ ((Z = zero_zero_complex))) => ((plus_plus_complex @ (divide1210191872omplex @ X @ Z) @ Y) = (divide1210191872omplex @ (plus_plus_complex @ X @ (times_times_complex @ Y @ Z)) @ Z)))))). % divide_add_eq_iff
thf(fact_131_add__divide__eq__iff, axiom,
    ((![Z : complex, X : complex, Y : complex]: ((~ ((Z = zero_zero_complex))) => ((plus_plus_complex @ X @ (divide1210191872omplex @ Y @ Z)) = (divide1210191872omplex @ (plus_plus_complex @ (times_times_complex @ X @ Z) @ Y) @ Z)))))). % add_divide_eq_iff
thf(fact_132_add__num__frac, axiom,
    ((![Y : complex, Z : complex, X : complex]: ((~ ((Y = zero_zero_complex))) => ((plus_plus_complex @ Z @ (divide1210191872omplex @ X @ Y)) = (divide1210191872omplex @ (plus_plus_complex @ X @ (times_times_complex @ Z @ Y)) @ Y)))))). % add_num_frac
thf(fact_133_add__frac__num, axiom,
    ((![Y : complex, X : complex, Z : complex]: ((~ ((Y = zero_zero_complex))) => ((plus_plus_complex @ (divide1210191872omplex @ X @ Y) @ Z) = (divide1210191872omplex @ (plus_plus_complex @ X @ (times_times_complex @ Z @ Y)) @ Y)))))). % add_frac_num
thf(fact_134_add__frac__eq, axiom,
    ((![Y : complex, Z : complex, X : complex, W : complex]: ((~ ((Y = zero_zero_complex))) => ((~ ((Z = zero_zero_complex))) => ((plus_plus_complex @ (divide1210191872omplex @ X @ Y) @ (divide1210191872omplex @ W @ Z)) = (divide1210191872omplex @ (plus_plus_complex @ (times_times_complex @ X @ Z) @ (times_times_complex @ W @ Y)) @ (times_times_complex @ Y @ Z)))))))). % add_frac_eq
thf(fact_135_add__divide__eq__if__simps_I1_J, axiom,
    ((![Z : complex, A2 : complex, B : complex]: (((Z = zero_zero_complex) => ((plus_plus_complex @ A2 @ (divide1210191872omplex @ B @ Z)) = A2)) & ((~ ((Z = zero_zero_complex))) => ((plus_plus_complex @ A2 @ (divide1210191872omplex @ B @ Z)) = (divide1210191872omplex @ (plus_plus_complex @ (times_times_complex @ A2 @ Z) @ B) @ Z))))))). % add_divide_eq_if_simps(1)
thf(fact_136_add__divide__eq__if__simps_I2_J, axiom,
    ((![Z : complex, A2 : complex, B : complex]: (((Z = zero_zero_complex) => ((plus_plus_complex @ (divide1210191872omplex @ A2 @ Z) @ B) = B)) & ((~ ((Z = zero_zero_complex))) => ((plus_plus_complex @ (divide1210191872omplex @ A2 @ Z) @ B) = (divide1210191872omplex @ (plus_plus_complex @ A2 @ (times_times_complex @ B @ Z)) @ Z))))))). % add_divide_eq_if_simps(2)
thf(fact_137_div__add__self2, axiom,
    ((![B : nat, A2 : nat]: ((~ ((B = zero_zero_nat))) => ((divide_divide_nat @ (plus_plus_nat @ A2 @ B) @ B) = (plus_plus_nat @ (divide_divide_nat @ A2 @ B) @ one_one_nat)))))). % div_add_self2
thf(fact_138_div__add__self1, axiom,
    ((![B : nat, A2 : nat]: ((~ ((B = zero_zero_nat))) => ((divide_divide_nat @ (plus_plus_nat @ B @ A2) @ B) = (plus_plus_nat @ (divide_divide_nat @ A2 @ B) @ one_one_nat)))))). % div_add_self1
thf(fact_139_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_140_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_141_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_142_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_143_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_144_left__add__twice, axiom,
    ((![A2 : complex, B : complex]: ((plus_plus_complex @ A2 @ (plus_plus_complex @ A2 @ B)) = (plus_plus_complex @ (times_times_complex @ (numera632737353omplex @ (bit0 @ one)) @ A2) @ B))))). % left_add_twice
thf(fact_145_left__add__twice, axiom,
    ((![A2 : nat, B : nat]: ((plus_plus_nat @ A2 @ (plus_plus_nat @ A2 @ B)) = (plus_plus_nat @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ A2) @ B))))). % left_add_twice
thf(fact_146_mult__2__right, axiom,
    ((![Z : complex]: ((times_times_complex @ Z @ (numera632737353omplex @ (bit0 @ one))) = (plus_plus_complex @ Z @ Z))))). % mult_2_right
thf(fact_147_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_148_mult__2, axiom,
    ((![Z : complex]: ((times_times_complex @ (numera632737353omplex @ (bit0 @ one)) @ Z) = (plus_plus_complex @ Z @ Z))))). % mult_2
thf(fact_149_mult__2, axiom,
    ((![Z : nat]: ((times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ Z) = (plus_plus_nat @ Z @ Z))))). % mult_2
thf(fact_150_zero__power2, axiom,
    (((power_power_nat @ zero_zero_nat @ (numeral_numeral_nat @ (bit0 @ one))) = zero_zero_nat))). % zero_power2
thf(fact_151_zero__power2, axiom,
    (((power_power_complex @ zero_zero_complex @ (numeral_numeral_nat @ (bit0 @ one))) = zero_zero_complex))). % zero_power2
thf(fact_152_power2__eq__square, axiom,
    ((![A2 : nat]: ((power_power_nat @ A2 @ (numeral_numeral_nat @ (bit0 @ one))) = (times_times_nat @ A2 @ A2))))). % power2_eq_square
thf(fact_153_power2__eq__square, axiom,
    ((![A2 : complex]: ((power_power_complex @ A2 @ (numeral_numeral_nat @ (bit0 @ one))) = (times_times_complex @ A2 @ A2))))). % power2_eq_square
thf(fact_154_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_155_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_156_one__power2, axiom,
    (((power_power_nat @ one_one_nat @ (numeral_numeral_nat @ (bit0 @ one))) = one_one_nat))). % one_power2
thf(fact_157_one__power2, axiom,
    (((power_power_complex @ one_one_complex @ (numeral_numeral_nat @ (bit0 @ one))) = one_one_complex))). % one_power2
thf(fact_158_power__even__eq, axiom,
    ((![A2 : complex, N : nat]: ((power_power_complex @ A2 @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N)) = (power_power_complex @ (power_power_complex @ A2 @ N) @ (numeral_numeral_nat @ (bit0 @ one))))))). % power_even_eq
thf(fact_159_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_160_exp__add__not__zero__imp__right, axiom,
    ((![M : nat, N : nat]: ((~ (((power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (plus_plus_nat @ M @ N)) = zero_zero_nat))) => (~ (((power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N) = zero_zero_nat))))))). % exp_add_not_zero_imp_right
thf(fact_161_exp__add__not__zero__imp__left, axiom,
    ((![M : nat, N : nat]: ((~ (((power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (plus_plus_nat @ M @ N)) = zero_zero_nat))) => (~ (((power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ M) = zero_zero_nat))))))). % exp_add_not_zero_imp_left
thf(fact_162_div__exp__eq, axiom,
    ((![A2 : nat, M : nat, N : nat]: ((divide_divide_nat @ (divide_divide_nat @ A2 @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ M)) @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N)) = (divide_divide_nat @ A2 @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (plus_plus_nat @ M @ N))))))). % div_exp_eq
thf(fact_163_nat__induct2, axiom,
    ((![P : nat > $o, N : nat]: ((P @ zero_zero_nat) => ((P @ one_one_nat) => ((![N3 : nat]: ((P @ N3) => (P @ (plus_plus_nat @ N3 @ (numeral_numeral_nat @ (bit0 @ one)))))) => (P @ N))))))). % nat_induct2
thf(fact_164_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_165_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_166_semiring__norm_I2_J, axiom,
    (((plus_plus_num @ one @ one) = (bit0 @ one)))). % semiring_norm(2)
thf(fact_167_div__self, axiom,
    ((![A2 : complex]: ((~ ((A2 = zero_zero_complex))) => ((divide1210191872omplex @ A2 @ A2) = one_one_complex))))). % div_self
thf(fact_168_div__self, axiom,
    ((![A2 : nat]: ((~ ((A2 = zero_zero_nat))) => ((divide_divide_nat @ A2 @ A2) = one_one_nat))))). % div_self
thf(fact_169_nonzero__mult__div__cancel__right, axiom,
    ((![B : complex, A2 : complex]: ((~ ((B = zero_zero_complex))) => ((divide1210191872omplex @ (times_times_complex @ A2 @ B) @ B) = A2))))). % nonzero_mult_div_cancel_right
thf(fact_170_nonzero__mult__div__cancel__right, axiom,
    ((![B : nat, A2 : nat]: ((~ ((B = zero_zero_nat))) => ((divide_divide_nat @ (times_times_nat @ A2 @ B) @ B) = A2))))). % nonzero_mult_div_cancel_right
thf(fact_171_nonzero__mult__div__cancel__left, axiom,
    ((![A2 : complex, B : complex]: ((~ ((A2 = zero_zero_complex))) => ((divide1210191872omplex @ (times_times_complex @ A2 @ B) @ A2) = B))))). % nonzero_mult_div_cancel_left
thf(fact_172_nonzero__mult__div__cancel__left, axiom,
    ((![A2 : nat, B : nat]: ((~ ((A2 = zero_zero_nat))) => ((divide_divide_nat @ (times_times_nat @ A2 @ B) @ A2) = B))))). % nonzero_mult_div_cancel_left
thf(fact_173_mult__cancel__right2, axiom,
    ((![A2 : complex, C : complex]: (((times_times_complex @ A2 @ C) = C) = (((C = zero_zero_complex)) | ((A2 = one_one_complex))))))). % mult_cancel_right2
thf(fact_174_mult__cancel__right1, axiom,
    ((![C : complex, B : complex]: ((C = (times_times_complex @ B @ C)) = (((C = zero_zero_complex)) | ((B = one_one_complex))))))). % mult_cancel_right1
thf(fact_175_mult__cancel__left2, axiom,
    ((![C : complex, A2 : complex]: (((times_times_complex @ C @ A2) = C) = (((C = zero_zero_complex)) | ((A2 = one_one_complex))))))). % mult_cancel_left2
thf(fact_176_mult__cancel__left1, axiom,
    ((![C : complex, B : complex]: ((C = (times_times_complex @ C @ B)) = (((C = zero_zero_complex)) | ((B = one_one_complex))))))). % mult_cancel_left1
thf(fact_177_DFT__lower, axiom,
    ((![M : nat, A2 : nat > complex, I2 : nat]: ((fFT_Mirabelle_DFT @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ M) @ A2 @ I2) = (plus_plus_complex @ (fFT_Mirabelle_DFT @ M @ (^[I : nat]: (A2 @ (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]: (A2 @ (plus_plus_nat @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ I) @ one_one_nat))) @ I2))))))). % DFT_lower
thf(fact_178_semiring__norm_I87_J, axiom,
    ((![M : num, N : num]: (((bit0 @ M) = (bit0 @ N)) = (M = N))))). % semiring_norm(87)
thf(fact_179_mult__zero__left, axiom,
    ((![A2 : nat]: ((times_times_nat @ zero_zero_nat @ A2) = zero_zero_nat)))). % mult_zero_left
thf(fact_180_mult__zero__left, axiom,
    ((![A2 : complex]: ((times_times_complex @ zero_zero_complex @ A2) = zero_zero_complex)))). % mult_zero_left
thf(fact_181_mult__zero__right, axiom,
    ((![A2 : nat]: ((times_times_nat @ A2 @ zero_zero_nat) = zero_zero_nat)))). % mult_zero_right
thf(fact_182_mult__zero__right, axiom,
    ((![A2 : complex]: ((times_times_complex @ A2 @ zero_zero_complex) = zero_zero_complex)))). % mult_zero_right
thf(fact_183_mult__eq__0__iff, axiom,
    ((![A2 : nat, B : nat]: (((times_times_nat @ A2 @ B) = zero_zero_nat) = (((A2 = zero_zero_nat)) | ((B = zero_zero_nat))))))). % mult_eq_0_iff
thf(fact_184_mult__eq__0__iff, axiom,
    ((![A2 : complex, B : complex]: (((times_times_complex @ A2 @ B) = zero_zero_complex) = (((A2 = zero_zero_complex)) | ((B = zero_zero_complex))))))). % mult_eq_0_iff
thf(fact_185_mult__cancel__left, axiom,
    ((![C : nat, A2 : nat, B : nat]: (((times_times_nat @ C @ A2) = (times_times_nat @ C @ B)) = (((C = zero_zero_nat)) | ((A2 = B))))))). % mult_cancel_left
thf(fact_186_mult__cancel__left, axiom,
    ((![C : complex, A2 : complex, B : complex]: (((times_times_complex @ C @ A2) = (times_times_complex @ C @ B)) = (((C = zero_zero_complex)) | ((A2 = B))))))). % mult_cancel_left
thf(fact_187_mult__cancel__right, axiom,
    ((![A2 : nat, C : nat, B : nat]: (((times_times_nat @ A2 @ C) = (times_times_nat @ B @ C)) = (((C = zero_zero_nat)) | ((A2 = B))))))). % mult_cancel_right
thf(fact_188_mult__cancel__right, axiom,
    ((![A2 : complex, C : complex, B : complex]: (((times_times_complex @ A2 @ C) = (times_times_complex @ B @ C)) = (((C = zero_zero_complex)) | ((A2 = B))))))). % mult_cancel_right
thf(fact_189_div__0, axiom,
    ((![A2 : complex]: ((divide1210191872omplex @ zero_zero_complex @ A2) = zero_zero_complex)))). % div_0
thf(fact_190_div__0, axiom,
    ((![A2 : nat]: ((divide_divide_nat @ zero_zero_nat @ A2) = zero_zero_nat)))). % div_0
thf(fact_191_div__by__0, axiom,
    ((![A2 : complex]: ((divide1210191872omplex @ A2 @ zero_zero_complex) = zero_zero_complex)))). % div_by_0
thf(fact_192_div__by__0, axiom,
    ((![A2 : nat]: ((divide_divide_nat @ A2 @ zero_zero_nat) = zero_zero_nat)))). % div_by_0
thf(fact_193_div__by__1, axiom,
    ((![A2 : complex]: ((divide1210191872omplex @ A2 @ one_one_complex) = A2)))). % div_by_1
thf(fact_194_div__by__1, axiom,
    ((![A2 : nat]: ((divide_divide_nat @ A2 @ one_one_nat) = A2)))). % div_by_1

% Conjectures (1)
thf(conj_0, conjecture,
    (((plus_plus_complex @ (groups59700922omplex @ (^[J : nat]: (divide1210191872omplex @ (a @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ J)) @ (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))))) @ (set_or562006527an_nat @ zero_zero_nat @ m)) @ (groups59700922omplex @ (^[J : nat]: (divide1210191872omplex @ (a @ (plus_plus_nat @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ J) @ one_one_nat)) @ (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))))) @ (set_or562006527an_nat @ zero_zero_nat @ m))) = (plus_plus_complex @ (groups59700922omplex @ (^[J : nat]: (divide1210191872omplex @ (a @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ J)) @ (power_power_complex @ (fFT_Mirabelle_root @ m) @ (times_times_nat @ i @ J)))) @ (set_or562006527an_nat @ zero_zero_nat @ m)) @ (times_times_complex @ (power_power_complex @ (divide1210191872omplex @ one_one_complex @ (fFT_Mirabelle_root @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ m))) @ i) @ (groups59700922omplex @ (^[J : nat]: (divide1210191872omplex @ (a @ (plus_plus_nat @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ J) @ one_one_nat)) @ (power_power_complex @ (fFT_Mirabelle_root @ m) @ (times_times_nat @ i @ J)))) @ (set_or562006527an_nat @ zero_zero_nat @ m))))))).
