% 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_279__3225462_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:10:12.471

% 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 (26)
thf(sy_c_FFT__Mirabelle__ulikgskiun_ODFT, type,
    fFT_Mirabelle_DFT : nat > (nat > complex) > nat > complex).
thf(sy_c_FFT__Mirabelle__ulikgskiun_Oroot, type,
    fFT_Mirabelle_root : nat > complex).
thf(sy_c_Groups_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_Nat_OSuc, type,
    suc : 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_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat, type,
    set_or562006527an_nat : nat > nat > set_nat).
thf(sy_c_member_001t__Nat__Onat, type,
    member_nat : nat > set_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 (188)
thf(fact_0_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_1_DFT__def, axiom,
    ((fFT_Mirabelle_DFT = (^[N : nat]: (^[A : nat > complex]: (^[I2 : nat]: (groups59700922omplex @ (^[J2 : nat]: (times_times_complex @ (power_power_complex @ (fFT_Mirabelle_root @ N) @ (times_times_nat @ I2 @ J2)) @ (A @ J2))) @ (set_or562006527an_nat @ zero_zero_nat @ N)))))))). % DFT_def
thf(fact_2_add__2__eq__Suc, axiom,
    ((![N2 : nat]: ((plus_plus_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N2) = (suc @ (suc @ N2)))))). % add_2_eq_Suc
thf(fact_3_add__2__eq__Suc_H, axiom,
    ((![N2 : nat]: ((plus_plus_nat @ N2 @ (numeral_numeral_nat @ (bit0 @ one))) = (suc @ (suc @ N2)))))). % add_2_eq_Suc'
thf(fact_4_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_5_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_6_mult__Suc__right, axiom,
    ((![M : nat, N2 : nat]: ((times_times_nat @ M @ (suc @ N2)) = (plus_plus_nat @ M @ (times_times_nat @ M @ N2)))))). % mult_Suc_right
thf(fact_7_mult__eq__1__iff, axiom,
    ((![M : nat, N2 : nat]: (((times_times_nat @ M @ N2) = (suc @ zero_zero_nat)) = (((M = (suc @ zero_zero_nat))) & ((N2 = (suc @ zero_zero_nat)))))))). % mult_eq_1_iff
thf(fact_8_one__eq__mult__iff, axiom,
    ((![M : nat, N2 : nat]: (((suc @ zero_zero_nat) = (times_times_nat @ M @ N2)) = (((M = (suc @ zero_zero_nat))) & ((N2 = (suc @ zero_zero_nat)))))))). % one_eq_mult_iff
thf(fact_9_power__Suc0__right, axiom,
    ((![A2 : complex]: ((power_power_complex @ A2 @ (suc @ zero_zero_nat)) = A2)))). % power_Suc0_right
thf(fact_10_power__Suc0__right, axiom,
    ((![A2 : nat]: ((power_power_nat @ A2 @ (suc @ zero_zero_nat)) = A2)))). % power_Suc0_right
thf(fact_11_power__zero__numeral, axiom,
    ((![K : num]: ((power_power_complex @ zero_zero_complex @ (numeral_numeral_nat @ K)) = zero_zero_complex)))). % power_zero_numeral
thf(fact_12_power__zero__numeral, axiom,
    ((![K : num]: ((power_power_nat @ zero_zero_nat @ (numeral_numeral_nat @ K)) = zero_zero_nat)))). % power_zero_numeral
thf(fact_13_power__0__Suc, axiom,
    ((![N2 : nat]: ((power_power_complex @ zero_zero_complex @ (suc @ N2)) = zero_zero_complex)))). % power_0_Suc
thf(fact_14_power__0__Suc, axiom,
    ((![N2 : nat]: ((power_power_nat @ zero_zero_nat @ (suc @ N2)) = zero_zero_nat)))). % power_0_Suc
thf(fact_15_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_16_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_17_numeral__eq__iff, axiom,
    ((![M : num, N2 : num]: (((numeral_numeral_nat @ M) = (numeral_numeral_nat @ N2)) = (M = N2))))). % numeral_eq_iff
thf(fact_18_old_Onat_Oinject, axiom,
    ((![Nat : nat, Nat2 : nat]: (((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))). % old.nat.inject
thf(fact_19_nat_Oinject, axiom,
    ((![X2 : nat, Y2 : nat]: (((suc @ X2) = (suc @ Y2)) = (X2 = Y2))))). % nat.inject
thf(fact_20_numeral__times__numeral, axiom,
    ((![M : num, N2 : num]: ((times_times_complex @ (numera632737353omplex @ M) @ (numera632737353omplex @ N2)) = (numera632737353omplex @ (times_times_num @ M @ N2)))))). % numeral_times_numeral
thf(fact_21_numeral__times__numeral, axiom,
    ((![M : num, N2 : num]: ((times_times_nat @ (numeral_numeral_nat @ M) @ (numeral_numeral_nat @ N2)) = (numeral_numeral_nat @ (times_times_num @ M @ N2)))))). % numeral_times_numeral
thf(fact_22_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_23_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_24_numeral__plus__numeral, axiom,
    ((![M : num, N2 : num]: ((plus_plus_complex @ (numera632737353omplex @ M) @ (numera632737353omplex @ N2)) = (numera632737353omplex @ (plus_plus_num @ M @ N2)))))). % numeral_plus_numeral
thf(fact_25_numeral__plus__numeral, axiom,
    ((![M : num, N2 : num]: ((plus_plus_nat @ (numeral_numeral_nat @ M) @ (numeral_numeral_nat @ N2)) = (numeral_numeral_nat @ (plus_plus_num @ M @ N2)))))). % numeral_plus_numeral
thf(fact_26_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_27_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_28_num__double, axiom,
    ((![N2 : num]: ((times_times_num @ (bit0 @ one) @ N2) = (bit0 @ N2))))). % num_double
thf(fact_29_nat__power__eq__Suc__0__iff, axiom,
    ((![X : nat, M : nat]: (((power_power_nat @ X @ M) = (suc @ zero_zero_nat)) = (((M = zero_zero_nat)) | ((X = (suc @ zero_zero_nat)))))))). % nat_power_eq_Suc_0_iff
thf(fact_30_power__Suc__0, axiom,
    ((![N2 : nat]: ((power_power_nat @ (suc @ zero_zero_nat) @ N2) = (suc @ zero_zero_nat))))). % power_Suc_0
thf(fact_31_power__mult__numeral, axiom,
    ((![A2 : complex, M : num, N2 : num]: ((power_power_complex @ (power_power_complex @ A2 @ (numeral_numeral_nat @ M)) @ (numeral_numeral_nat @ N2)) = (power_power_complex @ A2 @ (numeral_numeral_nat @ (times_times_num @ M @ N2))))))). % power_mult_numeral
thf(fact_32_power__mult__numeral, axiom,
    ((![A2 : nat, M : num, N2 : num]: ((power_power_nat @ (power_power_nat @ A2 @ (numeral_numeral_nat @ M)) @ (numeral_numeral_nat @ N2)) = (power_power_nat @ A2 @ (numeral_numeral_nat @ (times_times_num @ M @ N2))))))). % power_mult_numeral
thf(fact_33_Nat_Oadd__0__right, axiom,
    ((![M : nat]: ((plus_plus_nat @ M @ zero_zero_nat) = M)))). % Nat.add_0_right
thf(fact_34_add__is__0, axiom,
    ((![M : nat, N2 : nat]: (((plus_plus_nat @ M @ N2) = zero_zero_nat) = (((M = zero_zero_nat)) & ((N2 = zero_zero_nat))))))). % add_is_0
thf(fact_35_add__Suc__right, axiom,
    ((![M : nat, N2 : nat]: ((plus_plus_nat @ M @ (suc @ N2)) = (suc @ (plus_plus_nat @ M @ N2)))))). % add_Suc_right
thf(fact_36_mult__cancel2, axiom,
    ((![M : nat, K : nat, N2 : nat]: (((times_times_nat @ M @ K) = (times_times_nat @ N2 @ K)) = (((M = N2)) | ((K = zero_zero_nat))))))). % mult_cancel2
thf(fact_37_mult__cancel1, axiom,
    ((![K : nat, M : nat, N2 : nat]: (((times_times_nat @ K @ M) = (times_times_nat @ K @ N2)) = (((M = N2)) | ((K = zero_zero_nat))))))). % mult_cancel1
thf(fact_38_mult__0__right, axiom,
    ((![M : nat]: ((times_times_nat @ M @ zero_zero_nat) = zero_zero_nat)))). % mult_0_right
thf(fact_39_mult__is__0, axiom,
    ((![M : nat, N2 : nat]: (((times_times_nat @ M @ N2) = zero_zero_nat) = (((M = zero_zero_nat)) | ((N2 = zero_zero_nat))))))). % mult_is_0
thf(fact_40_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_41_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_42_power__add__numeral2, axiom,
    ((![A2 : complex, M : num, N2 : num, B : complex]: ((times_times_complex @ (power_power_complex @ A2 @ (numeral_numeral_nat @ M)) @ (times_times_complex @ (power_power_complex @ A2 @ (numeral_numeral_nat @ N2)) @ B)) = (times_times_complex @ (power_power_complex @ A2 @ (numeral_numeral_nat @ (plus_plus_num @ M @ N2))) @ B))))). % power_add_numeral2
thf(fact_43_power__add__numeral2, axiom,
    ((![A2 : nat, M : num, N2 : num, B : nat]: ((times_times_nat @ (power_power_nat @ A2 @ (numeral_numeral_nat @ M)) @ (times_times_nat @ (power_power_nat @ A2 @ (numeral_numeral_nat @ N2)) @ B)) = (times_times_nat @ (power_power_nat @ A2 @ (numeral_numeral_nat @ (plus_plus_num @ M @ N2))) @ B))))). % power_add_numeral2
thf(fact_44_power__add__numeral, axiom,
    ((![A2 : complex, M : num, N2 : num]: ((times_times_complex @ (power_power_complex @ A2 @ (numeral_numeral_nat @ M)) @ (power_power_complex @ A2 @ (numeral_numeral_nat @ N2))) = (power_power_complex @ A2 @ (numeral_numeral_nat @ (plus_plus_num @ M @ N2))))))). % power_add_numeral
thf(fact_45_power__add__numeral, axiom,
    ((![A2 : nat, M : num, N2 : num]: ((times_times_nat @ (power_power_nat @ A2 @ (numeral_numeral_nat @ M)) @ (power_power_nat @ A2 @ (numeral_numeral_nat @ N2))) = (power_power_nat @ A2 @ (numeral_numeral_nat @ (plus_plus_num @ M @ N2))))))). % power_add_numeral
thf(fact_46_Suc__numeral, axiom,
    ((![N2 : num]: ((suc @ (numeral_numeral_nat @ N2)) = (numeral_numeral_nat @ (plus_plus_num @ N2 @ one)))))). % Suc_numeral
thf(fact_47_add__One__commute, axiom,
    ((![N2 : num]: ((plus_plus_num @ one @ N2) = (plus_plus_num @ N2 @ one))))). % add_One_commute
thf(fact_48_root__nonzero, axiom,
    ((![N2 : nat]: (~ (((fFT_Mirabelle_root @ N2) = zero_zero_complex)))))). % root_nonzero
thf(fact_49_Suc__nat__number__of__add, axiom,
    ((![V : num, N2 : nat]: ((suc @ (plus_plus_nat @ (numeral_numeral_nat @ V) @ N2)) = (plus_plus_nat @ (numeral_numeral_nat @ (plus_plus_num @ V @ one)) @ N2))))). % Suc_nat_number_of_add
thf(fact_50_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_51_n__not__Suc__n, axiom,
    ((![N2 : nat]: (~ ((N2 = (suc @ N2))))))). % n_not_Suc_n
thf(fact_52_Suc__inject, axiom,
    ((![X : nat, Y : nat]: (((suc @ X) = (suc @ Y)) => (X = Y))))). % Suc_inject
thf(fact_53_zero__neq__numeral, axiom,
    ((![N2 : num]: (~ ((zero_zero_complex = (numera632737353omplex @ N2))))))). % zero_neq_numeral
thf(fact_54_zero__neq__numeral, axiom,
    ((![N2 : num]: (~ ((zero_zero_nat = (numeral_numeral_nat @ N2))))))). % zero_neq_numeral
thf(fact_55_power__not__zero, axiom,
    ((![A2 : complex, N2 : nat]: ((~ ((A2 = zero_zero_complex))) => (~ (((power_power_complex @ A2 @ N2) = zero_zero_complex))))))). % power_not_zero
thf(fact_56_power__not__zero, axiom,
    ((![A2 : nat, N2 : nat]: ((~ ((A2 = zero_zero_nat))) => (~ (((power_power_nat @ A2 @ N2) = zero_zero_nat))))))). % power_not_zero
thf(fact_57_power__commuting__commutes, axiom,
    ((![X : complex, Y : complex, N2 : nat]: (((times_times_complex @ X @ Y) = (times_times_complex @ Y @ X)) => ((times_times_complex @ (power_power_complex @ X @ N2) @ Y) = (times_times_complex @ Y @ (power_power_complex @ X @ N2))))))). % power_commuting_commutes
thf(fact_58_power__commuting__commutes, axiom,
    ((![X : nat, Y : nat, N2 : nat]: (((times_times_nat @ X @ Y) = (times_times_nat @ Y @ X)) => ((times_times_nat @ (power_power_nat @ X @ N2) @ Y) = (times_times_nat @ Y @ (power_power_nat @ X @ N2))))))). % power_commuting_commutes
thf(fact_59_power__mult__distrib, axiom,
    ((![A2 : complex, B : complex, N2 : nat]: ((power_power_complex @ (times_times_complex @ A2 @ B) @ N2) = (times_times_complex @ (power_power_complex @ A2 @ N2) @ (power_power_complex @ B @ N2)))))). % power_mult_distrib
thf(fact_60_power__mult__distrib, axiom,
    ((![A2 : nat, B : nat, N2 : nat]: ((power_power_nat @ (times_times_nat @ A2 @ B) @ N2) = (times_times_nat @ (power_power_nat @ A2 @ N2) @ (power_power_nat @ B @ N2)))))). % power_mult_distrib
thf(fact_61_power__commutes, axiom,
    ((![A2 : complex, N2 : nat]: ((times_times_complex @ (power_power_complex @ A2 @ N2) @ A2) = (times_times_complex @ A2 @ (power_power_complex @ A2 @ N2)))))). % power_commutes
thf(fact_62_power__commutes, axiom,
    ((![A2 : nat, N2 : nat]: ((times_times_nat @ (power_power_nat @ A2 @ N2) @ A2) = (times_times_nat @ A2 @ (power_power_nat @ A2 @ N2)))))). % power_commutes
thf(fact_63_not0__implies__Suc, axiom,
    ((![N2 : nat]: ((~ ((N2 = zero_zero_nat))) => (?[M2 : nat]: (N2 = (suc @ M2))))))). % not0_implies_Suc
thf(fact_64_old_Onat_Oinducts, axiom,
    ((![P : nat > $o, Nat : nat]: ((P @ zero_zero_nat) => ((![Nat3 : nat]: ((P @ Nat3) => (P @ (suc @ Nat3)))) => (P @ Nat)))))). % old.nat.inducts
thf(fact_65_old_Onat_Oexhaust, axiom,
    ((![Y : nat]: ((~ ((Y = zero_zero_nat))) => (~ ((![Nat3 : nat]: (~ ((Y = (suc @ Nat3))))))))))). % old.nat.exhaust
thf(fact_66_Zero__not__Suc, axiom,
    ((![M : nat]: (~ ((zero_zero_nat = (suc @ M))))))). % Zero_not_Suc
thf(fact_67_Zero__neq__Suc, axiom,
    ((![M : nat]: (~ ((zero_zero_nat = (suc @ M))))))). % Zero_neq_Suc
thf(fact_68_Suc__neq__Zero, axiom,
    ((![M : nat]: (~ (((suc @ M) = zero_zero_nat)))))). % Suc_neq_Zero
thf(fact_69_zero__induct, axiom,
    ((![P : nat > $o, K : nat]: ((P @ K) => ((![N3 : nat]: ((P @ (suc @ N3)) => (P @ N3))) => (P @ zero_zero_nat)))))). % zero_induct
thf(fact_70_diff__induct, axiom,
    ((![P : nat > nat > $o, M : nat, N2 : nat]: ((![X3 : nat]: (P @ X3 @ zero_zero_nat)) => ((![Y3 : nat]: (P @ zero_zero_nat @ (suc @ Y3))) => ((![X3 : nat, Y3 : nat]: ((P @ X3 @ Y3) => (P @ (suc @ X3) @ (suc @ Y3)))) => (P @ M @ N2))))))). % diff_induct
thf(fact_71_nat__induct, axiom,
    ((![P : nat > $o, N2 : nat]: ((P @ zero_zero_nat) => ((![N3 : nat]: ((P @ N3) => (P @ (suc @ N3)))) => (P @ N2)))))). % nat_induct
thf(fact_72_nat_OdiscI, axiom,
    ((![Nat : nat, X2 : nat]: ((Nat = (suc @ X2)) => (~ ((Nat = zero_zero_nat))))))). % nat.discI
thf(fact_73_old_Onat_Odistinct_I1_J, axiom,
    ((![Nat2 : nat]: (~ ((zero_zero_nat = (suc @ Nat2))))))). % old.nat.distinct(1)
thf(fact_74_old_Onat_Odistinct_I2_J, axiom,
    ((![Nat2 : nat]: (~ (((suc @ Nat2) = zero_zero_nat)))))). % old.nat.distinct(2)
thf(fact_75_nat_Odistinct_I1_J, axiom,
    ((![X2 : nat]: (~ ((zero_zero_nat = (suc @ X2))))))). % nat.distinct(1)
thf(fact_76_add__eq__self__zero, axiom,
    ((![M : nat, N2 : nat]: (((plus_plus_nat @ M @ N2) = M) => (N2 = zero_zero_nat))))). % add_eq_self_zero
thf(fact_77_plus__nat_Oadd__0, axiom,
    ((![N2 : nat]: ((plus_plus_nat @ zero_zero_nat @ N2) = N2)))). % plus_nat.add_0
thf(fact_78_add__Suc, axiom,
    ((![M : nat, N2 : nat]: ((plus_plus_nat @ (suc @ M) @ N2) = (suc @ (plus_plus_nat @ M @ N2)))))). % add_Suc
thf(fact_79_nat__arith_Osuc1, axiom,
    ((![A3 : nat, K : nat, A2 : nat]: ((A3 = (plus_plus_nat @ K @ A2)) => ((suc @ A3) = (plus_plus_nat @ K @ (suc @ A2))))))). % nat_arith.suc1
thf(fact_80_add__Suc__shift, axiom,
    ((![M : nat, N2 : nat]: ((plus_plus_nat @ (suc @ M) @ N2) = (plus_plus_nat @ M @ (suc @ N2)))))). % add_Suc_shift
thf(fact_81_power__mult, axiom,
    ((![A2 : complex, M : nat, N2 : nat]: ((power_power_complex @ A2 @ (times_times_nat @ M @ N2)) = (power_power_complex @ (power_power_complex @ A2 @ M) @ N2))))). % power_mult
thf(fact_82_power__mult, axiom,
    ((![A2 : nat, M : nat, N2 : nat]: ((power_power_nat @ A2 @ (times_times_nat @ M @ N2)) = (power_power_nat @ (power_power_nat @ A2 @ M) @ N2))))). % power_mult
thf(fact_83_mult__0, axiom,
    ((![N2 : nat]: ((times_times_nat @ zero_zero_nat @ N2) = zero_zero_nat)))). % mult_0
thf(fact_84_Suc__mult__cancel1, axiom,
    ((![K : nat, M : nat, N2 : nat]: (((times_times_nat @ (suc @ K) @ M) = (times_times_nat @ (suc @ K) @ N2)) = (M = N2))))). % Suc_mult_cancel1
thf(fact_85_add__mult__distrib2, axiom,
    ((![K : nat, M : nat, N2 : nat]: ((times_times_nat @ K @ (plus_plus_nat @ M @ N2)) = (plus_plus_nat @ (times_times_nat @ K @ M) @ (times_times_nat @ K @ N2)))))). % add_mult_distrib2
thf(fact_86_add__mult__distrib, axiom,
    ((![M : nat, N2 : nat, K : nat]: ((times_times_nat @ (plus_plus_nat @ M @ N2) @ K) = (plus_plus_nat @ (times_times_nat @ M @ K) @ (times_times_nat @ N2 @ K)))))). % add_mult_distrib
thf(fact_87_mult__numeral__1__right, axiom,
    ((![A2 : complex]: ((times_times_complex @ A2 @ (numera632737353omplex @ one)) = A2)))). % mult_numeral_1_right
thf(fact_88_mult__numeral__1__right, axiom,
    ((![A2 : nat]: ((times_times_nat @ A2 @ (numeral_numeral_nat @ one)) = A2)))). % mult_numeral_1_right
thf(fact_89_mult__numeral__1, axiom,
    ((![A2 : complex]: ((times_times_complex @ (numera632737353omplex @ one) @ A2) = A2)))). % mult_numeral_1
thf(fact_90_mult__numeral__1, axiom,
    ((![A2 : nat]: ((times_times_nat @ (numeral_numeral_nat @ one) @ A2) = A2)))). % mult_numeral_1
thf(fact_91_numeral__Bit0, axiom,
    ((![N2 : num]: ((numera632737353omplex @ (bit0 @ N2)) = (plus_plus_complex @ (numera632737353omplex @ N2) @ (numera632737353omplex @ N2)))))). % numeral_Bit0
thf(fact_92_numeral__Bit0, axiom,
    ((![N2 : num]: ((numeral_numeral_nat @ (bit0 @ N2)) = (plus_plus_nat @ (numeral_numeral_nat @ N2) @ (numeral_numeral_nat @ N2)))))). % numeral_Bit0
thf(fact_93_power__Suc2, axiom,
    ((![A2 : complex, N2 : nat]: ((power_power_complex @ A2 @ (suc @ N2)) = (times_times_complex @ (power_power_complex @ A2 @ N2) @ A2))))). % power_Suc2
thf(fact_94_power__Suc2, axiom,
    ((![A2 : nat, N2 : nat]: ((power_power_nat @ A2 @ (suc @ N2)) = (times_times_nat @ (power_power_nat @ A2 @ N2) @ A2))))). % power_Suc2
thf(fact_95_power__Suc, axiom,
    ((![A2 : complex, N2 : nat]: ((power_power_complex @ A2 @ (suc @ N2)) = (times_times_complex @ A2 @ (power_power_complex @ A2 @ N2)))))). % power_Suc
thf(fact_96_power__Suc, axiom,
    ((![A2 : nat, N2 : nat]: ((power_power_nat @ A2 @ (suc @ N2)) = (times_times_nat @ A2 @ (power_power_nat @ A2 @ N2)))))). % power_Suc
thf(fact_97_power__add, axiom,
    ((![A2 : complex, M : nat, N2 : nat]: ((power_power_complex @ A2 @ (plus_plus_nat @ M @ N2)) = (times_times_complex @ (power_power_complex @ A2 @ M) @ (power_power_complex @ A2 @ N2)))))). % power_add
thf(fact_98_power__add, axiom,
    ((![A2 : nat, M : nat, N2 : nat]: ((power_power_nat @ A2 @ (plus_plus_nat @ M @ N2)) = (times_times_nat @ (power_power_nat @ A2 @ M) @ (power_power_nat @ A2 @ N2)))))). % power_add
thf(fact_99_one__is__add, axiom,
    ((![M : nat, N2 : nat]: (((suc @ zero_zero_nat) = (plus_plus_nat @ M @ N2)) = (((((M = (suc @ zero_zero_nat))) & ((N2 = zero_zero_nat)))) | ((((M = zero_zero_nat)) & ((N2 = (suc @ zero_zero_nat)))))))))). % one_is_add
thf(fact_100_add__is__1, axiom,
    ((![M : nat, N2 : nat]: (((plus_plus_nat @ M @ N2) = (suc @ zero_zero_nat)) = (((((M = (suc @ zero_zero_nat))) & ((N2 = zero_zero_nat)))) | ((((M = zero_zero_nat)) & ((N2 = (suc @ zero_zero_nat)))))))))). % add_is_1
thf(fact_101_mult__Suc, axiom,
    ((![M : nat, N2 : nat]: ((times_times_nat @ (suc @ M) @ N2) = (plus_plus_nat @ N2 @ (times_times_nat @ M @ N2)))))). % mult_Suc
thf(fact_102_numeral__code_I2_J, axiom,
    ((![N2 : num]: ((numera632737353omplex @ (bit0 @ N2)) = (plus_plus_complex @ (numera632737353omplex @ N2) @ (numera632737353omplex @ N2)))))). % numeral_code(2)
thf(fact_103_numeral__code_I2_J, axiom,
    ((![N2 : num]: ((numeral_numeral_nat @ (bit0 @ N2)) = (plus_plus_nat @ (numeral_numeral_nat @ N2) @ (numeral_numeral_nat @ N2)))))). % numeral_code(2)
thf(fact_104_numeral__1__eq__Suc__0, axiom,
    (((numeral_numeral_nat @ one) = (suc @ zero_zero_nat)))). % numeral_1_eq_Suc_0
thf(fact_105_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_106_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_107_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_108_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_109_mult__2__right, axiom,
    ((![Z : complex]: ((times_times_complex @ Z @ (numera632737353omplex @ (bit0 @ one))) = (plus_plus_complex @ Z @ Z))))). % mult_2_right
thf(fact_110_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_111_mult__2, axiom,
    ((![Z : complex]: ((times_times_complex @ (numera632737353omplex @ (bit0 @ one)) @ Z) = (plus_plus_complex @ Z @ Z))))). % mult_2
thf(fact_112_mult__2, axiom,
    ((![Z : nat]: ((times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ Z) = (plus_plus_nat @ Z @ Z))))). % mult_2
thf(fact_113_zero__power2, axiom,
    (((power_power_complex @ zero_zero_complex @ (numeral_numeral_nat @ (bit0 @ one))) = zero_zero_complex))). % zero_power2
thf(fact_114_zero__power2, axiom,
    (((power_power_nat @ zero_zero_nat @ (numeral_numeral_nat @ (bit0 @ one))) = zero_zero_nat))). % zero_power2
thf(fact_115_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_116_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_117_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_118_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_119_numeral__2__eq__2, axiom,
    (((numeral_numeral_nat @ (bit0 @ one)) = (suc @ (suc @ zero_zero_nat))))). % numeral_2_eq_2
thf(fact_120_power__even__eq, axiom,
    ((![A2 : complex, N2 : nat]: ((power_power_complex @ A2 @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N2)) = (power_power_complex @ (power_power_complex @ A2 @ N2) @ (numeral_numeral_nat @ (bit0 @ one))))))). % power_even_eq
thf(fact_121_power__even__eq, axiom,
    ((![A2 : nat, N2 : nat]: ((power_power_nat @ A2 @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N2)) = (power_power_nat @ (power_power_nat @ A2 @ N2) @ (numeral_numeral_nat @ (bit0 @ one))))))). % power_even_eq
thf(fact_122_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_123_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_124_power__odd__eq, axiom,
    ((![A2 : complex, N2 : nat]: ((power_power_complex @ A2 @ (suc @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N2))) = (times_times_complex @ A2 @ (power_power_complex @ (power_power_complex @ A2 @ N2) @ (numeral_numeral_nat @ (bit0 @ one)))))))). % power_odd_eq
thf(fact_125_power__odd__eq, axiom,
    ((![A2 : nat, N2 : nat]: ((power_power_nat @ A2 @ (suc @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N2))) = (times_times_nat @ A2 @ (power_power_nat @ (power_power_nat @ A2 @ N2) @ (numeral_numeral_nat @ (bit0 @ one)))))))). % power_odd_eq
thf(fact_126_calculation, axiom,
    (((groups59700922omplex @ (^[J2 : nat]: (times_times_complex @ (power_power_complex @ (fFT_Mirabelle_root @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ m)) @ (times_times_nat @ i @ J2)) @ (a @ J2))) @ (set_or562006527an_nat @ zero_zero_nat @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ m))) = (plus_plus_complex @ (groups59700922omplex @ (^[J2 : nat]: (times_times_complex @ (power_power_complex @ (fFT_Mirabelle_root @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ m)) @ (times_times_nat @ i @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ J2))) @ (a @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ J2)))) @ (set_or562006527an_nat @ zero_zero_nat @ m)) @ (groups59700922omplex @ (^[J2 : nat]: (times_times_complex @ (power_power_complex @ (fFT_Mirabelle_root @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ m)) @ (times_times_nat @ i @ (plus_plus_nat @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ J2) @ one_one_nat))) @ (a @ (plus_plus_nat @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ J2) @ one_one_nat)))) @ (set_or562006527an_nat @ zero_zero_nat @ m)))))). % calculation
thf(fact_127_exp__add__not__zero__imp__right, axiom,
    ((![M : nat, N2 : nat]: ((~ (((power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (plus_plus_nat @ M @ N2)) = zero_zero_nat))) => (~ (((power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N2) = zero_zero_nat))))))). % exp_add_not_zero_imp_right
thf(fact_128_exp__add__not__zero__imp__left, axiom,
    ((![M : nat, N2 : nat]: ((~ (((power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (plus_plus_nat @ M @ N2)) = zero_zero_nat))) => (~ (((power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ M) = zero_zero_nat))))))). % exp_add_not_zero_imp_left
thf(fact_129_sum_Oneutral__const, axiom,
    ((![A3 : set_nat]: ((groups59700922omplex @ (^[Uu : nat]: zero_zero_complex) @ A3) = zero_zero_complex)))). % sum.neutral_const
thf(fact_130_sum_OatLeast0__lessThan__Suc, axiom,
    ((![G : nat > nat, N2 : nat]: ((groups1842438620at_nat @ G @ (set_or562006527an_nat @ zero_zero_nat @ (suc @ N2))) = (plus_plus_nat @ (groups1842438620at_nat @ G @ (set_or562006527an_nat @ zero_zero_nat @ N2)) @ (G @ N2)))))). % sum.atLeast0_lessThan_Suc
thf(fact_131_sum_OatLeast0__lessThan__Suc, axiom,
    ((![G : nat > complex, N2 : nat]: ((groups59700922omplex @ G @ (set_or562006527an_nat @ zero_zero_nat @ (suc @ N2))) = (plus_plus_complex @ (groups59700922omplex @ G @ (set_or562006527an_nat @ zero_zero_nat @ N2)) @ (G @ N2)))))). % sum.atLeast0_lessThan_Suc
thf(fact_132_sum__shift__lb__Suc0__0__upt, axiom,
    ((![F : nat > nat, K : nat]: (((F @ zero_zero_nat) = zero_zero_nat) => ((groups1842438620at_nat @ F @ (set_or562006527an_nat @ (suc @ zero_zero_nat) @ K)) = (groups1842438620at_nat @ F @ (set_or562006527an_nat @ zero_zero_nat @ K))))))). % sum_shift_lb_Suc0_0_upt
thf(fact_133_sum__shift__lb__Suc0__0__upt, axiom,
    ((![F : nat > complex, K : nat]: (((F @ zero_zero_nat) = zero_zero_complex) => ((groups59700922omplex @ F @ (set_or562006527an_nat @ (suc @ zero_zero_nat) @ K)) = (groups59700922omplex @ F @ (set_or562006527an_nat @ zero_zero_nat @ K))))))). % sum_shift_lb_Suc0_0_upt
thf(fact_134_double__not__eq__Suc__double, axiom,
    ((![M : nat, N2 : nat]: (~ (((times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ M) = (suc @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N2)))))))). % double_not_eq_Suc_double
thf(fact_135_Suc__double__not__eq__double, axiom,
    ((![M : nat, N2 : nat]: (~ (((suc @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ M)) = (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N2))))))). % Suc_double_not_eq_double
thf(fact_136_semiring__norm_I2_J, axiom,
    (((plus_plus_num @ one @ one) = (bit0 @ one)))). % semiring_norm(2)
thf(fact_137_semiring__norm_I83_J, axiom,
    ((![N2 : num]: (~ ((one = (bit0 @ N2))))))). % semiring_norm(83)
thf(fact_138_semiring__norm_I87_J, axiom,
    ((![M : num, N2 : num]: (((bit0 @ M) = (bit0 @ N2)) = (M = N2))))). % semiring_norm(87)
thf(fact_139_power__one, axiom,
    ((![N2 : nat]: ((power_power_complex @ one_one_complex @ N2) = one_one_complex)))). % power_one
thf(fact_140_power__one, axiom,
    ((![N2 : nat]: ((power_power_nat @ one_one_nat @ N2) = one_one_nat)))). % power_one
thf(fact_141_semiring__norm_I85_J, axiom,
    ((![M : num]: (~ (((bit0 @ M) = one)))))). % semiring_norm(85)
thf(fact_142_power__one__right, axiom,
    ((![A2 : complex]: ((power_power_complex @ A2 @ one_one_nat) = A2)))). % power_one_right
thf(fact_143_power__one__right, axiom,
    ((![A2 : nat]: ((power_power_nat @ A2 @ one_one_nat) = A2)))). % power_one_right
thf(fact_144_nat__1__eq__mult__iff, axiom,
    ((![M : nat, N2 : nat]: ((one_one_nat = (times_times_nat @ M @ N2)) = (((M = one_one_nat)) & ((N2 = one_one_nat))))))). % nat_1_eq_mult_iff
thf(fact_145_nat__mult__eq__1__iff, axiom,
    ((![M : nat, N2 : nat]: (((times_times_nat @ M @ N2) = one_one_nat) = (((M = one_one_nat)) & ((N2 = one_one_nat))))))). % nat_mult_eq_1_iff
thf(fact_146_semiring__norm_I6_J, axiom,
    ((![M : num, N2 : num]: ((plus_plus_num @ (bit0 @ M) @ (bit0 @ N2)) = (bit0 @ (plus_plus_num @ M @ N2)))))). % semiring_norm(6)
thf(fact_147_semiring__norm_I13_J, axiom,
    ((![M : num, N2 : num]: ((times_times_num @ (bit0 @ M) @ (bit0 @ N2)) = (bit0 @ (bit0 @ (times_times_num @ M @ N2))))))). % semiring_norm(13)
thf(fact_148_semiring__norm_I12_J, axiom,
    ((![N2 : num]: ((times_times_num @ one @ N2) = N2)))). % semiring_norm(12)
thf(fact_149_semiring__norm_I11_J, axiom,
    ((![M : num]: ((times_times_num @ M @ one) = M)))). % semiring_norm(11)
thf(fact_150_numeral__eq__one__iff, axiom,
    ((![N2 : num]: (((numeral_numeral_nat @ N2) = one_one_nat) = (N2 = one))))). % numeral_eq_one_iff
thf(fact_151_one__eq__numeral__iff, axiom,
    ((![N2 : num]: ((one_one_nat = (numeral_numeral_nat @ N2)) = (one = N2))))). % one_eq_numeral_iff
thf(fact_152_one__add__one, axiom,
    (((plus_plus_complex @ one_one_complex @ one_one_complex) = (numera632737353omplex @ (bit0 @ one))))). % one_add_one
thf(fact_153_one__add__one, axiom,
    (((plus_plus_nat @ one_one_nat @ one_one_nat) = (numeral_numeral_nat @ (bit0 @ one))))). % one_add_one
thf(fact_154_Suc__1, axiom,
    (((suc @ one_one_nat) = (numeral_numeral_nat @ (bit0 @ one))))). % Suc_1
thf(fact_155_numeral__plus__one, axiom,
    ((![N2 : num]: ((plus_plus_complex @ (numera632737353omplex @ N2) @ one_one_complex) = (numera632737353omplex @ (plus_plus_num @ N2 @ one)))))). % numeral_plus_one
thf(fact_156_numeral__plus__one, axiom,
    ((![N2 : num]: ((plus_plus_nat @ (numeral_numeral_nat @ N2) @ one_one_nat) = (numeral_numeral_nat @ (plus_plus_num @ N2 @ one)))))). % numeral_plus_one
thf(fact_157_one__plus__numeral, axiom,
    ((![N2 : num]: ((plus_plus_complex @ one_one_complex @ (numera632737353omplex @ N2)) = (numera632737353omplex @ (plus_plus_num @ one @ N2)))))). % one_plus_numeral
thf(fact_158_one__plus__numeral, axiom,
    ((![N2 : num]: ((plus_plus_nat @ one_one_nat @ (numeral_numeral_nat @ N2)) = (numeral_numeral_nat @ (plus_plus_num @ one @ N2)))))). % one_plus_numeral
thf(fact_159_nat__mult__1, axiom,
    ((![N2 : nat]: ((times_times_nat @ one_one_nat @ N2) = N2)))). % nat_mult_1
thf(fact_160_nat__mult__1__right, axiom,
    ((![N2 : nat]: ((times_times_nat @ N2 @ one_one_nat) = N2)))). % nat_mult_1_right
thf(fact_161_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_162_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_163_numeral__One, axiom,
    (((numeral_numeral_nat @ one) = one_one_nat))). % numeral_One
thf(fact_164_left__right__inverse__power, axiom,
    ((![X : complex, Y : complex, N2 : nat]: (((times_times_complex @ X @ Y) = one_one_complex) => ((times_times_complex @ (power_power_complex @ X @ N2) @ (power_power_complex @ Y @ N2)) = one_one_complex))))). % left_right_inverse_power
thf(fact_165_left__right__inverse__power, axiom,
    ((![X : nat, Y : nat, N2 : nat]: (((times_times_nat @ X @ Y) = one_one_nat) => ((times_times_nat @ (power_power_nat @ X @ N2) @ (power_power_nat @ Y @ N2)) = one_one_nat))))). % left_right_inverse_power
thf(fact_166_power__0, axiom,
    ((![A2 : complex]: ((power_power_complex @ A2 @ zero_zero_nat) = one_one_complex)))). % power_0
thf(fact_167_power__0, axiom,
    ((![A2 : nat]: ((power_power_nat @ A2 @ zero_zero_nat) = one_one_nat)))). % power_0
thf(fact_168_numerals_I1_J, axiom,
    (((numeral_numeral_nat @ one) = one_one_nat))). % numerals(1)
thf(fact_169_One__nat__def, axiom,
    ((one_one_nat = (suc @ zero_zero_nat)))). % One_nat_def
thf(fact_170_Suc__eq__plus1, axiom,
    ((suc = (^[N : nat]: (plus_plus_nat @ N @ one_one_nat))))). % Suc_eq_plus1
thf(fact_171_plus__1__eq__Suc, axiom,
    (((plus_plus_nat @ one_one_nat) = suc))). % plus_1_eq_Suc
thf(fact_172_Suc__eq__plus1__left, axiom,
    ((suc = (plus_plus_nat @ one_one_nat)))). % Suc_eq_plus1_left
thf(fact_173_mult__eq__self__implies__10, axiom,
    ((![M : nat, N2 : nat]: ((M = (times_times_nat @ M @ N2)) => ((N2 = one_one_nat) | (M = zero_zero_nat)))))). % mult_eq_self_implies_10
thf(fact_174_power__0__left, axiom,
    ((![N2 : nat]: (((N2 = zero_zero_nat) => ((power_power_complex @ zero_zero_complex @ N2) = one_one_complex)) & ((~ ((N2 = zero_zero_nat))) => ((power_power_complex @ zero_zero_complex @ N2) = zero_zero_complex)))))). % power_0_left
thf(fact_175_power__0__left, axiom,
    ((![N2 : nat]: (((N2 = zero_zero_nat) => ((power_power_nat @ zero_zero_nat @ N2) = one_one_nat)) & ((~ ((N2 = zero_zero_nat))) => ((power_power_nat @ zero_zero_nat @ N2) = zero_zero_nat)))))). % power_0_left
thf(fact_176_sum_Ocong, axiom,
    ((![A3 : set_nat, B2 : set_nat, G : nat > complex, H : nat > complex]: ((A3 = B2) => ((![X3 : nat]: ((member_nat @ X3 @ B2) => ((G @ X3) = (H @ X3)))) => ((groups59700922omplex @ G @ A3) = (groups59700922omplex @ H @ B2))))))). % sum.cong
thf(fact_177_sum_Oeq__general, axiom,
    ((![B2 : set_nat, A3 : set_nat, H : nat > nat, Gamma : nat > complex, Phi : nat > complex]: ((![Y3 : nat]: ((member_nat @ Y3 @ B2) => (?[X4 : nat]: (((member_nat @ X4 @ A3) & ((H @ X4) = Y3)) & (![Ya : nat]: (((member_nat @ Ya @ A3) & ((H @ Ya) = Y3)) => (Ya = X4))))))) => ((![X3 : nat]: ((member_nat @ X3 @ A3) => ((member_nat @ (H @ X3) @ B2) & ((Gamma @ (H @ X3)) = (Phi @ X3))))) => ((groups59700922omplex @ Phi @ A3) = (groups59700922omplex @ Gamma @ B2))))))). % sum.eq_general
thf(fact_178_sum_Oeq__general__inverses, axiom,
    ((![B2 : set_nat, K : nat > nat, A3 : set_nat, H : nat > nat, Gamma : nat > complex, Phi : nat > complex]: ((![Y3 : nat]: ((member_nat @ Y3 @ B2) => ((member_nat @ (K @ Y3) @ A3) & ((H @ (K @ Y3)) = Y3)))) => ((![X3 : nat]: ((member_nat @ X3 @ A3) => ((member_nat @ (H @ X3) @ B2) & (((K @ (H @ X3)) = X3) & ((Gamma @ (H @ X3)) = (Phi @ X3)))))) => ((groups59700922omplex @ Phi @ A3) = (groups59700922omplex @ Gamma @ B2))))))). % sum.eq_general_inverses
thf(fact_179_sum_Oreindex__bij__witness, axiom,
    ((![S : set_nat, I : nat > nat, J : nat > nat, T : set_nat, H : nat > complex, G : nat > complex]: ((![A4 : nat]: ((member_nat @ A4 @ S) => ((I @ (J @ A4)) = A4))) => ((![A4 : nat]: ((member_nat @ A4 @ S) => (member_nat @ (J @ A4) @ T))) => ((![B3 : nat]: ((member_nat @ B3 @ T) => ((J @ (I @ B3)) = B3))) => ((![B3 : nat]: ((member_nat @ B3 @ T) => (member_nat @ (I @ B3) @ S))) => ((![A4 : nat]: ((member_nat @ A4 @ S) => ((H @ (J @ A4)) = (G @ A4)))) => ((groups59700922omplex @ G @ S) = (groups59700922omplex @ H @ T)))))))))). % sum.reindex_bij_witness
thf(fact_180_nat__induct2, axiom,
    ((![P : nat > $o, N2 : nat]: ((P @ zero_zero_nat) => ((P @ one_one_nat) => ((![N3 : nat]: ((P @ N3) => (P @ (plus_plus_nat @ N3 @ (numeral_numeral_nat @ (bit0 @ one)))))) => (P @ N2))))))). % nat_induct2
thf(fact_181_one__power2, axiom,
    (((power_power_complex @ one_one_complex @ (numeral_numeral_nat @ (bit0 @ one))) = one_one_complex))). % one_power2
thf(fact_182_one__power2, axiom,
    (((power_power_nat @ one_one_nat @ (numeral_numeral_nat @ (bit0 @ one))) = one_one_nat))). % one_power2
thf(fact_183_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_184_sum_Oswap, axiom,
    ((![G : nat > nat > complex, B2 : set_nat, A3 : set_nat]: ((groups59700922omplex @ (^[I2 : nat]: (groups59700922omplex @ (G @ I2) @ B2)) @ A3) = (groups59700922omplex @ (^[J2 : nat]: (groups59700922omplex @ (^[I2 : nat]: (G @ I2 @ J2)) @ A3)) @ B2))))). % sum.swap
thf(fact_185_sum_Oneutral, axiom,
    ((![A3 : set_nat, G : nat > complex]: ((![X3 : nat]: ((member_nat @ X3 @ A3) => ((G @ X3) = zero_zero_complex))) => ((groups59700922omplex @ G @ A3) = zero_zero_complex))))). % sum.neutral
thf(fact_186_sum_Onot__neutral__contains__not__neutral, axiom,
    ((![G : nat > complex, A3 : set_nat]: ((~ (((groups59700922omplex @ G @ A3) = zero_zero_complex))) => (~ ((![A4 : nat]: ((member_nat @ A4 @ A3) => ((G @ A4) = zero_zero_complex))))))))). % sum.not_neutral_contains_not_neutral
thf(fact_187_nat__mult__eq__cancel__disj, axiom,
    ((![K : nat, M : nat, N2 : nat]: (((times_times_nat @ K @ M) = (times_times_nat @ K @ N2)) = (((K = zero_zero_nat)) | ((M = N2))))))). % nat_mult_eq_cancel_disj

% Conjectures (1)
thf(conj_0, conjecture,
    (((groups59700922omplex @ (^[J2 : nat]: (times_times_complex @ (power_power_complex @ (fFT_Mirabelle_root @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ m)) @ (plus_plus_nat @ i @ (times_times_nat @ i @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ J2)))) @ (a @ (suc @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ J2))))) @ (set_or562006527an_nat @ zero_zero_nat @ m)) = (groups59700922omplex @ (^[N : nat]: (times_times_complex @ (power_power_complex @ (fFT_Mirabelle_root @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ m)) @ i) @ (times_times_complex @ (power_power_complex @ (fFT_Mirabelle_root @ m) @ (times_times_nat @ i @ N)) @ (a @ (suc @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N)))))) @ (set_or562006527an_nat @ zero_zero_nat @ m))))).
