% 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_483__3228332_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:11:28.261

% Could-be-implicit typings (4)
thf(ty_n_t__Complex__Ocomplex, type,
    complex : $tType).
thf(ty_n_t__Real__Oreal, type,
    real : $tType).
thf(ty_n_t__Num__Onum, type,
    num : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).

% Explicit typings (30)
thf(sy_c_FFT__Mirabelle__ulikgskiun_ODFT, type,
    fFT_Mirabelle_DFT : nat > (nat > complex) > nat > complex).
thf(sy_c_FFT__Mirabelle__ulikgskiun_OFFT, type,
    fFT_Mirabelle_FFT : 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_Oone__class_Oone_001t__Real__Oreal, type,
    one_one_real : real).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Complex__Ocomplex, type,
    plus_plus_complex : complex > complex > complex).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat, type,
    plus_plus_nat : nat > nat > nat).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Num__Onum, type,
    plus_plus_num : num > num > num).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal, type,
    plus_plus_real : real > real > real).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Complex__Ocomplex, type,
    times_times_complex : complex > complex > complex).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat, type,
    times_times_nat : nat > nat > nat).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Num__Onum, type,
    times_times_num : num > num > num).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal, type,
    times_times_real : real > real > real).
thf(sy_c_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_Num_Onumeral__class_Onumeral_001t__Real__Oreal, type,
    numeral_numeral_real : num > real).
thf(sy_c_Num_Opow, type,
    pow : num > num > num).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat, type,
    ord_less_nat : nat > nat > $o).
thf(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum, type,
    ord_less_num : num > num > $o).
thf(sy_c_Orderings_Oord__class_Oless_001t__Real__Oreal, type,
    ord_less_real : real > real > $o).
thf(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex, type,
    power_power_complex : complex > nat > complex).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat, type,
    power_power_nat : nat > nat > nat).
thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal, type,
    power_power_real : real > nat > real).
thf(sy_v_a____, type,
    a : nat > complex).
thf(sy_v_i____, type,
    i : nat).
thf(sy_v_ka____, type,
    ka : nat).

% Relevant facts (240)
thf(fact_0_True, axiom,
    ((ord_less_nat @ i @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ ka)))). % True
thf(fact_1_i, axiom,
    ((ord_less_nat @ i @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (suc @ ka))))). % i
thf(fact_2_Suc_Ohyps, axiom,
    ((![I : nat, A : nat > complex]: ((ord_less_nat @ I @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ ka)) => ((fFT_Mirabelle_DFT @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ ka) @ A @ I) = (fFT_Mirabelle_FFT @ ka @ A @ I)))))). % Suc.hyps
thf(fact_3_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_4_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_5_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_6_distrib__left__numeral, axiom,
    ((![V : num, B : real, C : real]: ((times_times_real @ (numeral_numeral_real @ V) @ (plus_plus_real @ B @ C)) = (plus_plus_real @ (times_times_real @ (numeral_numeral_real @ V) @ B) @ (times_times_real @ (numeral_numeral_real @ V) @ C)))))). % distrib_left_numeral
thf(fact_7_distrib__right__numeral, axiom,
    ((![A : complex, B : complex, V : num]: ((times_times_complex @ (plus_plus_complex @ A @ B) @ (numera632737353omplex @ V)) = (plus_plus_complex @ (times_times_complex @ A @ (numera632737353omplex @ V)) @ (times_times_complex @ B @ (numera632737353omplex @ V))))))). % distrib_right_numeral
thf(fact_8_distrib__right__numeral, axiom,
    ((![A : nat, B : nat, V : num]: ((times_times_nat @ (plus_plus_nat @ A @ B) @ (numeral_numeral_nat @ V)) = (plus_plus_nat @ (times_times_nat @ A @ (numeral_numeral_nat @ V)) @ (times_times_nat @ B @ (numeral_numeral_nat @ V))))))). % distrib_right_numeral
thf(fact_9_distrib__right__numeral, axiom,
    ((![A : real, B : real, V : num]: ((times_times_real @ (plus_plus_real @ A @ B) @ (numeral_numeral_real @ V)) = (plus_plus_real @ (times_times_real @ A @ (numeral_numeral_real @ V)) @ (times_times_real @ B @ (numeral_numeral_real @ V))))))). % distrib_right_numeral
thf(fact_10_power__odd__eq, axiom,
    ((![A : nat, N : nat]: ((power_power_nat @ A @ (suc @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N))) = (times_times_nat @ A @ (power_power_nat @ (power_power_nat @ A @ N) @ (numeral_numeral_nat @ (bit0 @ one)))))))). % power_odd_eq
thf(fact_11_power__odd__eq, axiom,
    ((![A : complex, N : nat]: ((power_power_complex @ A @ (suc @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N))) = (times_times_complex @ A @ (power_power_complex @ (power_power_complex @ A @ N) @ (numeral_numeral_nat @ (bit0 @ one)))))))). % power_odd_eq
thf(fact_12_power__odd__eq, axiom,
    ((![A : real, N : nat]: ((power_power_real @ A @ (suc @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N))) = (times_times_real @ A @ (power_power_real @ (power_power_real @ A @ N) @ (numeral_numeral_nat @ (bit0 @ one)))))))). % power_odd_eq
thf(fact_13_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_14_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_15_power2__sum, axiom,
    ((![X : real, Y : real]: ((power_power_real @ (plus_plus_real @ X @ Y) @ (numeral_numeral_nat @ (bit0 @ one))) = (plus_plus_real @ (plus_plus_real @ (power_power_real @ X @ (numeral_numeral_nat @ (bit0 @ one))) @ (power_power_real @ Y @ (numeral_numeral_nat @ (bit0 @ one)))) @ (times_times_real @ (times_times_real @ (numeral_numeral_real @ (bit0 @ one)) @ X) @ Y)))))). % power2_sum
thf(fact_16_lessI, axiom,
    ((![N : nat]: (ord_less_nat @ N @ (suc @ N))))). % lessI
thf(fact_17_Suc__mono, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (ord_less_nat @ (suc @ M) @ (suc @ N)))))). % Suc_mono
thf(fact_18_Suc__less__eq, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ (suc @ M) @ (suc @ N)) = (ord_less_nat @ M @ N))))). % Suc_less_eq
thf(fact_19_Suc__double__not__eq__double, axiom,
    ((![M : nat, N : nat]: (~ (((suc @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ M)) = (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N))))))). % Suc_double_not_eq_double
thf(fact_20_double__not__eq__Suc__double, axiom,
    ((![M : nat, N : nat]: (~ (((times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ M) = (suc @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N)))))))). % double_not_eq_Suc_double
thf(fact_21_power__even__eq, axiom,
    ((![A : nat, N : nat]: ((power_power_nat @ A @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N)) = (power_power_nat @ (power_power_nat @ A @ N) @ (numeral_numeral_nat @ (bit0 @ one))))))). % power_even_eq
thf(fact_22_power__even__eq, axiom,
    ((![A : complex, N : nat]: ((power_power_complex @ A @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N)) = (power_power_complex @ (power_power_complex @ A @ N) @ (numeral_numeral_nat @ (bit0 @ one))))))). % power_even_eq
thf(fact_23_power__even__eq, axiom,
    ((![A : real, N : nat]: ((power_power_real @ A @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N)) = (power_power_real @ (power_power_real @ A @ N) @ (numeral_numeral_nat @ (bit0 @ one))))))). % power_even_eq
thf(fact_24_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_25_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_26_power4__eq__xxxx, axiom,
    ((![X : real]: ((power_power_real @ X @ (numeral_numeral_nat @ (bit0 @ (bit0 @ one)))) = (times_times_real @ (times_times_real @ (times_times_real @ X @ X) @ X) @ X))))). % power4_eq_xxxx
thf(fact_27_power2__eq__square, axiom,
    ((![A : nat]: ((power_power_nat @ A @ (numeral_numeral_nat @ (bit0 @ one))) = (times_times_nat @ A @ A))))). % power2_eq_square
thf(fact_28_power2__eq__square, axiom,
    ((![A : complex]: ((power_power_complex @ A @ (numeral_numeral_nat @ (bit0 @ one))) = (times_times_complex @ A @ A))))). % power2_eq_square
thf(fact_29_power2__eq__square, axiom,
    ((![A : real]: ((power_power_real @ A @ (numeral_numeral_nat @ (bit0 @ one))) = (times_times_real @ A @ A))))). % power2_eq_square
thf(fact_30_numeral__eq__iff, axiom,
    ((![M : num, N : num]: (((numeral_numeral_nat @ M) = (numeral_numeral_nat @ N)) = (M = N))))). % numeral_eq_iff
thf(fact_31_numeral__eq__iff, axiom,
    ((![M : num, N : num]: (((numeral_numeral_real @ M) = (numeral_numeral_real @ N)) = (M = N))))). % numeral_eq_iff
thf(fact_32_numeral__eq__iff, axiom,
    ((![M : num, N : num]: (((numera632737353omplex @ M) = (numera632737353omplex @ N)) = (M = N))))). % numeral_eq_iff
thf(fact_33_add__Suc__right, axiom,
    ((![M : nat, N : nat]: ((plus_plus_nat @ M @ (suc @ N)) = (suc @ (plus_plus_nat @ M @ N)))))). % add_Suc_right
thf(fact_34_old_Onat_Oinject, axiom,
    ((![Nat : nat, Nat2 : nat]: (((suc @ Nat) = (suc @ Nat2)) = (Nat = Nat2))))). % old.nat.inject
thf(fact_35_nat_Oinject, axiom,
    ((![X2 : nat, Y2 : nat]: (((suc @ X2) = (suc @ Y2)) = (X2 = Y2))))). % nat.inject
thf(fact_36_nat__add__left__cancel__less, axiom,
    ((![K : nat, M : nat, N : nat]: ((ord_less_nat @ (plus_plus_nat @ K @ M) @ (plus_plus_nat @ K @ N)) = (ord_less_nat @ M @ N))))). % nat_add_left_cancel_less
thf(fact_37_numeral__less__iff, axiom,
    ((![M : num, N : num]: ((ord_less_nat @ (numeral_numeral_nat @ M) @ (numeral_numeral_nat @ N)) = (ord_less_num @ M @ N))))). % numeral_less_iff
thf(fact_38_numeral__less__iff, axiom,
    ((![M : num, N : num]: ((ord_less_real @ (numeral_numeral_real @ M) @ (numeral_numeral_real @ N)) = (ord_less_num @ M @ N))))). % numeral_less_iff
thf(fact_39_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_40_numeral__times__numeral, axiom,
    ((![M : num, N : num]: ((times_times_real @ (numeral_numeral_real @ M) @ (numeral_numeral_real @ N)) = (numeral_numeral_real @ (times_times_num @ M @ N)))))). % numeral_times_numeral
thf(fact_41_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_42_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_43_mult__numeral__left__semiring__numeral, axiom,
    ((![V : num, W : num, Z : real]: ((times_times_real @ (numeral_numeral_real @ V) @ (times_times_real @ (numeral_numeral_real @ W) @ Z)) = (times_times_real @ (numeral_numeral_real @ (times_times_num @ V @ W)) @ Z))))). % mult_numeral_left_semiring_numeral
thf(fact_44_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_45_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_46_numeral__plus__numeral, axiom,
    ((![M : num, N : num]: ((plus_plus_real @ (numeral_numeral_real @ M) @ (numeral_numeral_real @ N)) = (numeral_numeral_real @ (plus_plus_num @ M @ N)))))). % numeral_plus_numeral
thf(fact_47_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_48_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_49_add__numeral__left, axiom,
    ((![V : num, W : num, Z : real]: ((plus_plus_real @ (numeral_numeral_real @ V) @ (plus_plus_real @ (numeral_numeral_real @ W) @ Z)) = (plus_plus_real @ (numeral_numeral_real @ (plus_plus_num @ V @ W)) @ Z))))). % add_numeral_left
thf(fact_50_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_51_num__double, axiom,
    ((![N : num]: ((times_times_num @ (bit0 @ one) @ N) = (bit0 @ N))))). % num_double
thf(fact_52_power__mult__numeral, axiom,
    ((![A : nat, M : num, N : num]: ((power_power_nat @ (power_power_nat @ A @ (numeral_numeral_nat @ M)) @ (numeral_numeral_nat @ N)) = (power_power_nat @ A @ (numeral_numeral_nat @ (times_times_num @ M @ N))))))). % power_mult_numeral
thf(fact_53_power__mult__numeral, axiom,
    ((![A : complex, M : num, N : num]: ((power_power_complex @ (power_power_complex @ A @ (numeral_numeral_nat @ M)) @ (numeral_numeral_nat @ N)) = (power_power_complex @ A @ (numeral_numeral_nat @ (times_times_num @ M @ N))))))). % power_mult_numeral
thf(fact_54_power__mult__numeral, axiom,
    ((![A : real, M : num, N : num]: ((power_power_real @ (power_power_real @ A @ (numeral_numeral_nat @ M)) @ (numeral_numeral_nat @ N)) = (power_power_real @ A @ (numeral_numeral_nat @ (times_times_num @ M @ N))))))). % power_mult_numeral
thf(fact_55_mult__Suc__right, axiom,
    ((![M : nat, N : nat]: ((times_times_nat @ M @ (suc @ N)) = (plus_plus_nat @ M @ (times_times_nat @ M @ N)))))). % mult_Suc_right
thf(fact_56_power__add__numeral2, axiom,
    ((![A : nat, M : num, N : num, B : nat]: ((times_times_nat @ (power_power_nat @ A @ (numeral_numeral_nat @ M)) @ (times_times_nat @ (power_power_nat @ A @ (numeral_numeral_nat @ N)) @ B)) = (times_times_nat @ (power_power_nat @ A @ (numeral_numeral_nat @ (plus_plus_num @ M @ N))) @ B))))). % power_add_numeral2
thf(fact_57_power__add__numeral2, axiom,
    ((![A : complex, M : num, N : num, B : complex]: ((times_times_complex @ (power_power_complex @ A @ (numeral_numeral_nat @ M)) @ (times_times_complex @ (power_power_complex @ A @ (numeral_numeral_nat @ N)) @ B)) = (times_times_complex @ (power_power_complex @ A @ (numeral_numeral_nat @ (plus_plus_num @ M @ N))) @ B))))). % power_add_numeral2
thf(fact_58_power__add__numeral2, axiom,
    ((![A : real, M : num, N : num, B : real]: ((times_times_real @ (power_power_real @ A @ (numeral_numeral_nat @ M)) @ (times_times_real @ (power_power_real @ A @ (numeral_numeral_nat @ N)) @ B)) = (times_times_real @ (power_power_real @ A @ (numeral_numeral_nat @ (plus_plus_num @ M @ N))) @ B))))). % power_add_numeral2
thf(fact_59_power__add__numeral, axiom,
    ((![A : nat, M : num, N : num]: ((times_times_nat @ (power_power_nat @ A @ (numeral_numeral_nat @ M)) @ (power_power_nat @ A @ (numeral_numeral_nat @ N))) = (power_power_nat @ A @ (numeral_numeral_nat @ (plus_plus_num @ M @ N))))))). % power_add_numeral
thf(fact_60_power__add__numeral, axiom,
    ((![A : complex, M : num, N : num]: ((times_times_complex @ (power_power_complex @ A @ (numeral_numeral_nat @ M)) @ (power_power_complex @ A @ (numeral_numeral_nat @ N))) = (power_power_complex @ A @ (numeral_numeral_nat @ (plus_plus_num @ M @ N))))))). % power_add_numeral
thf(fact_61_power__add__numeral, axiom,
    ((![A : real, M : num, N : num]: ((times_times_real @ (power_power_real @ A @ (numeral_numeral_nat @ M)) @ (power_power_real @ A @ (numeral_numeral_nat @ N))) = (power_power_real @ A @ (numeral_numeral_nat @ (plus_plus_num @ M @ N))))))). % power_add_numeral
thf(fact_62_Suc__numeral, axiom,
    ((![N : num]: ((suc @ (numeral_numeral_nat @ N)) = (numeral_numeral_nat @ (plus_plus_num @ N @ one)))))). % Suc_numeral
thf(fact_63_add__2__eq__Suc_H, axiom,
    ((![N : nat]: ((plus_plus_nat @ N @ (numeral_numeral_nat @ (bit0 @ one))) = (suc @ (suc @ N)))))). % add_2_eq_Suc'
thf(fact_64_add__2__eq__Suc, axiom,
    ((![N : nat]: ((plus_plus_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N) = (suc @ (suc @ N)))))). % add_2_eq_Suc
thf(fact_65_add__One__commute, axiom,
    ((![N : num]: ((plus_plus_num @ one @ N) = (plus_plus_num @ N @ one))))). % add_One_commute
thf(fact_66_add__Suc, axiom,
    ((![M : nat, N : nat]: ((plus_plus_nat @ (suc @ M) @ N) = (suc @ (plus_plus_nat @ M @ N)))))). % add_Suc
thf(fact_67_nat__arith_Osuc1, axiom,
    ((![A2 : nat, K : nat, A : nat]: ((A2 = (plus_plus_nat @ K @ A)) => ((suc @ A2) = (plus_plus_nat @ K @ (suc @ A))))))). % nat_arith.suc1
thf(fact_68_add__Suc__shift, axiom,
    ((![M : nat, N : nat]: ((plus_plus_nat @ (suc @ M) @ N) = (plus_plus_nat @ M @ (suc @ N)))))). % add_Suc_shift
thf(fact_69_less__add__eq__less, axiom,
    ((![K : nat, L : nat, M : nat, N : nat]: ((ord_less_nat @ K @ L) => (((plus_plus_nat @ M @ L) = (plus_plus_nat @ K @ N)) => (ord_less_nat @ M @ N)))))). % less_add_eq_less
thf(fact_70_trans__less__add2, axiom,
    ((![I : nat, J : nat, M : nat]: ((ord_less_nat @ I @ J) => (ord_less_nat @ I @ (plus_plus_nat @ M @ J)))))). % trans_less_add2
thf(fact_71_trans__less__add1, axiom,
    ((![I : nat, J : nat, M : nat]: ((ord_less_nat @ I @ J) => (ord_less_nat @ I @ (plus_plus_nat @ J @ M)))))). % trans_less_add1
thf(fact_72_add__less__mono1, axiom,
    ((![I : nat, J : nat, K : nat]: ((ord_less_nat @ I @ J) => (ord_less_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ K)))))). % add_less_mono1
thf(fact_73_not__add__less2, axiom,
    ((![J : nat, I : nat]: (~ ((ord_less_nat @ (plus_plus_nat @ J @ I) @ I)))))). % not_add_less2
thf(fact_74_not__add__less1, axiom,
    ((![I : nat, J : nat]: (~ ((ord_less_nat @ (plus_plus_nat @ I @ J) @ I)))))). % not_add_less1
thf(fact_75_add__less__mono, axiom,
    ((![I : nat, J : nat, K : nat, L : nat]: ((ord_less_nat @ I @ J) => ((ord_less_nat @ K @ L) => (ord_less_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J @ L))))))). % add_less_mono
thf(fact_76_add__lessD1, axiom,
    ((![I : nat, J : nat, K : nat]: ((ord_less_nat @ (plus_plus_nat @ I @ J) @ K) => (ord_less_nat @ I @ K))))). % add_lessD1
thf(fact_77_add__mult__distrib2, axiom,
    ((![K : nat, M : nat, N : nat]: ((times_times_nat @ K @ (plus_plus_nat @ M @ N)) = (plus_plus_nat @ (times_times_nat @ K @ M) @ (times_times_nat @ K @ N)))))). % add_mult_distrib2
thf(fact_78_add__mult__distrib, axiom,
    ((![M : nat, N : nat, K : nat]: ((times_times_nat @ (plus_plus_nat @ M @ N) @ K) = (plus_plus_nat @ (times_times_nat @ M @ K) @ (times_times_nat @ N @ K)))))). % add_mult_distrib
thf(fact_79_Suc__nat__number__of__add, axiom,
    ((![V : num, N : nat]: ((suc @ (plus_plus_nat @ (numeral_numeral_nat @ V) @ N)) = (plus_plus_nat @ (numeral_numeral_nat @ (plus_plus_num @ V @ one)) @ N))))). % Suc_nat_number_of_add
thf(fact_80_power__add, axiom,
    ((![A : nat, M : nat, N : nat]: ((power_power_nat @ A @ (plus_plus_nat @ M @ N)) = (times_times_nat @ (power_power_nat @ A @ M) @ (power_power_nat @ A @ N)))))). % power_add
thf(fact_81_power__add, axiom,
    ((![A : complex, M : nat, N : nat]: ((power_power_complex @ A @ (plus_plus_nat @ M @ N)) = (times_times_complex @ (power_power_complex @ A @ M) @ (power_power_complex @ A @ N)))))). % power_add
thf(fact_82_power__add, axiom,
    ((![A : real, M : nat, N : nat]: ((power_power_real @ A @ (plus_plus_nat @ M @ N)) = (times_times_real @ (power_power_real @ A @ M) @ (power_power_real @ A @ N)))))). % power_add
thf(fact_83_less__imp__Suc__add, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (?[K2 : nat]: (N = (suc @ (plus_plus_nat @ M @ K2)))))))). % less_imp_Suc_add
thf(fact_84_less__iff__Suc__add, axiom,
    ((ord_less_nat = (^[M2 : nat]: (^[N2 : nat]: (?[K3 : nat]: (N2 = (suc @ (plus_plus_nat @ M2 @ K3))))))))). % less_iff_Suc_add
thf(fact_85_less__add__Suc2, axiom,
    ((![I : nat, M : nat]: (ord_less_nat @ I @ (suc @ (plus_plus_nat @ M @ I)))))). % less_add_Suc2
thf(fact_86_less__add__Suc1, axiom,
    ((![I : nat, M : nat]: (ord_less_nat @ I @ (suc @ (plus_plus_nat @ I @ M)))))). % less_add_Suc1
thf(fact_87_less__natE, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((![Q : nat]: (~ ((N = (suc @ (plus_plus_nat @ M @ Q)))))))))))). % less_natE
thf(fact_88_mult__Suc, axiom,
    ((![M : nat, N : nat]: ((times_times_nat @ (suc @ M) @ N) = (plus_plus_nat @ N @ (times_times_nat @ M @ N)))))). % mult_Suc
thf(fact_89_is__num__normalize_I1_J, axiom,
    ((![A : complex, B : complex, C : complex]: ((plus_plus_complex @ (plus_plus_complex @ A @ B) @ C) = (plus_plus_complex @ A @ (plus_plus_complex @ B @ C)))))). % is_num_normalize(1)
thf(fact_90_is__num__normalize_I1_J, axiom,
    ((![A : real, B : real, C : real]: ((plus_plus_real @ (plus_plus_real @ A @ B) @ C) = (plus_plus_real @ A @ (plus_plus_real @ B @ C)))))). % is_num_normalize(1)
thf(fact_91_n__not__Suc__n, axiom,
    ((![N : nat]: (~ ((N = (suc @ N))))))). % n_not_Suc_n
thf(fact_92_Suc__inject, axiom,
    ((![X : nat, Y : nat]: (((suc @ X) = (suc @ Y)) => (X = Y))))). % Suc_inject
thf(fact_93_linorder__neqE__nat, axiom,
    ((![X : nat, Y : nat]: ((~ ((X = Y))) => ((~ ((ord_less_nat @ X @ Y))) => (ord_less_nat @ Y @ X)))))). % linorder_neqE_nat
thf(fact_94_infinite__descent, axiom,
    ((![P : nat > $o, N : nat]: ((![N3 : nat]: ((~ ((P @ N3))) => (?[M3 : nat]: ((ord_less_nat @ M3 @ N3) & (~ ((P @ M3))))))) => (P @ N))))). % infinite_descent
thf(fact_95_nat__less__induct, axiom,
    ((![P : nat > $o, N : nat]: ((![N3 : nat]: ((![M3 : nat]: ((ord_less_nat @ M3 @ N3) => (P @ M3))) => (P @ N3))) => (P @ N))))). % nat_less_induct
thf(fact_96_less__irrefl__nat, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_irrefl_nat
thf(fact_97_less__not__refl3, axiom,
    ((![S : nat, T : nat]: ((ord_less_nat @ S @ T) => (~ ((S = T))))))). % less_not_refl3
thf(fact_98_less__not__refl2, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ N @ M) => (~ ((M = N))))))). % less_not_refl2
thf(fact_99_less__not__refl, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_not_refl
thf(fact_100_nat__neq__iff, axiom,
    ((![M : nat, N : nat]: ((~ ((M = N))) = (((ord_less_nat @ M @ N)) | ((ord_less_nat @ N @ M))))))). % nat_neq_iff
thf(fact_101_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_102_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_103_power__commuting__commutes, axiom,
    ((![X : real, Y : real, N : nat]: (((times_times_real @ X @ Y) = (times_times_real @ Y @ X)) => ((times_times_real @ (power_power_real @ X @ N) @ Y) = (times_times_real @ Y @ (power_power_real @ X @ N))))))). % power_commuting_commutes
thf(fact_104_power__mult__distrib, axiom,
    ((![A : nat, B : nat, N : nat]: ((power_power_nat @ (times_times_nat @ A @ B) @ N) = (times_times_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ B @ N)))))). % power_mult_distrib
thf(fact_105_power__mult__distrib, axiom,
    ((![A : complex, B : complex, N : nat]: ((power_power_complex @ (times_times_complex @ A @ B) @ N) = (times_times_complex @ (power_power_complex @ A @ N) @ (power_power_complex @ B @ N)))))). % power_mult_distrib
thf(fact_106_power__mult__distrib, axiom,
    ((![A : real, B : real, N : nat]: ((power_power_real @ (times_times_real @ A @ B) @ N) = (times_times_real @ (power_power_real @ A @ N) @ (power_power_real @ B @ N)))))). % power_mult_distrib
thf(fact_107_power__commutes, axiom,
    ((![A : nat, N : nat]: ((times_times_nat @ (power_power_nat @ A @ N) @ A) = (times_times_nat @ A @ (power_power_nat @ A @ N)))))). % power_commutes
thf(fact_108_power__commutes, axiom,
    ((![A : complex, N : nat]: ((times_times_complex @ (power_power_complex @ A @ N) @ A) = (times_times_complex @ A @ (power_power_complex @ A @ N)))))). % power_commutes
thf(fact_109_power__commutes, axiom,
    ((![A : real, N : nat]: ((times_times_real @ (power_power_real @ A @ N) @ A) = (times_times_real @ A @ (power_power_real @ A @ N)))))). % power_commutes
thf(fact_110_not__less__less__Suc__eq, axiom,
    ((![N : nat, M : nat]: ((~ ((ord_less_nat @ N @ M))) => ((ord_less_nat @ N @ (suc @ M)) = (N = M)))))). % not_less_less_Suc_eq
thf(fact_111_strict__inc__induct, axiom,
    ((![I : nat, J : nat, P : nat > $o]: ((ord_less_nat @ I @ J) => ((![I2 : nat]: ((J = (suc @ I2)) => (P @ I2))) => ((![I2 : nat]: ((ord_less_nat @ I2 @ J) => ((P @ (suc @ I2)) => (P @ I2)))) => (P @ I))))))). % strict_inc_induct
thf(fact_112_less__Suc__induct, axiom,
    ((![I : nat, J : nat, P : nat > nat > $o]: ((ord_less_nat @ I @ J) => ((![I2 : nat]: (P @ I2 @ (suc @ I2))) => ((![I2 : nat, J2 : nat, K2 : nat]: ((ord_less_nat @ I2 @ J2) => ((ord_less_nat @ J2 @ K2) => ((P @ I2 @ J2) => ((P @ J2 @ K2) => (P @ I2 @ K2)))))) => (P @ I @ J))))))). % less_Suc_induct
thf(fact_113_less__trans__Suc, axiom,
    ((![I : nat, J : nat, K : nat]: ((ord_less_nat @ I @ J) => ((ord_less_nat @ J @ K) => (ord_less_nat @ (suc @ I) @ K)))))). % less_trans_Suc
thf(fact_114_Suc__less__SucD, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ (suc @ M) @ (suc @ N)) => (ord_less_nat @ M @ N))))). % Suc_less_SucD
thf(fact_115_less__antisym, axiom,
    ((![N : nat, M : nat]: ((~ ((ord_less_nat @ N @ M))) => ((ord_less_nat @ N @ (suc @ M)) => (M = N)))))). % less_antisym
thf(fact_116_Suc__less__eq2, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ (suc @ N) @ M) = (?[M4 : nat]: (((M = (suc @ M4))) & ((ord_less_nat @ N @ M4)))))))). % Suc_less_eq2
thf(fact_117_All__less__Suc, axiom,
    ((![N : nat, P : nat > $o]: ((![I3 : nat]: (((ord_less_nat @ I3 @ (suc @ N))) => ((P @ I3)))) = (((P @ N)) & ((![I3 : nat]: (((ord_less_nat @ I3 @ N)) => ((P @ I3)))))))))). % All_less_Suc
thf(fact_118_not__less__eq, axiom,
    ((![M : nat, N : nat]: ((~ ((ord_less_nat @ M @ N))) = (ord_less_nat @ N @ (suc @ M)))))). % not_less_eq
thf(fact_119_less__Suc__eq, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ (suc @ N)) = (((ord_less_nat @ M @ N)) | ((M = N))))))). % less_Suc_eq
thf(fact_120_Ex__less__Suc, axiom,
    ((![N : nat, P : nat > $o]: ((?[I3 : nat]: (((ord_less_nat @ I3 @ (suc @ N))) & ((P @ I3)))) = (((P @ N)) | ((?[I3 : nat]: (((ord_less_nat @ I3 @ N)) & ((P @ I3)))))))))). % Ex_less_Suc
thf(fact_121_less__SucI, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (ord_less_nat @ M @ (suc @ N)))))). % less_SucI
thf(fact_122_less__SucE, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ (suc @ N)) => ((~ ((ord_less_nat @ M @ N))) => (M = N)))))). % less_SucE
thf(fact_123_Suc__lessI, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => ((~ (((suc @ M) = N))) => (ord_less_nat @ (suc @ M) @ N)))))). % Suc_lessI
thf(fact_124_Suc__lessE, axiom,
    ((![I : nat, K : nat]: ((ord_less_nat @ (suc @ I) @ K) => (~ ((![J2 : nat]: ((ord_less_nat @ I @ J2) => (~ ((K = (suc @ J2)))))))))))). % Suc_lessE
thf(fact_125_Suc__lessD, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ (suc @ M) @ N) => (ord_less_nat @ M @ N))))). % Suc_lessD
thf(fact_126_Nat_OlessE, axiom,
    ((![I : nat, K : nat]: ((ord_less_nat @ I @ K) => ((~ ((K = (suc @ I)))) => (~ ((![J2 : nat]: ((ord_less_nat @ I @ J2) => (~ ((K = (suc @ J2))))))))))))). % Nat.lessE
thf(fact_127_power__mult, axiom,
    ((![A : nat, M : nat, N : nat]: ((power_power_nat @ A @ (times_times_nat @ M @ N)) = (power_power_nat @ (power_power_nat @ A @ M) @ N))))). % power_mult
thf(fact_128_power__mult, axiom,
    ((![A : complex, M : nat, N : nat]: ((power_power_complex @ A @ (times_times_nat @ M @ N)) = (power_power_complex @ (power_power_complex @ A @ M) @ N))))). % power_mult
thf(fact_129_power__mult, axiom,
    ((![A : real, M : nat, N : nat]: ((power_power_real @ A @ (times_times_nat @ M @ N)) = (power_power_real @ (power_power_real @ A @ M) @ N))))). % power_mult
thf(fact_130_Suc__mult__cancel1, axiom,
    ((![K : nat, M : nat, N : nat]: (((times_times_nat @ (suc @ K) @ M) = (times_times_nat @ (suc @ K) @ N)) = (M = N))))). % Suc_mult_cancel1
thf(fact_131_mult__numeral__1__right, axiom,
    ((![A : nat]: ((times_times_nat @ A @ (numeral_numeral_nat @ one)) = A)))). % mult_numeral_1_right
thf(fact_132_mult__numeral__1__right, axiom,
    ((![A : real]: ((times_times_real @ A @ (numeral_numeral_real @ one)) = A)))). % mult_numeral_1_right
thf(fact_133_mult__numeral__1__right, axiom,
    ((![A : complex]: ((times_times_complex @ A @ (numera632737353omplex @ one)) = A)))). % mult_numeral_1_right
thf(fact_134_mult__numeral__1, axiom,
    ((![A : nat]: ((times_times_nat @ (numeral_numeral_nat @ one) @ A) = A)))). % mult_numeral_1
thf(fact_135_mult__numeral__1, axiom,
    ((![A : real]: ((times_times_real @ (numeral_numeral_real @ one) @ A) = A)))). % mult_numeral_1
thf(fact_136_mult__numeral__1, axiom,
    ((![A : complex]: ((times_times_complex @ (numera632737353omplex @ one) @ A) = A)))). % mult_numeral_1
thf(fact_137_numeral__Bit0, axiom,
    ((![N : num]: ((numeral_numeral_nat @ (bit0 @ N)) = (plus_plus_nat @ (numeral_numeral_nat @ N) @ (numeral_numeral_nat @ N)))))). % numeral_Bit0
thf(fact_138_numeral__Bit0, axiom,
    ((![N : num]: ((numeral_numeral_real @ (bit0 @ N)) = (plus_plus_real @ (numeral_numeral_real @ N) @ (numeral_numeral_real @ N)))))). % numeral_Bit0
thf(fact_139_numeral__Bit0, axiom,
    ((![N : num]: ((numera632737353omplex @ (bit0 @ N)) = (plus_plus_complex @ (numera632737353omplex @ N) @ (numera632737353omplex @ N)))))). % numeral_Bit0
thf(fact_140_power__Suc2, axiom,
    ((![A : nat, N : nat]: ((power_power_nat @ A @ (suc @ N)) = (times_times_nat @ (power_power_nat @ A @ N) @ A))))). % power_Suc2
thf(fact_141_power__Suc2, axiom,
    ((![A : complex, N : nat]: ((power_power_complex @ A @ (suc @ N)) = (times_times_complex @ (power_power_complex @ A @ N) @ A))))). % power_Suc2
thf(fact_142_power__Suc2, axiom,
    ((![A : real, N : nat]: ((power_power_real @ A @ (suc @ N)) = (times_times_real @ (power_power_real @ A @ N) @ A))))). % power_Suc2
thf(fact_143_power__Suc, axiom,
    ((![A : nat, N : nat]: ((power_power_nat @ A @ (suc @ N)) = (times_times_nat @ A @ (power_power_nat @ A @ N)))))). % power_Suc
thf(fact_144_power__Suc, axiom,
    ((![A : complex, N : nat]: ((power_power_complex @ A @ (suc @ N)) = (times_times_complex @ A @ (power_power_complex @ A @ N)))))). % power_Suc
thf(fact_145_power__Suc, axiom,
    ((![A : real, N : nat]: ((power_power_real @ A @ (suc @ N)) = (times_times_real @ A @ (power_power_real @ A @ N)))))). % power_Suc
thf(fact_146_lift__Suc__mono__less__iff, axiom,
    ((![F : nat > nat, N : nat, M : nat]: ((![N3 : nat]: (ord_less_nat @ (F @ N3) @ (F @ (suc @ N3)))) => ((ord_less_nat @ (F @ N) @ (F @ M)) = (ord_less_nat @ N @ M)))))). % lift_Suc_mono_less_iff
thf(fact_147_lift__Suc__mono__less__iff, axiom,
    ((![F : nat > num, N : nat, M : nat]: ((![N3 : nat]: (ord_less_num @ (F @ N3) @ (F @ (suc @ N3)))) => ((ord_less_num @ (F @ N) @ (F @ M)) = (ord_less_nat @ N @ M)))))). % lift_Suc_mono_less_iff
thf(fact_148_lift__Suc__mono__less, axiom,
    ((![F : nat > nat, N : nat, N4 : nat]: ((![N3 : nat]: (ord_less_nat @ (F @ N3) @ (F @ (suc @ N3)))) => ((ord_less_nat @ N @ N4) => (ord_less_nat @ (F @ N) @ (F @ N4))))))). % lift_Suc_mono_less
thf(fact_149_lift__Suc__mono__less, axiom,
    ((![F : nat > num, N : nat, N4 : nat]: ((![N3 : nat]: (ord_less_num @ (F @ N3) @ (F @ (suc @ N3)))) => ((ord_less_nat @ N @ N4) => (ord_less_num @ (F @ N) @ (F @ N4))))))). % lift_Suc_mono_less
thf(fact_150_Suc__mult__less__cancel1, axiom,
    ((![K : nat, M : nat, N : nat]: ((ord_less_nat @ (times_times_nat @ (suc @ K) @ M) @ (times_times_nat @ (suc @ K) @ N)) = (ord_less_nat @ M @ N))))). % Suc_mult_less_cancel1
thf(fact_151_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_152_numeral__code_I2_J, axiom,
    ((![N : num]: ((numeral_numeral_real @ (bit0 @ N)) = (plus_plus_real @ (numeral_numeral_real @ N) @ (numeral_numeral_real @ N)))))). % numeral_code(2)
thf(fact_153_numeral__code_I2_J, axiom,
    ((![N : num]: ((numera632737353omplex @ (bit0 @ N)) = (plus_plus_complex @ (numera632737353omplex @ N) @ (numera632737353omplex @ N)))))). % numeral_code(2)
thf(fact_154_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_155_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_156_power__numeral__even, axiom,
    ((![Z : real, W : num]: ((power_power_real @ Z @ (numeral_numeral_nat @ (bit0 @ W))) = (times_times_real @ (power_power_real @ Z @ (numeral_numeral_nat @ W)) @ (power_power_real @ Z @ (numeral_numeral_nat @ W))))))). % power_numeral_even
thf(fact_157_left__add__twice, axiom,
    ((![A : nat, B : nat]: ((plus_plus_nat @ A @ (plus_plus_nat @ A @ B)) = (plus_plus_nat @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ A) @ B))))). % left_add_twice
thf(fact_158_left__add__twice, axiom,
    ((![A : real, B : real]: ((plus_plus_real @ A @ (plus_plus_real @ A @ B)) = (plus_plus_real @ (times_times_real @ (numeral_numeral_real @ (bit0 @ one)) @ A) @ B))))). % left_add_twice
thf(fact_159_left__add__twice, axiom,
    ((![A : complex, B : complex]: ((plus_plus_complex @ A @ (plus_plus_complex @ A @ B)) = (plus_plus_complex @ (times_times_complex @ (numera632737353omplex @ (bit0 @ one)) @ A) @ B))))). % left_add_twice
thf(fact_160_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_161_mult__2__right, axiom,
    ((![Z : real]: ((times_times_real @ Z @ (numeral_numeral_real @ (bit0 @ one))) = (plus_plus_real @ Z @ Z))))). % mult_2_right
thf(fact_162_mult__2__right, axiom,
    ((![Z : complex]: ((times_times_complex @ Z @ (numera632737353omplex @ (bit0 @ one))) = (plus_plus_complex @ Z @ Z))))). % mult_2_right
thf(fact_163_mult__2, axiom,
    ((![Z : nat]: ((times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ Z) = (plus_plus_nat @ Z @ Z))))). % mult_2
thf(fact_164_mult__2, axiom,
    ((![Z : real]: ((times_times_real @ (numeral_numeral_real @ (bit0 @ one)) @ Z) = (plus_plus_real @ Z @ Z))))). % mult_2
thf(fact_165_mult__2, axiom,
    ((![Z : complex]: ((times_times_complex @ (numera632737353omplex @ (bit0 @ one)) @ Z) = (plus_plus_complex @ Z @ Z))))). % mult_2
thf(fact_166_semiring__norm_I2_J, axiom,
    (((plus_plus_num @ one @ one) = (bit0 @ one)))). % semiring_norm(2)
thf(fact_167_semiring__norm_I83_J, axiom,
    ((![N : num]: (~ ((one = (bit0 @ N))))))). % semiring_norm(83)
thf(fact_168_semiring__norm_I85_J, axiom,
    ((![M : num]: (~ (((bit0 @ M) = one)))))). % semiring_norm(85)
thf(fact_169_add__less__cancel__right, axiom,
    ((![A : real, C : real, B : real]: ((ord_less_real @ (plus_plus_real @ A @ C) @ (plus_plus_real @ B @ C)) = (ord_less_real @ A @ B))))). % add_less_cancel_right
thf(fact_170_add__less__cancel__right, axiom,
    ((![A : nat, C : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ C)) = (ord_less_nat @ A @ B))))). % add_less_cancel_right
thf(fact_171_add__less__cancel__left, axiom,
    ((![C : real, A : real, B : real]: ((ord_less_real @ (plus_plus_real @ C @ A) @ (plus_plus_real @ C @ B)) = (ord_less_real @ A @ B))))). % add_less_cancel_left
thf(fact_172_add__less__cancel__left, axiom,
    ((![C : nat, A : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ C @ A) @ (plus_plus_nat @ C @ B)) = (ord_less_nat @ A @ B))))). % add_less_cancel_left
thf(fact_173_DFT__lower, axiom,
    ((![M : nat, A : nat > complex, I : nat]: ((fFT_Mirabelle_DFT @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ M) @ A @ I) = (plus_plus_complex @ (fFT_Mirabelle_DFT @ M @ (^[I3 : nat]: (A @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ I3))) @ I) @ (times_times_complex @ (power_power_complex @ (fFT_Mirabelle_root @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ M)) @ I) @ (fFT_Mirabelle_DFT @ M @ (^[I3 : nat]: (A @ (plus_plus_nat @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ I3) @ one_one_nat))) @ I))))))). % DFT_lower
thf(fact_174_four__x__squared, axiom,
    ((![X : real]: ((times_times_real @ (numeral_numeral_real @ (bit0 @ (bit0 @ one))) @ (power_power_real @ X @ (numeral_numeral_nat @ (bit0 @ one)))) = (power_power_real @ (times_times_real @ (numeral_numeral_real @ (bit0 @ one)) @ X) @ (numeral_numeral_nat @ (bit0 @ one))))))). % four_x_squared
thf(fact_175_power__numeral, axiom,
    ((![K : num, L : num]: ((power_power_nat @ (numeral_numeral_nat @ K) @ (numeral_numeral_nat @ L)) = (numeral_numeral_nat @ (pow @ K @ L)))))). % power_numeral
thf(fact_176_power__numeral, axiom,
    ((![K : num, L : num]: ((power_power_real @ (numeral_numeral_real @ K) @ (numeral_numeral_nat @ L)) = (numeral_numeral_real @ (pow @ K @ L)))))). % power_numeral
thf(fact_177_power__numeral, axiom,
    ((![K : num, L : num]: ((power_power_complex @ (numera632737353omplex @ K) @ (numeral_numeral_nat @ L)) = (numera632737353omplex @ (pow @ K @ L)))))). % power_numeral
thf(fact_178_semiring__norm_I11_J, axiom,
    ((![M : num]: ((times_times_num @ M @ one) = M)))). % semiring_norm(11)
thf(fact_179_add__left__cancel, axiom,
    ((![A : complex, B : complex, C : complex]: (((plus_plus_complex @ A @ B) = (plus_plus_complex @ A @ C)) = (B = C))))). % add_left_cancel
thf(fact_180_add__left__cancel, axiom,
    ((![A : nat, B : nat, C : nat]: (((plus_plus_nat @ A @ B) = (plus_plus_nat @ A @ C)) = (B = C))))). % add_left_cancel
thf(fact_181_add__left__cancel, axiom,
    ((![A : real, B : real, C : real]: (((plus_plus_real @ A @ B) = (plus_plus_real @ A @ C)) = (B = C))))). % add_left_cancel
thf(fact_182_add__right__cancel, axiom,
    ((![B : complex, A : complex, C : complex]: (((plus_plus_complex @ B @ A) = (plus_plus_complex @ C @ A)) = (B = C))))). % add_right_cancel
thf(fact_183_add__right__cancel, axiom,
    ((![B : nat, A : nat, C : nat]: (((plus_plus_nat @ B @ A) = (plus_plus_nat @ C @ A)) = (B = C))))). % add_right_cancel
thf(fact_184_add__right__cancel, axiom,
    ((![B : real, A : real, C : real]: (((plus_plus_real @ B @ A) = (plus_plus_real @ C @ A)) = (B = C))))). % add_right_cancel
thf(fact_185_semiring__norm_I87_J, axiom,
    ((![M : num, N : num]: (((bit0 @ M) = (bit0 @ N)) = (M = N))))). % semiring_norm(87)
thf(fact_186_mult_Oleft__neutral, axiom,
    ((![A : nat]: ((times_times_nat @ one_one_nat @ A) = A)))). % mult.left_neutral
thf(fact_187_mult_Oleft__neutral, axiom,
    ((![A : complex]: ((times_times_complex @ one_one_complex @ A) = A)))). % mult.left_neutral
thf(fact_188_mult_Oleft__neutral, axiom,
    ((![A : real]: ((times_times_real @ one_one_real @ A) = A)))). % mult.left_neutral
thf(fact_189_mult_Oright__neutral, axiom,
    ((![A : nat]: ((times_times_nat @ A @ one_one_nat) = A)))). % mult.right_neutral
thf(fact_190_mult_Oright__neutral, axiom,
    ((![A : complex]: ((times_times_complex @ A @ one_one_complex) = A)))). % mult.right_neutral
thf(fact_191_mult_Oright__neutral, axiom,
    ((![A : real]: ((times_times_real @ A @ one_one_real) = A)))). % mult.right_neutral
thf(fact_192_power__one, axiom,
    ((![N : nat]: ((power_power_nat @ one_one_nat @ N) = one_one_nat)))). % power_one
thf(fact_193_power__one, axiom,
    ((![N : nat]: ((power_power_complex @ one_one_complex @ N) = one_one_complex)))). % power_one
thf(fact_194_power__one, axiom,
    ((![N : nat]: ((power_power_real @ one_one_real @ N) = one_one_real)))). % power_one
thf(fact_195_power__one__right, axiom,
    ((![A : nat]: ((power_power_nat @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_196_power__one__right, axiom,
    ((![A : complex]: ((power_power_complex @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_197_power__one__right, axiom,
    ((![A : real]: ((power_power_real @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_198_nat__1__eq__mult__iff, axiom,
    ((![M : nat, N : nat]: ((one_one_nat = (times_times_nat @ M @ N)) = (((M = one_one_nat)) & ((N = one_one_nat))))))). % nat_1_eq_mult_iff
thf(fact_199_nat__mult__eq__1__iff, axiom,
    ((![M : nat, N : nat]: (((times_times_nat @ M @ N) = one_one_nat) = (((M = one_one_nat)) & ((N = one_one_nat))))))). % nat_mult_eq_1_iff
thf(fact_200_semiring__norm_I6_J, axiom,
    ((![M : num, N : num]: ((plus_plus_num @ (bit0 @ M) @ (bit0 @ N)) = (bit0 @ (plus_plus_num @ M @ N)))))). % semiring_norm(6)
thf(fact_201_semiring__norm_I13_J, axiom,
    ((![M : num, N : num]: ((times_times_num @ (bit0 @ M) @ (bit0 @ N)) = (bit0 @ (bit0 @ (times_times_num @ M @ N))))))). % semiring_norm(13)
thf(fact_202_semiring__norm_I12_J, axiom,
    ((![N : num]: ((times_times_num @ one @ N) = N)))). % semiring_norm(12)
thf(fact_203_semiring__norm_I78_J, axiom,
    ((![M : num, N : num]: ((ord_less_num @ (bit0 @ M) @ (bit0 @ N)) = (ord_less_num @ M @ N))))). % semiring_norm(78)
thf(fact_204_semiring__norm_I75_J, axiom,
    ((![M : num]: (~ ((ord_less_num @ M @ one)))))). % semiring_norm(75)
thf(fact_205_power__inject__exp, axiom,
    ((![A : real, M : nat, N : nat]: ((ord_less_real @ one_one_real @ A) => (((power_power_real @ A @ M) = (power_power_real @ A @ N)) = (M = N)))))). % power_inject_exp
thf(fact_206_power__inject__exp, axiom,
    ((![A : nat, M : nat, N : nat]: ((ord_less_nat @ one_one_nat @ A) => (((power_power_nat @ A @ M) = (power_power_nat @ A @ N)) = (M = N)))))). % power_inject_exp
thf(fact_207_numeral__eq__one__iff, axiom,
    ((![N : num]: (((numeral_numeral_nat @ N) = one_one_nat) = (N = one))))). % numeral_eq_one_iff
thf(fact_208_numeral__eq__one__iff, axiom,
    ((![N : num]: (((numeral_numeral_real @ N) = one_one_real) = (N = one))))). % numeral_eq_one_iff
thf(fact_209_numeral__eq__one__iff, axiom,
    ((![N : num]: (((numera632737353omplex @ N) = one_one_complex) = (N = one))))). % numeral_eq_one_iff
thf(fact_210_one__eq__numeral__iff, axiom,
    ((![N : num]: ((one_one_nat = (numeral_numeral_nat @ N)) = (one = N))))). % one_eq_numeral_iff
thf(fact_211_one__eq__numeral__iff, axiom,
    ((![N : num]: ((one_one_real = (numeral_numeral_real @ N)) = (one = N))))). % one_eq_numeral_iff
thf(fact_212_one__eq__numeral__iff, axiom,
    ((![N : num]: ((one_one_complex = (numera632737353omplex @ N)) = (one = N))))). % one_eq_numeral_iff
thf(fact_213_semiring__norm_I76_J, axiom,
    ((![N : num]: (ord_less_num @ one @ (bit0 @ N))))). % semiring_norm(76)
thf(fact_214_power__strict__increasing__iff, axiom,
    ((![B : real, X : nat, Y : nat]: ((ord_less_real @ one_one_real @ B) => ((ord_less_real @ (power_power_real @ B @ X) @ (power_power_real @ B @ Y)) = (ord_less_nat @ X @ Y)))))). % power_strict_increasing_iff
thf(fact_215_power__strict__increasing__iff, axiom,
    ((![B : nat, X : nat, Y : nat]: ((ord_less_nat @ one_one_nat @ B) => ((ord_less_nat @ (power_power_nat @ B @ X) @ (power_power_nat @ B @ Y)) = (ord_less_nat @ X @ Y)))))). % power_strict_increasing_iff
thf(fact_216_one__add__one, axiom,
    (((plus_plus_nat @ one_one_nat @ one_one_nat) = (numeral_numeral_nat @ (bit0 @ one))))). % one_add_one
thf(fact_217_one__add__one, axiom,
    (((plus_plus_real @ one_one_real @ one_one_real) = (numeral_numeral_real @ (bit0 @ one))))). % one_add_one
thf(fact_218_one__add__one, axiom,
    (((plus_plus_complex @ one_one_complex @ one_one_complex) = (numera632737353omplex @ (bit0 @ one))))). % one_add_one
thf(fact_219_Suc__1, axiom,
    (((suc @ one_one_nat) = (numeral_numeral_nat @ (bit0 @ one))))). % Suc_1
thf(fact_220_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_221_numeral__plus__one, axiom,
    ((![N : num]: ((plus_plus_real @ (numeral_numeral_real @ N) @ one_one_real) = (numeral_numeral_real @ (plus_plus_num @ N @ one)))))). % numeral_plus_one
thf(fact_222_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_223_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_224_one__plus__numeral, axiom,
    ((![N : num]: ((plus_plus_real @ one_one_real @ (numeral_numeral_real @ N)) = (numeral_numeral_real @ (plus_plus_num @ one @ N)))))). % one_plus_numeral
thf(fact_225_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_226_one__less__numeral__iff, axiom,
    ((![N : num]: ((ord_less_nat @ one_one_nat @ (numeral_numeral_nat @ N)) = (ord_less_num @ one @ N))))). % one_less_numeral_iff
thf(fact_227_one__less__numeral__iff, axiom,
    ((![N : num]: ((ord_less_real @ one_one_real @ (numeral_numeral_real @ N)) = (ord_less_num @ one @ N))))). % one_less_numeral_iff
thf(fact_228_one__reorient, axiom,
    ((![X : nat]: ((one_one_nat = X) = (X = one_one_nat))))). % one_reorient
thf(fact_229_comm__monoid__mult__class_Omult__1, axiom,
    ((![A : nat]: ((times_times_nat @ one_one_nat @ A) = A)))). % comm_monoid_mult_class.mult_1
thf(fact_230_comm__monoid__mult__class_Omult__1, axiom,
    ((![A : complex]: ((times_times_complex @ one_one_complex @ A) = A)))). % comm_monoid_mult_class.mult_1
thf(fact_231_comm__monoid__mult__class_Omult__1, axiom,
    ((![A : real]: ((times_times_real @ one_one_real @ A) = A)))). % comm_monoid_mult_class.mult_1
thf(fact_232_mult_Ocomm__neutral, axiom,
    ((![A : nat]: ((times_times_nat @ A @ one_one_nat) = A)))). % mult.comm_neutral
thf(fact_233_mult_Ocomm__neutral, axiom,
    ((![A : complex]: ((times_times_complex @ A @ one_one_complex) = A)))). % mult.comm_neutral
thf(fact_234_mult_Ocomm__neutral, axiom,
    ((![A : real]: ((times_times_real @ A @ one_one_real) = A)))). % mult.comm_neutral
thf(fact_235_less__numeral__extra_I4_J, axiom,
    ((~ ((ord_less_nat @ one_one_nat @ one_one_nat))))). % less_numeral_extra(4)
thf(fact_236_Suc__eq__plus1__left, axiom,
    ((suc = (plus_plus_nat @ one_one_nat)))). % Suc_eq_plus1_left
thf(fact_237_plus__1__eq__Suc, axiom,
    (((plus_plus_nat @ one_one_nat) = suc))). % plus_1_eq_Suc
thf(fact_238_Suc__eq__plus1, axiom,
    ((suc = (^[N2 : nat]: (plus_plus_nat @ N2 @ one_one_nat))))). % Suc_eq_plus1
thf(fact_239_nat__mult__1, axiom,
    ((![N : nat]: ((times_times_nat @ one_one_nat @ N) = N)))). % nat_mult_1

% Conjectures (2)
thf(conj_0, hypothesis,
    ($true)).
thf(conj_1, conjecture,
    (((fFT_Mirabelle_DFT @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ ka)) @ a @ i) = (plus_plus_complex @ (fFT_Mirabelle_FFT @ ka @ (^[I3 : nat]: (a @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ I3))) @ i) @ (times_times_complex @ (power_power_complex @ (fFT_Mirabelle_root @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ ka))) @ i) @ (fFT_Mirabelle_FFT @ ka @ (^[I3 : nat]: (a @ (suc @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ I3)))) @ i)))))).
