% 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_352__3226042_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:10:38.901

% 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 (29)
thf(sy_c_FFT__Mirabelle__ulikgskiun_Oroot, type,
    fFT_Mirabelle_root : nat > complex).
thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex, type,
    one_one_complex : complex).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat, type,
    one_one_nat : nat).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Complex__Ocomplex, type,
    plus_plus_complex : complex > complex > complex).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat, type,
    plus_plus_nat : nat > nat > nat).
thf(sy_c_Groups_Oplus__class_Oplus_001t__Num__Onum, type,
    plus_plus_num : num > num > num).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Complex__Ocomplex, type,
    times_times_complex : complex > complex > complex).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat, type,
    times_times_nat : nat > nat > nat).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Num__Onum, type,
    times_times_num : num > num > num).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Complex__Ocomplex, type,
    zero_zero_complex : complex).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Complex__Ocomplex, type,
    groups59700922omplex : (nat > complex) > set_nat > complex).
thf(sy_c_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Nat__Onat, type,
    groups1842438620at_nat : (nat > nat) > set_nat > nat).
thf(sy_c_Num_Onum_OBit0, type,
    bit0 : num > num).
thf(sy_c_Num_Onum_OOne, type,
    one : num).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Complex__Ocomplex, type,
    numera632737353omplex : num > complex).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat, type,
    numeral_numeral_nat : num > nat).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat, type,
    ord_less_nat : nat > nat > $o).
thf(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum, type,
    ord_less_num : num > num > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat, type,
    ord_less_eq_nat : nat > nat > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum, type,
    ord_less_eq_num : num > num > $o).
thf(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex, type,
    power_power_complex : complex > nat > complex).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat, type,
    power_power_nat : nat > nat > nat).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex, type,
    divide1210191872omplex : complex > complex > complex).
thf(sy_c_Rings_Odivide__class_Odivide_001t__Nat__Onat, type,
    divide_divide_nat : nat > nat > nat).
thf(sy_c_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat, type,
    set_or562006527an_nat : nat > nat > set_nat).
thf(sy_v_a, type,
    a : nat > complex).
thf(sy_v_i, type,
    i : nat).
thf(sy_v_m, type,
    m : nat).

% Relevant facts (122)
thf(fact_0_mbound, axiom,
    ((ord_less_nat @ zero_zero_nat @ m))). % mbound
thf(fact_1_ibound, axiom,
    ((ord_less_eq_nat @ m @ i))). % ibound
thf(fact_2_sum__splice, axiom,
    ((![F : nat > nat, N : nat]: ((groups1842438620at_nat @ F @ (set_or562006527an_nat @ zero_zero_nat @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N))) = (plus_plus_nat @ (groups1842438620at_nat @ (^[I : nat]: (F @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ I))) @ (set_or562006527an_nat @ zero_zero_nat @ N)) @ (groups1842438620at_nat @ (^[I : nat]: (F @ (plus_plus_nat @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ I) @ one_one_nat))) @ (set_or562006527an_nat @ zero_zero_nat @ N))))))). % sum_splice
thf(fact_3_sum__splice, axiom,
    ((![F : nat > complex, N : nat]: ((groups59700922omplex @ F @ (set_or562006527an_nat @ zero_zero_nat @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N))) = (plus_plus_complex @ (groups59700922omplex @ (^[I : nat]: (F @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ I))) @ (set_or562006527an_nat @ zero_zero_nat @ N)) @ (groups59700922omplex @ (^[I : nat]: (F @ (plus_plus_nat @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ I) @ one_one_nat))) @ (set_or562006527an_nat @ zero_zero_nat @ N))))))). % sum_splice
thf(fact_4_root__cancel1, axiom,
    ((![M : nat, I2 : nat, J : nat]: ((power_power_complex @ (fFT_Mirabelle_root @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ M)) @ (times_times_nat @ I2 @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ J))) = (power_power_complex @ (fFT_Mirabelle_root @ M) @ (times_times_nat @ I2 @ J)))))). % root_cancel1
thf(fact_5_bits__1__div__2, axiom,
    (((divide_divide_nat @ one_one_nat @ (numeral_numeral_nat @ (bit0 @ one))) = zero_zero_nat))). % bits_1_div_2
thf(fact_6_one__div__two__eq__zero, axiom,
    (((divide_divide_nat @ one_one_nat @ (numeral_numeral_nat @ (bit0 @ one))) = zero_zero_nat))). % one_div_two_eq_zero
thf(fact_7_zero__eq__power2, axiom,
    ((![A : complex]: (((power_power_complex @ A @ (numeral_numeral_nat @ (bit0 @ one))) = zero_zero_complex) = (A = zero_zero_complex))))). % zero_eq_power2
thf(fact_8_zero__eq__power2, axiom,
    ((![A : nat]: (((power_power_nat @ A @ (numeral_numeral_nat @ (bit0 @ one))) = zero_zero_nat) = (A = zero_zero_nat))))). % zero_eq_power2
thf(fact_9_one__add__one, axiom,
    (((plus_plus_complex @ one_one_complex @ one_one_complex) = (numera632737353omplex @ (bit0 @ one))))). % one_add_one
thf(fact_10_one__add__one, axiom,
    (((plus_plus_nat @ one_one_nat @ one_one_nat) = (numeral_numeral_nat @ (bit0 @ one))))). % one_add_one
thf(fact_11_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_12_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_13_one__plus__numeral, axiom,
    ((![N : num]: ((plus_plus_nat @ one_one_nat @ (numeral_numeral_nat @ N)) = (numeral_numeral_nat @ (plus_plus_num @ one @ N)))))). % one_plus_numeral
thf(fact_14_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_15_nonzero__divide__mult__cancel__left, axiom,
    ((![A : complex, B : complex]: ((~ ((A = zero_zero_complex))) => ((divide1210191872omplex @ A @ (times_times_complex @ A @ B)) = (divide1210191872omplex @ one_one_complex @ B)))))). % nonzero_divide_mult_cancel_left
thf(fact_16_nonzero__divide__mult__cancel__right, axiom,
    ((![B : complex, A : complex]: ((~ ((B = zero_zero_complex))) => ((divide1210191872omplex @ B @ (times_times_complex @ A @ B)) = (divide1210191872omplex @ one_one_complex @ A)))))). % nonzero_divide_mult_cancel_right
thf(fact_17_div__mult__self1, axiom,
    ((![B : nat, A : nat, C : nat]: ((~ ((B = zero_zero_nat))) => ((divide_divide_nat @ (plus_plus_nat @ A @ (times_times_nat @ C @ B)) @ B) = (plus_plus_nat @ C @ (divide_divide_nat @ A @ B))))))). % div_mult_self1
thf(fact_18_div__mult__self2, axiom,
    ((![B : nat, A : nat, C : nat]: ((~ ((B = zero_zero_nat))) => ((divide_divide_nat @ (plus_plus_nat @ A @ (times_times_nat @ B @ C)) @ B) = (plus_plus_nat @ C @ (divide_divide_nat @ A @ B))))))). % div_mult_self2
thf(fact_19_div__mult__self3, axiom,
    ((![B : nat, C : nat, A : nat]: ((~ ((B = zero_zero_nat))) => ((divide_divide_nat @ (plus_plus_nat @ (times_times_nat @ C @ B) @ A) @ B) = (plus_plus_nat @ C @ (divide_divide_nat @ A @ B))))))). % div_mult_self3
thf(fact_20_numeral__eq__iff, axiom,
    ((![M : num, N : num]: (((numeral_numeral_nat @ M) = (numeral_numeral_nat @ N)) = (M = N))))). % numeral_eq_iff
thf(fact_21_numeral__eq__iff, axiom,
    ((![M : num, N : num]: (((numera632737353omplex @ M) = (numera632737353omplex @ N)) = (M = N))))). % numeral_eq_iff
thf(fact_22_numeral__le__iff, axiom,
    ((![M : num, N : num]: ((ord_less_eq_nat @ (numeral_numeral_nat @ M) @ (numeral_numeral_nat @ N)) = (ord_less_eq_num @ M @ N))))). % numeral_le_iff
thf(fact_23_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_24_division__ring__divide__zero, axiom,
    ((![A : complex]: ((divide1210191872omplex @ A @ zero_zero_complex) = zero_zero_complex)))). % division_ring_divide_zero
thf(fact_25_bits__div__by__0, axiom,
    ((![A : nat]: ((divide_divide_nat @ A @ zero_zero_nat) = zero_zero_nat)))). % bits_div_by_0
thf(fact_26_divide__cancel__right, axiom,
    ((![A : complex, C : complex, B : complex]: (((divide1210191872omplex @ A @ C) = (divide1210191872omplex @ B @ C)) = (((C = zero_zero_complex)) | ((A = B))))))). % divide_cancel_right
thf(fact_27_bits__div__0, axiom,
    ((![A : nat]: ((divide_divide_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % bits_div_0
thf(fact_28_divide__cancel__left, axiom,
    ((![C : complex, A : complex, B : complex]: (((divide1210191872omplex @ C @ A) = (divide1210191872omplex @ C @ B)) = (((C = zero_zero_complex)) | ((A = B))))))). % divide_cancel_left
thf(fact_29_divide__eq__0__iff, axiom,
    ((![A : complex, B : complex]: (((divide1210191872omplex @ A @ B) = zero_zero_complex) = (((A = zero_zero_complex)) | ((B = zero_zero_complex))))))). % divide_eq_0_iff
thf(fact_30_numeral__times__numeral, axiom,
    ((![M : num, N : num]: ((times_times_nat @ (numeral_numeral_nat @ M) @ (numeral_numeral_nat @ N)) = (numeral_numeral_nat @ (times_times_num @ M @ N)))))). % numeral_times_numeral
thf(fact_31_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_32_mult__numeral__left__semiring__numeral, axiom,
    ((![V : num, W : num, Z : nat]: ((times_times_nat @ (numeral_numeral_nat @ V) @ (times_times_nat @ (numeral_numeral_nat @ W) @ Z)) = (times_times_nat @ (numeral_numeral_nat @ (times_times_num @ V @ W)) @ Z))))). % mult_numeral_left_semiring_numeral
thf(fact_33_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_34_times__divide__eq__right, axiom,
    ((![A : complex, B : complex, C : complex]: ((times_times_complex @ A @ (divide1210191872omplex @ B @ C)) = (divide1210191872omplex @ (times_times_complex @ A @ B) @ C))))). % times_divide_eq_right
thf(fact_35_divide__divide__eq__right, axiom,
    ((![A : complex, B : complex, C : complex]: ((divide1210191872omplex @ A @ (divide1210191872omplex @ B @ C)) = (divide1210191872omplex @ (times_times_complex @ A @ C) @ B))))). % divide_divide_eq_right
thf(fact_36_divide__divide__eq__left, axiom,
    ((![A : complex, B : complex, C : complex]: ((divide1210191872omplex @ (divide1210191872omplex @ A @ B) @ C) = (divide1210191872omplex @ A @ (times_times_complex @ B @ C)))))). % divide_divide_eq_left
thf(fact_37_times__divide__eq__left, axiom,
    ((![B : complex, C : complex, A : complex]: ((times_times_complex @ (divide1210191872omplex @ B @ C) @ A) = (divide1210191872omplex @ (times_times_complex @ B @ A) @ C))))). % times_divide_eq_left
thf(fact_38_bits__div__by__1, axiom,
    ((![A : nat]: ((divide_divide_nat @ A @ one_one_nat) = A)))). % bits_div_by_1
thf(fact_39_power__one, axiom,
    ((![N : nat]: ((power_power_complex @ one_one_complex @ N) = one_one_complex)))). % power_one
thf(fact_40_power__one, axiom,
    ((![N : nat]: ((power_power_nat @ one_one_nat @ N) = one_one_nat)))). % power_one
thf(fact_41_num__double, axiom,
    ((![N : num]: ((times_times_num @ (bit0 @ one) @ N) = (bit0 @ N))))). % num_double
thf(fact_42_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_43_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_44_div__less, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => ((divide_divide_nat @ M @ N) = zero_zero_nat))))). % div_less
thf(fact_45_nat__zero__less__power__iff, axiom,
    ((![X : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ (power_power_nat @ X @ N)) = (((ord_less_nat @ zero_zero_nat @ X)) | ((N = zero_zero_nat))))))). % nat_zero_less_power_iff
thf(fact_46_power__one__right, axiom,
    ((![A : complex]: ((power_power_complex @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_47_power__one__right, axiom,
    ((![A : nat]: ((power_power_nat @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_48_root0, axiom,
    (((fFT_Mirabelle_root @ zero_zero_nat) = one_one_complex))). % root0
thf(fact_49_root1, axiom,
    (((fFT_Mirabelle_root @ one_one_nat) = one_one_complex))). % root1
thf(fact_50_div__mult__mult1__if, axiom,
    ((![C : nat, A : nat, B : nat]: (((C = zero_zero_nat) => ((divide_divide_nat @ (times_times_nat @ C @ A) @ (times_times_nat @ C @ B)) = zero_zero_nat)) & ((~ ((C = zero_zero_nat))) => ((divide_divide_nat @ (times_times_nat @ C @ A) @ (times_times_nat @ C @ B)) = (divide_divide_nat @ A @ B))))))). % div_mult_mult1_if
thf(fact_51_div__mult__mult2, axiom,
    ((![C : nat, A : nat, B : nat]: ((~ ((C = zero_zero_nat))) => ((divide_divide_nat @ (times_times_nat @ A @ C) @ (times_times_nat @ B @ C)) = (divide_divide_nat @ A @ B)))))). % div_mult_mult2
thf(fact_52_div__mult__mult1, axiom,
    ((![C : nat, A : nat, B : nat]: ((~ ((C = zero_zero_nat))) => ((divide_divide_nat @ (times_times_nat @ C @ A) @ (times_times_nat @ C @ B)) = (divide_divide_nat @ A @ B)))))). % div_mult_mult1
thf(fact_53_nonzero__mult__divide__mult__cancel__right2, axiom,
    ((![C : complex, A : complex, B : complex]: ((~ ((C = zero_zero_complex))) => ((divide1210191872omplex @ (times_times_complex @ A @ C) @ (times_times_complex @ C @ B)) = (divide1210191872omplex @ A @ B)))))). % nonzero_mult_divide_mult_cancel_right2
thf(fact_54_nonzero__mult__divide__mult__cancel__right, axiom,
    ((![C : complex, A : complex, B : complex]: ((~ ((C = zero_zero_complex))) => ((divide1210191872omplex @ (times_times_complex @ A @ C) @ (times_times_complex @ B @ C)) = (divide1210191872omplex @ A @ B)))))). % nonzero_mult_divide_mult_cancel_right
thf(fact_55_nonzero__mult__divide__mult__cancel__left2, axiom,
    ((![C : complex, A : complex, B : complex]: ((~ ((C = zero_zero_complex))) => ((divide1210191872omplex @ (times_times_complex @ C @ A) @ (times_times_complex @ B @ C)) = (divide1210191872omplex @ A @ B)))))). % nonzero_mult_divide_mult_cancel_left2
thf(fact_56_nonzero__mult__divide__mult__cancel__left, axiom,
    ((![C : complex, A : complex, B : complex]: ((~ ((C = zero_zero_complex))) => ((divide1210191872omplex @ (times_times_complex @ C @ A) @ (times_times_complex @ C @ B)) = (divide1210191872omplex @ A @ B)))))). % nonzero_mult_divide_mult_cancel_left
thf(fact_57_mult__divide__mult__cancel__left__if, axiom,
    ((![C : complex, A : complex, B : complex]: (((C = zero_zero_complex) => ((divide1210191872omplex @ (times_times_complex @ C @ A) @ (times_times_complex @ C @ B)) = zero_zero_complex)) & ((~ ((C = zero_zero_complex))) => ((divide1210191872omplex @ (times_times_complex @ C @ A) @ (times_times_complex @ C @ B)) = (divide1210191872omplex @ A @ B))))))). % mult_divide_mult_cancel_left_if
thf(fact_58_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_59_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_60_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_61_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_62_divide__self__if, axiom,
    ((![A : complex]: (((A = zero_zero_complex) => ((divide1210191872omplex @ A @ A) = zero_zero_complex)) & ((~ ((A = zero_zero_complex))) => ((divide1210191872omplex @ A @ A) = one_one_complex)))))). % divide_self_if
thf(fact_63_divide__self, axiom,
    ((![A : complex]: ((~ ((A = zero_zero_complex))) => ((divide1210191872omplex @ A @ A) = one_one_complex))))). % divide_self
thf(fact_64_one__eq__divide__iff, axiom,
    ((![A : complex, B : complex]: ((one_one_complex = (divide1210191872omplex @ A @ B)) = (((~ ((B = zero_zero_complex)))) & ((A = B))))))). % one_eq_divide_iff
thf(fact_65_divide__eq__1__iff, axiom,
    ((![A : complex, B : complex]: (((divide1210191872omplex @ A @ B) = one_one_complex) = (((~ ((B = zero_zero_complex)))) & ((A = B))))))). % divide_eq_1_iff
thf(fact_66_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_67_one__eq__numeral__iff, axiom,
    ((![N : num]: ((one_one_nat = (numeral_numeral_nat @ N)) = (one = N))))). % one_eq_numeral_iff
thf(fact_68_one__eq__numeral__iff, axiom,
    ((![N : num]: ((one_one_complex = (numera632737353omplex @ N)) = (one = N))))). % one_eq_numeral_iff
thf(fact_69_numeral__eq__one__iff, axiom,
    ((![N : num]: (((numeral_numeral_nat @ N) = one_one_nat) = (N = one))))). % numeral_eq_one_iff
thf(fact_70_numeral__eq__one__iff, axiom,
    ((![N : num]: (((numera632737353omplex @ N) = one_one_complex) = (N = one))))). % numeral_eq_one_iff
thf(fact_71_power__zero__numeral, axiom,
    ((![K : num]: ((power_power_complex @ zero_zero_complex @ (numeral_numeral_nat @ K)) = zero_zero_complex)))). % power_zero_numeral
thf(fact_72_power__zero__numeral, axiom,
    ((![K : num]: ((power_power_nat @ zero_zero_nat @ (numeral_numeral_nat @ K)) = zero_zero_nat)))). % power_zero_numeral
thf(fact_73_div__mult__self1__is__m, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((divide_divide_nat @ (times_times_nat @ N @ M) @ N) = M))))). % div_mult_self1_is_m
thf(fact_74_div__mult__self__is__m, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((divide_divide_nat @ (times_times_nat @ M @ N) @ N) = M))))). % div_mult_self_is_m
thf(fact_75_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_76_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_77_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_78_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_79_eq__divide__eq__numeral1_I1_J, axiom,
    ((![A : complex, B : complex, W : num]: ((A = (divide1210191872omplex @ B @ (numera632737353omplex @ W))) = (((((~ (((numera632737353omplex @ W) = zero_zero_complex)))) => (((times_times_complex @ A @ (numera632737353omplex @ W)) = B)))) & (((((numera632737353omplex @ W) = zero_zero_complex)) => ((A = zero_zero_complex))))))))). % eq_divide_eq_numeral1(1)
thf(fact_80_divide__eq__eq__numeral1_I1_J, axiom,
    ((![B : complex, W : num, A : complex]: (((divide1210191872omplex @ B @ (numera632737353omplex @ W)) = A) = (((((~ (((numera632737353omplex @ W) = zero_zero_complex)))) => ((B = (times_times_complex @ A @ (numera632737353omplex @ W)))))) & (((((numera632737353omplex @ W) = zero_zero_complex)) => ((A = zero_zero_complex))))))))). % divide_eq_eq_numeral1(1)
thf(fact_81_div__mult__self4, axiom,
    ((![B : nat, C : nat, A : nat]: ((~ ((B = zero_zero_nat))) => ((divide_divide_nat @ (plus_plus_nat @ (times_times_nat @ B @ C) @ A) @ B) = (plus_plus_nat @ C @ (divide_divide_nat @ A @ B))))))). % div_mult_self4
thf(fact_82_numeral__le__one__iff, axiom,
    ((![N : num]: ((ord_less_eq_nat @ (numeral_numeral_nat @ N) @ one_one_nat) = (ord_less_eq_num @ N @ one))))). % numeral_le_one_iff
thf(fact_83_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_84_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_85_power__eq__0__iff, axiom,
    ((![A : complex, N : nat]: (((power_power_complex @ A @ N) = zero_zero_complex) = (((A = zero_zero_complex)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % power_eq_0_iff
thf(fact_86_power__eq__0__iff, axiom,
    ((![A : nat, N : nat]: (((power_power_nat @ A @ N) = zero_zero_nat) = (((A = zero_zero_nat)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % power_eq_0_iff
thf(fact_87_add__self__div__2, axiom,
    ((![M : nat]: ((divide_divide_nat @ (plus_plus_nat @ M @ M) @ (numeral_numeral_nat @ (bit0 @ one))) = M)))). % add_self_div_2
thf(fact_88_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_89_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_90_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_91_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_92_power__strict__decreasing__iff, axiom,
    ((![B : nat, M : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ B) => ((ord_less_nat @ B @ one_one_nat) => ((ord_less_nat @ (power_power_nat @ B @ M) @ (power_power_nat @ B @ N)) = (ord_less_nat @ N @ M))))))). % power_strict_decreasing_iff
thf(fact_93_power__increasing__iff, axiom,
    ((![B : nat, X : nat, Y : nat]: ((ord_less_nat @ one_one_nat @ B) => ((ord_less_eq_nat @ (power_power_nat @ B @ X) @ (power_power_nat @ B @ Y)) = (ord_less_eq_nat @ X @ Y)))))). % power_increasing_iff
thf(fact_94_power__mono__iff, axiom,
    ((![A : nat, B : nat, N : nat]: ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_eq_nat @ zero_zero_nat @ B) => ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_eq_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ B @ N)) = (ord_less_eq_nat @ A @ B)))))))). % power_mono_iff
thf(fact_95_power__decreasing__iff, axiom,
    ((![B : nat, M : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ B) => ((ord_less_nat @ B @ one_one_nat) => ((ord_less_eq_nat @ (power_power_nat @ B @ M) @ (power_power_nat @ B @ N)) = (ord_less_eq_nat @ N @ M))))))). % power_decreasing_iff
thf(fact_96_power2__eq__iff__nonneg, axiom,
    ((![X : nat, Y : nat]: ((ord_less_eq_nat @ zero_zero_nat @ X) => ((ord_less_eq_nat @ zero_zero_nat @ Y) => (((power_power_nat @ X @ (numeral_numeral_nat @ (bit0 @ one))) = (power_power_nat @ Y @ (numeral_numeral_nat @ (bit0 @ one)))) = (X = Y))))))). % power2_eq_iff_nonneg
thf(fact_97_nat__power__less__imp__less, axiom,
    ((![I2 : nat, M : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ I2) => ((ord_less_nat @ (power_power_nat @ I2 @ M) @ (power_power_nat @ I2 @ N)) => (ord_less_nat @ M @ N)))))). % nat_power_less_imp_less
thf(fact_98_div__le__mono, axiom,
    ((![M : nat, N : nat, K : nat]: ((ord_less_eq_nat @ M @ N) => (ord_less_eq_nat @ (divide_divide_nat @ M @ K) @ (divide_divide_nat @ N @ K)))))). % div_le_mono
thf(fact_99_Euclidean__Division_Odiv__eq__0__iff, axiom,
    ((![M : nat, N : nat]: (((divide_divide_nat @ M @ N) = zero_zero_nat) = (((ord_less_nat @ M @ N)) | ((N = zero_zero_nat))))))). % Euclidean_Division.div_eq_0_iff
thf(fact_100_div__le__mono2, axiom,
    ((![M : nat, N : nat, K : nat]: ((ord_less_nat @ zero_zero_nat @ M) => ((ord_less_eq_nat @ M @ N) => (ord_less_eq_nat @ (divide_divide_nat @ K @ N) @ (divide_divide_nat @ K @ M))))))). % div_le_mono2
thf(fact_101_div__le__dividend, axiom,
    ((![M : nat, N : nat]: (ord_less_eq_nat @ (divide_divide_nat @ M @ N) @ M)))). % div_le_dividend
thf(fact_102_div__greater__zero__iff, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ (divide_divide_nat @ M @ N)) = (((ord_less_eq_nat @ N @ M)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % div_greater_zero_iff
thf(fact_103_less__mult__imp__div__less, axiom,
    ((![M : nat, I2 : nat, N : nat]: ((ord_less_nat @ M @ (times_times_nat @ I2 @ N)) => (ord_less_nat @ (divide_divide_nat @ M @ N) @ I2))))). % less_mult_imp_div_less
thf(fact_104_div__times__less__eq__dividend, axiom,
    ((![M : nat, N : nat]: (ord_less_eq_nat @ (times_times_nat @ (divide_divide_nat @ M @ N) @ N) @ M)))). % div_times_less_eq_dividend
thf(fact_105_times__div__less__eq__dividend, axiom,
    ((![N : nat, M : nat]: (ord_less_eq_nat @ (times_times_nat @ N @ (divide_divide_nat @ M @ N)) @ M)))). % times_div_less_eq_dividend
thf(fact_106_power__le__imp__le__exp, axiom,
    ((![A : nat, M : nat, N : nat]: ((ord_less_nat @ one_one_nat @ A) => ((ord_less_eq_nat @ (power_power_nat @ A @ M) @ (power_power_nat @ A @ N)) => (ord_less_eq_nat @ M @ N)))))). % power_le_imp_le_exp
thf(fact_107_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_nat @ zero_zero_nat @ zero_zero_nat))))). % less_numeral_extra(3)
thf(fact_108_le__numeral__extra_I3_J, axiom,
    ((ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat))). % le_numeral_extra(3)
thf(fact_109_less__numeral__extra_I4_J, axiom,
    ((~ ((ord_less_nat @ one_one_nat @ one_one_nat))))). % less_numeral_extra(4)
thf(fact_110_add__One__commute, axiom,
    ((![N : num]: ((plus_plus_num @ one @ N) = (plus_plus_num @ N @ one))))). % add_One_commute
thf(fact_111_le__numeral__extra_I4_J, axiom,
    ((ord_less_eq_nat @ one_one_nat @ one_one_nat))). % le_numeral_extra(4)
thf(fact_112_div__mult2__eq, axiom,
    ((![M : nat, N : nat, Q : nat]: ((divide_divide_nat @ M @ (times_times_nat @ N @ Q)) = (divide_divide_nat @ (divide_divide_nat @ M @ N) @ Q))))). % div_mult2_eq
thf(fact_113_power__less__imp__less__base, axiom,
    ((![A : nat, N : nat, B : nat]: ((ord_less_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ B @ N)) => ((ord_less_eq_nat @ zero_zero_nat @ B) => (ord_less_nat @ A @ B)))))). % power_less_imp_less_base
thf(fact_114_power__strict__increasing, axiom,
    ((![N : nat, N2 : nat, A : nat]: ((ord_less_nat @ N @ N2) => ((ord_less_nat @ one_one_nat @ A) => (ord_less_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ A @ N2))))))). % power_strict_increasing
thf(fact_115_power__less__imp__less__exp, axiom,
    ((![A : nat, M : nat, N : nat]: ((ord_less_nat @ one_one_nat @ A) => ((ord_less_nat @ (power_power_nat @ A @ M) @ (power_power_nat @ A @ N)) => (ord_less_nat @ M @ N)))))). % power_less_imp_less_exp
thf(fact_116_power__increasing, axiom,
    ((![N : nat, N2 : nat, A : nat]: ((ord_less_eq_nat @ N @ N2) => ((ord_less_eq_nat @ one_one_nat @ A) => (ord_less_eq_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ A @ N2))))))). % power_increasing
thf(fact_117_div__eq__dividend__iff, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ M) => (((divide_divide_nat @ M @ N) = M) = (N = one_one_nat)))))). % div_eq_dividend_iff
thf(fact_118_div__less__dividend, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ one_one_nat @ N) => ((ord_less_nat @ zero_zero_nat @ M) => (ord_less_nat @ (divide_divide_nat @ M @ N) @ M)))))). % div_less_dividend
thf(fact_119_power__strict__mono, axiom,
    ((![A : nat, B : nat, N : nat]: ((ord_less_nat @ A @ B) => ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_nat @ zero_zero_nat @ N) => (ord_less_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ B @ N)))))))). % power_strict_mono
thf(fact_120_power__strict__decreasing, axiom,
    ((![N : nat, N2 : nat, A : nat]: ((ord_less_nat @ N @ N2) => ((ord_less_nat @ zero_zero_nat @ A) => ((ord_less_nat @ A @ one_one_nat) => (ord_less_nat @ (power_power_nat @ A @ N2) @ (power_power_nat @ A @ N)))))))). % power_strict_decreasing
thf(fact_121_power__decreasing, axiom,
    ((![N : nat, N2 : nat, A : nat]: ((ord_less_eq_nat @ N @ N2) => ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_eq_nat @ A @ one_one_nat) => (ord_less_eq_nat @ (power_power_nat @ A @ N2) @ (power_power_nat @ A @ N)))))))). % power_decreasing

% Conjectures (1)
thf(conj_0, conjecture,
    (((groups59700922omplex @ (^[J2 : nat]: (divide1210191872omplex @ (a @ J2) @ (power_power_complex @ (fFT_Mirabelle_root @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ m)) @ (times_times_nat @ i @ J2)))) @ (set_or562006527an_nat @ zero_zero_nat @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ m))) = (plus_plus_complex @ (groups59700922omplex @ (^[J2 : nat]: (divide1210191872omplex @ (a @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ J2)) @ (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))))) @ (set_or562006527an_nat @ zero_zero_nat @ m)) @ (groups59700922omplex @ (^[J2 : nat]: (divide1210191872omplex @ (a @ (plus_plus_nat @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ J2) @ one_one_nat)) @ (power_power_complex @ (fFT_Mirabelle_root @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ m)) @ (times_times_nat @ i @ (plus_plus_nat @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ J2) @ one_one_nat))))) @ (set_or562006527an_nat @ zero_zero_nat @ m)))))).
