% 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_276__3225426_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:10:09.800

% 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 (25)
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_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 (225)
thf(fact_0_calculation, axiom,
    (((groups59700922omplex @ (^[J : nat]: (times_times_complex @ (power_power_complex @ (fFT_Mirabelle_root @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ m)) @ (times_times_nat @ i @ J)) @ (a @ J))) @ (set_or562006527an_nat @ zero_zero_nat @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ m))) = (plus_plus_complex @ (groups59700922omplex @ (^[J : 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)) @ J))) @ (a @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ J)))) @ (set_or562006527an_nat @ zero_zero_nat @ m)) @ (groups59700922omplex @ (^[J : 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)) @ J) @ one_one_nat))) @ (a @ (plus_plus_nat @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ J) @ one_one_nat)))) @ (set_or562006527an_nat @ zero_zero_nat @ m)))))). % calculation
thf(fact_1_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_2_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_3_root__cancel1, axiom,
    ((![M : nat, I2 : nat, J2 : nat]: ((power_power_complex @ (fFT_Mirabelle_root @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ M)) @ (times_times_nat @ I2 @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ J2))) = (power_power_complex @ (fFT_Mirabelle_root @ M) @ (times_times_nat @ I2 @ J2)))))). % root_cancel1
thf(fact_4_DFT__def, axiom,
    ((fFT_Mirabelle_DFT = (^[N2 : nat]: (^[A : nat > complex]: (^[I : nat]: (groups59700922omplex @ (^[J : nat]: (times_times_complex @ (power_power_complex @ (fFT_Mirabelle_root @ N2) @ (times_times_nat @ I @ J)) @ (A @ J))) @ (set_or562006527an_nat @ zero_zero_nat @ N2)))))))). % DFT_def
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_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_7_one__add__one, axiom,
    (((plus_plus_complex @ one_one_complex @ one_one_complex) = (numera632737353omplex @ (bit0 @ one))))). % one_add_one
thf(fact_8_one__add__one, axiom,
    (((plus_plus_nat @ one_one_nat @ one_one_nat) = (numeral_numeral_nat @ (bit0 @ one))))). % one_add_one
thf(fact_9_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_10_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_11_one__plus__numeral, axiom,
    ((![N : num]: ((plus_plus_complex @ one_one_complex @ (numera632737353omplex @ N)) = (numera632737353omplex @ (plus_plus_num @ one @ N)))))). % one_plus_numeral
thf(fact_12_one__plus__numeral, axiom,
    ((![N : num]: ((plus_plus_nat @ one_one_nat @ (numeral_numeral_nat @ N)) = (numeral_numeral_nat @ (plus_plus_num @ one @ N)))))). % one_plus_numeral
thf(fact_13_power__zero__numeral, axiom,
    ((![K : num]: ((power_power_nat @ zero_zero_nat @ (numeral_numeral_nat @ K)) = zero_zero_nat)))). % power_zero_numeral
thf(fact_14_power__zero__numeral, axiom,
    ((![K : num]: ((power_power_complex @ zero_zero_complex @ (numeral_numeral_nat @ K)) = zero_zero_complex)))). % power_zero_numeral
thf(fact_15_numeral__eq__one__iff, axiom,
    ((![N : num]: (((numeral_numeral_nat @ N) = one_one_nat) = (N = one))))). % numeral_eq_one_iff
thf(fact_16_numeral__eq__one__iff, axiom,
    ((![N : num]: (((numera632737353omplex @ N) = one_one_complex) = (N = one))))). % numeral_eq_one_iff
thf(fact_17_one__eq__numeral__iff, axiom,
    ((![N : num]: ((one_one_nat = (numeral_numeral_nat @ N)) = (one = N))))). % one_eq_numeral_iff
thf(fact_18_one__eq__numeral__iff, axiom,
    ((![N : num]: ((one_one_complex = (numera632737353omplex @ N)) = (one = N))))). % one_eq_numeral_iff
thf(fact_19_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_20_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_21_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_22_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_23_mult__cancel__left1, axiom,
    ((![C : complex, B : complex]: ((C = (times_times_complex @ C @ B)) = (((C = zero_zero_complex)) | ((B = one_one_complex))))))). % mult_cancel_left1
thf(fact_24_numeral__eq__iff, axiom,
    ((![M : num, N : num]: (((numeral_numeral_nat @ M) = (numeral_numeral_nat @ N)) = (M = N))))). % numeral_eq_iff
thf(fact_25_numeral__eq__iff, axiom,
    ((![M : num, N : num]: (((numera632737353omplex @ M) = (numera632737353omplex @ N)) = (M = N))))). % numeral_eq_iff
thf(fact_26_mult__cancel__right, axiom,
    ((![A2 : complex, C : complex, B : complex]: (((times_times_complex @ A2 @ C) = (times_times_complex @ B @ C)) = (((C = zero_zero_complex)) | ((A2 = B))))))). % mult_cancel_right
thf(fact_27_mult__cancel__right, axiom,
    ((![A2 : nat, C : nat, B : nat]: (((times_times_nat @ A2 @ C) = (times_times_nat @ B @ C)) = (((C = zero_zero_nat)) | ((A2 = B))))))). % mult_cancel_right
thf(fact_28_mult__cancel__left, axiom,
    ((![C : complex, A2 : complex, B : complex]: (((times_times_complex @ C @ A2) = (times_times_complex @ C @ B)) = (((C = zero_zero_complex)) | ((A2 = B))))))). % mult_cancel_left
thf(fact_29_mult__cancel__left, axiom,
    ((![C : nat, A2 : nat, B : nat]: (((times_times_nat @ C @ A2) = (times_times_nat @ C @ B)) = (((C = zero_zero_nat)) | ((A2 = B))))))). % mult_cancel_left
thf(fact_30_mult__eq__0__iff, axiom,
    ((![A2 : complex, B : complex]: (((times_times_complex @ A2 @ B) = zero_zero_complex) = (((A2 = zero_zero_complex)) | ((B = zero_zero_complex))))))). % mult_eq_0_iff
thf(fact_31_mult__eq__0__iff, axiom,
    ((![A2 : nat, B : nat]: (((times_times_nat @ A2 @ B) = zero_zero_nat) = (((A2 = zero_zero_nat)) | ((B = zero_zero_nat))))))). % mult_eq_0_iff
thf(fact_32_mult__zero__right, axiom,
    ((![A2 : complex]: ((times_times_complex @ A2 @ zero_zero_complex) = zero_zero_complex)))). % mult_zero_right
thf(fact_33_mult__zero__right, axiom,
    ((![A2 : nat]: ((times_times_nat @ A2 @ zero_zero_nat) = zero_zero_nat)))). % mult_zero_right
thf(fact_34_mult__zero__left, axiom,
    ((![A2 : complex]: ((times_times_complex @ zero_zero_complex @ A2) = zero_zero_complex)))). % mult_zero_left
thf(fact_35_mult__zero__left, axiom,
    ((![A2 : nat]: ((times_times_nat @ zero_zero_nat @ A2) = zero_zero_nat)))). % mult_zero_left
thf(fact_36_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_37_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_38_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_39_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_40_power__one, axiom,
    ((![N : nat]: ((power_power_complex @ one_one_complex @ N) = one_one_complex)))). % power_one
thf(fact_41_power__one, axiom,
    ((![N : nat]: ((power_power_nat @ one_one_nat @ N) = one_one_nat)))). % power_one
thf(fact_42_num__double, axiom,
    ((![N : num]: ((times_times_num @ (bit0 @ one) @ N) = (bit0 @ N))))). % num_double
thf(fact_43_power__mult__numeral, axiom,
    ((![A2 : complex, M : num, N : num]: ((power_power_complex @ (power_power_complex @ A2 @ (numeral_numeral_nat @ M)) @ (numeral_numeral_nat @ N)) = (power_power_complex @ A2 @ (numeral_numeral_nat @ (times_times_num @ M @ N))))))). % power_mult_numeral
thf(fact_44_power__mult__numeral, axiom,
    ((![A2 : nat, M : num, N : num]: ((power_power_nat @ (power_power_nat @ A2 @ (numeral_numeral_nat @ M)) @ (numeral_numeral_nat @ N)) = (power_power_nat @ A2 @ (numeral_numeral_nat @ (times_times_num @ M @ N))))))). % power_mult_numeral
thf(fact_45_power__one__right, axiom,
    ((![A2 : complex]: ((power_power_complex @ A2 @ one_one_nat) = A2)))). % power_one_right
thf(fact_46_power__one__right, axiom,
    ((![A2 : nat]: ((power_power_nat @ A2 @ one_one_nat) = A2)))). % power_one_right
thf(fact_47_root0, axiom,
    (((fFT_Mirabelle_root @ zero_zero_nat) = one_one_complex))). % root0
thf(fact_48_root1, axiom,
    (((fFT_Mirabelle_root @ one_one_nat) = one_one_complex))). % root1
thf(fact_49_mult__cancel__right2, axiom,
    ((![A2 : complex, C : complex]: (((times_times_complex @ A2 @ C) = C) = (((C = zero_zero_complex)) | ((A2 = one_one_complex))))))). % mult_cancel_right2
thf(fact_50_mult__cancel__right1, axiom,
    ((![C : complex, B : complex]: ((C = (times_times_complex @ B @ C)) = (((C = zero_zero_complex)) | ((B = one_one_complex))))))). % mult_cancel_right1
thf(fact_51_mult__cancel__left2, axiom,
    ((![C : complex, A2 : complex]: (((times_times_complex @ C @ A2) = C) = (((C = zero_zero_complex)) | ((A2 = one_one_complex))))))). % mult_cancel_left2
thf(fact_52_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_53_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_54_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_55_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_56_power__add__numeral2, axiom,
    ((![A2 : complex, M : num, N : num, B : complex]: ((times_times_complex @ (power_power_complex @ A2 @ (numeral_numeral_nat @ M)) @ (times_times_complex @ (power_power_complex @ A2 @ (numeral_numeral_nat @ N)) @ B)) = (times_times_complex @ (power_power_complex @ A2 @ (numeral_numeral_nat @ (plus_plus_num @ M @ N))) @ B))))). % power_add_numeral2
thf(fact_57_power__add__numeral2, axiom,
    ((![A2 : nat, M : num, N : num, B : nat]: ((times_times_nat @ (power_power_nat @ A2 @ (numeral_numeral_nat @ M)) @ (times_times_nat @ (power_power_nat @ A2 @ (numeral_numeral_nat @ N)) @ B)) = (times_times_nat @ (power_power_nat @ A2 @ (numeral_numeral_nat @ (plus_plus_num @ M @ N))) @ B))))). % power_add_numeral2
thf(fact_58_power__add__numeral, axiom,
    ((![A2 : complex, M : num, N : num]: ((times_times_complex @ (power_power_complex @ A2 @ (numeral_numeral_nat @ M)) @ (power_power_complex @ A2 @ (numeral_numeral_nat @ N))) = (power_power_complex @ A2 @ (numeral_numeral_nat @ (plus_plus_num @ M @ N))))))). % power_add_numeral
thf(fact_59_power__add__numeral, axiom,
    ((![A2 : nat, M : num, N : num]: ((times_times_nat @ (power_power_nat @ A2 @ (numeral_numeral_nat @ M)) @ (power_power_nat @ A2 @ (numeral_numeral_nat @ N))) = (power_power_nat @ A2 @ (numeral_numeral_nat @ (plus_plus_num @ M @ N))))))). % power_add_numeral
thf(fact_60_add__One__commute, axiom,
    ((![N : num]: ((plus_plus_num @ one @ N) = (plus_plus_num @ N @ one))))). % add_One_commute
thf(fact_61_root__nonzero, axiom,
    ((![N : nat]: (~ (((fFT_Mirabelle_root @ N) = zero_zero_complex)))))). % root_nonzero
thf(fact_62_root__unity, axiom,
    ((![N : nat]: ((power_power_complex @ (fFT_Mirabelle_root @ N) @ N) = one_one_complex)))). % root_unity
thf(fact_63_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_64_mult__right__cancel, axiom,
    ((![C : complex, A2 : complex, B : complex]: ((~ ((C = zero_zero_complex))) => (((times_times_complex @ A2 @ C) = (times_times_complex @ B @ C)) = (A2 = B)))))). % mult_right_cancel
thf(fact_65_mult__right__cancel, axiom,
    ((![C : nat, A2 : nat, B : nat]: ((~ ((C = zero_zero_nat))) => (((times_times_nat @ A2 @ C) = (times_times_nat @ B @ C)) = (A2 = B)))))). % mult_right_cancel
thf(fact_66_mult__left__cancel, axiom,
    ((![C : complex, A2 : complex, B : complex]: ((~ ((C = zero_zero_complex))) => (((times_times_complex @ C @ A2) = (times_times_complex @ C @ B)) = (A2 = B)))))). % mult_left_cancel
thf(fact_67_mult__left__cancel, axiom,
    ((![C : nat, A2 : nat, B : nat]: ((~ ((C = zero_zero_nat))) => (((times_times_nat @ C @ A2) = (times_times_nat @ C @ B)) = (A2 = B)))))). % mult_left_cancel
thf(fact_68_no__zero__divisors, axiom,
    ((![A2 : complex, B : complex]: ((~ ((A2 = zero_zero_complex))) => ((~ ((B = zero_zero_complex))) => (~ (((times_times_complex @ A2 @ B) = zero_zero_complex)))))))). % no_zero_divisors
thf(fact_69_no__zero__divisors, axiom,
    ((![A2 : nat, B : nat]: ((~ ((A2 = zero_zero_nat))) => ((~ ((B = zero_zero_nat))) => (~ (((times_times_nat @ A2 @ B) = zero_zero_nat)))))))). % no_zero_divisors
thf(fact_70_divisors__zero, axiom,
    ((![A2 : complex, B : complex]: (((times_times_complex @ A2 @ B) = zero_zero_complex) => ((A2 = zero_zero_complex) | (B = zero_zero_complex)))))). % divisors_zero
thf(fact_71_divisors__zero, axiom,
    ((![A2 : nat, B : nat]: (((times_times_nat @ A2 @ B) = zero_zero_nat) => ((A2 = zero_zero_nat) | (B = zero_zero_nat)))))). % divisors_zero
thf(fact_72_mult__not__zero, axiom,
    ((![A2 : complex, B : complex]: ((~ (((times_times_complex @ A2 @ B) = zero_zero_complex))) => ((~ ((A2 = zero_zero_complex))) & (~ ((B = zero_zero_complex)))))))). % mult_not_zero
thf(fact_73_mult__not__zero, axiom,
    ((![A2 : nat, B : nat]: ((~ (((times_times_nat @ A2 @ B) = zero_zero_nat))) => ((~ ((A2 = zero_zero_nat))) & (~ ((B = zero_zero_nat)))))))). % mult_not_zero
thf(fact_74_zero__neq__numeral, axiom,
    ((![N : num]: (~ ((zero_zero_nat = (numeral_numeral_nat @ N))))))). % zero_neq_numeral
thf(fact_75_zero__neq__numeral, axiom,
    ((![N : num]: (~ ((zero_zero_complex = (numera632737353omplex @ N))))))). % zero_neq_numeral
thf(fact_76_zero__neq__one, axiom,
    ((~ ((zero_zero_nat = one_one_nat))))). % zero_neq_one
thf(fact_77_zero__neq__one, axiom,
    ((~ ((zero_zero_complex = one_one_complex))))). % zero_neq_one
thf(fact_78_combine__common__factor, axiom,
    ((![A2 : complex, E : complex, B : complex, C : complex]: ((plus_plus_complex @ (times_times_complex @ A2 @ E) @ (plus_plus_complex @ (times_times_complex @ B @ E) @ C)) = (plus_plus_complex @ (times_times_complex @ (plus_plus_complex @ A2 @ B) @ E) @ C))))). % combine_common_factor
thf(fact_79_combine__common__factor, axiom,
    ((![A2 : nat, E : nat, B : nat, C : nat]: ((plus_plus_nat @ (times_times_nat @ A2 @ E) @ (plus_plus_nat @ (times_times_nat @ B @ E) @ C)) = (plus_plus_nat @ (times_times_nat @ (plus_plus_nat @ A2 @ B) @ E) @ C))))). % combine_common_factor
thf(fact_80_distrib__right, axiom,
    ((![A2 : complex, B : complex, C : complex]: ((times_times_complex @ (plus_plus_complex @ A2 @ B) @ C) = (plus_plus_complex @ (times_times_complex @ A2 @ C) @ (times_times_complex @ B @ C)))))). % distrib_right
thf(fact_81_distrib__right, axiom,
    ((![A2 : nat, B : nat, C : nat]: ((times_times_nat @ (plus_plus_nat @ A2 @ B) @ C) = (plus_plus_nat @ (times_times_nat @ A2 @ C) @ (times_times_nat @ B @ C)))))). % distrib_right
thf(fact_82_distrib__left, axiom,
    ((![A2 : complex, B : complex, C : complex]: ((times_times_complex @ A2 @ (plus_plus_complex @ B @ C)) = (plus_plus_complex @ (times_times_complex @ A2 @ B) @ (times_times_complex @ A2 @ C)))))). % distrib_left
thf(fact_83_distrib__left, axiom,
    ((![A2 : nat, B : nat, C : nat]: ((times_times_nat @ A2 @ (plus_plus_nat @ B @ C)) = (plus_plus_nat @ (times_times_nat @ A2 @ B) @ (times_times_nat @ A2 @ C)))))). % distrib_left
thf(fact_84_comm__semiring__class_Odistrib, axiom,
    ((![A2 : complex, B : complex, C : complex]: ((times_times_complex @ (plus_plus_complex @ A2 @ B) @ C) = (plus_plus_complex @ (times_times_complex @ A2 @ C) @ (times_times_complex @ B @ C)))))). % comm_semiring_class.distrib
thf(fact_85_comm__semiring__class_Odistrib, axiom,
    ((![A2 : nat, B : nat, C : nat]: ((times_times_nat @ (plus_plus_nat @ A2 @ B) @ C) = (plus_plus_nat @ (times_times_nat @ A2 @ C) @ (times_times_nat @ B @ C)))))). % comm_semiring_class.distrib
thf(fact_86_ring__class_Oring__distribs_I1_J, axiom,
    ((![A2 : complex, B : complex, C : complex]: ((times_times_complex @ A2 @ (plus_plus_complex @ B @ C)) = (plus_plus_complex @ (times_times_complex @ A2 @ B) @ (times_times_complex @ A2 @ C)))))). % ring_class.ring_distribs(1)
thf(fact_87_ring__class_Oring__distribs_I2_J, axiom,
    ((![A2 : complex, B : complex, C : complex]: ((times_times_complex @ (plus_plus_complex @ A2 @ B) @ C) = (plus_plus_complex @ (times_times_complex @ A2 @ C) @ (times_times_complex @ B @ C)))))). % ring_class.ring_distribs(2)
thf(fact_88_power__not__zero, axiom,
    ((![A2 : complex, N : nat]: ((~ ((A2 = zero_zero_complex))) => (~ (((power_power_complex @ A2 @ N) = zero_zero_complex))))))). % power_not_zero
thf(fact_89_power__not__zero, axiom,
    ((![A2 : nat, N : nat]: ((~ ((A2 = zero_zero_nat))) => (~ (((power_power_nat @ A2 @ N) = zero_zero_nat))))))). % power_not_zero
thf(fact_90_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_91_power__commuting__commutes, axiom,
    ((![X : nat, Y : nat, N : nat]: (((times_times_nat @ X @ Y) = (times_times_nat @ Y @ X)) => ((times_times_nat @ (power_power_nat @ X @ N) @ Y) = (times_times_nat @ Y @ (power_power_nat @ X @ N))))))). % power_commuting_commutes
thf(fact_92_power__mult__distrib, axiom,
    ((![A2 : complex, B : complex, N : nat]: ((power_power_complex @ (times_times_complex @ A2 @ B) @ N) = (times_times_complex @ (power_power_complex @ A2 @ N) @ (power_power_complex @ B @ N)))))). % power_mult_distrib
thf(fact_93_power__mult__distrib, axiom,
    ((![A2 : nat, B : nat, N : nat]: ((power_power_nat @ (times_times_nat @ A2 @ B) @ N) = (times_times_nat @ (power_power_nat @ A2 @ N) @ (power_power_nat @ B @ N)))))). % power_mult_distrib
thf(fact_94_power__commutes, axiom,
    ((![A2 : complex, N : nat]: ((times_times_complex @ (power_power_complex @ A2 @ N) @ A2) = (times_times_complex @ A2 @ (power_power_complex @ A2 @ N)))))). % power_commutes
thf(fact_95_power__commutes, axiom,
    ((![A2 : nat, N : nat]: ((times_times_nat @ (power_power_nat @ A2 @ N) @ A2) = (times_times_nat @ A2 @ (power_power_nat @ A2 @ N)))))). % power_commutes
thf(fact_96_power__mult, axiom,
    ((![A2 : complex, M : nat, N : nat]: ((power_power_complex @ A2 @ (times_times_nat @ M @ N)) = (power_power_complex @ (power_power_complex @ A2 @ M) @ N))))). % power_mult
thf(fact_97_power__mult, axiom,
    ((![A2 : nat, M : nat, N : nat]: ((power_power_nat @ A2 @ (times_times_nat @ M @ N)) = (power_power_nat @ (power_power_nat @ A2 @ M) @ N))))). % power_mult
thf(fact_98_lambda__zero, axiom,
    (((^[H : complex]: zero_zero_complex) = (times_times_complex @ zero_zero_complex)))). % lambda_zero
thf(fact_99_lambda__zero, axiom,
    (((^[H : nat]: zero_zero_nat) = (times_times_nat @ zero_zero_nat)))). % lambda_zero
thf(fact_100_lambda__one, axiom,
    (((^[X2 : complex]: X2) = (times_times_complex @ one_one_complex)))). % lambda_one
thf(fact_101_lambda__one, axiom,
    (((^[X2 : nat]: X2) = (times_times_nat @ one_one_nat)))). % lambda_one
thf(fact_102_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_103_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_104_mult__numeral__1__right, axiom,
    ((![A2 : nat]: ((times_times_nat @ A2 @ (numeral_numeral_nat @ one)) = A2)))). % mult_numeral_1_right
thf(fact_105_mult__numeral__1__right, axiom,
    ((![A2 : complex]: ((times_times_complex @ A2 @ (numera632737353omplex @ one)) = A2)))). % mult_numeral_1_right
thf(fact_106_mult__numeral__1, axiom,
    ((![A2 : nat]: ((times_times_nat @ (numeral_numeral_nat @ one) @ A2) = A2)))). % mult_numeral_1
thf(fact_107_mult__numeral__1, axiom,
    ((![A2 : complex]: ((times_times_complex @ (numera632737353omplex @ one) @ A2) = A2)))). % mult_numeral_1
thf(fact_108_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_109_numeral__Bit0, axiom,
    ((![N : num]: ((numera632737353omplex @ (bit0 @ N)) = (plus_plus_complex @ (numera632737353omplex @ N) @ (numera632737353omplex @ N)))))). % numeral_Bit0
thf(fact_110_numeral__One, axiom,
    (((numeral_numeral_nat @ one) = one_one_nat))). % numeral_One
thf(fact_111_numeral__One, axiom,
    (((numera632737353omplex @ one) = one_one_complex))). % numeral_One
thf(fact_112_left__right__inverse__power, axiom,
    ((![X : complex, Y : complex, N : nat]: (((times_times_complex @ X @ Y) = one_one_complex) => ((times_times_complex @ (power_power_complex @ X @ N) @ (power_power_complex @ Y @ N)) = one_one_complex))))). % left_right_inverse_power
thf(fact_113_left__right__inverse__power, axiom,
    ((![X : nat, Y : nat, N : nat]: (((times_times_nat @ X @ Y) = one_one_nat) => ((times_times_nat @ (power_power_nat @ X @ N) @ (power_power_nat @ Y @ N)) = one_one_nat))))). % left_right_inverse_power
thf(fact_114_power__0, axiom,
    ((![A2 : complex]: ((power_power_complex @ A2 @ zero_zero_nat) = one_one_complex)))). % power_0
thf(fact_115_power__0, axiom,
    ((![A2 : nat]: ((power_power_nat @ A2 @ zero_zero_nat) = one_one_nat)))). % power_0
thf(fact_116_power__add, axiom,
    ((![A2 : complex, M : nat, N : nat]: ((power_power_complex @ A2 @ (plus_plus_nat @ M @ N)) = (times_times_complex @ (power_power_complex @ A2 @ M) @ (power_power_complex @ A2 @ N)))))). % power_add
thf(fact_117_power__add, axiom,
    ((![A2 : nat, M : nat, N : nat]: ((power_power_nat @ A2 @ (plus_plus_nat @ M @ N)) = (times_times_nat @ (power_power_nat @ A2 @ M) @ (power_power_nat @ A2 @ N)))))). % power_add
thf(fact_118_numerals_I1_J, axiom,
    (((numeral_numeral_nat @ one) = one_one_nat))). % numerals(1)
thf(fact_119_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_120_numeral__code_I2_J, axiom,
    ((![N : num]: ((numera632737353omplex @ (bit0 @ N)) = (plus_plus_complex @ (numera632737353omplex @ N) @ (numera632737353omplex @ N)))))). % numeral_code(2)
thf(fact_121_power__0__left, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((power_power_complex @ zero_zero_complex @ N) = one_one_complex)) & ((~ ((N = zero_zero_nat))) => ((power_power_complex @ zero_zero_complex @ N) = zero_zero_complex)))))). % power_0_left
thf(fact_122_power__0__left, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((power_power_nat @ zero_zero_nat @ N) = one_one_nat)) & ((~ ((N = zero_zero_nat))) => ((power_power_nat @ zero_zero_nat @ N) = zero_zero_nat)))))). % power_0_left
thf(fact_123_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_124_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_125_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_126_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_127_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_128_mult__2__right, axiom,
    ((![Z : complex]: ((times_times_complex @ Z @ (numera632737353omplex @ (bit0 @ one))) = (plus_plus_complex @ Z @ Z))))). % mult_2_right
thf(fact_129_mult__2, axiom,
    ((![Z : nat]: ((times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ Z) = (plus_plus_nat @ Z @ Z))))). % mult_2
thf(fact_130_mult__2, axiom,
    ((![Z : complex]: ((times_times_complex @ (numera632737353omplex @ (bit0 @ one)) @ Z) = (plus_plus_complex @ Z @ Z))))). % mult_2
thf(fact_131_zero__power2, axiom,
    (((power_power_complex @ zero_zero_complex @ (numeral_numeral_nat @ (bit0 @ one))) = zero_zero_complex))). % zero_power2
thf(fact_132_zero__power2, axiom,
    (((power_power_nat @ zero_zero_nat @ (numeral_numeral_nat @ (bit0 @ one))) = zero_zero_nat))). % zero_power2
thf(fact_133_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_134_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_135_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_136_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_137_one__power2, axiom,
    (((power_power_complex @ one_one_complex @ (numeral_numeral_nat @ (bit0 @ one))) = one_one_complex))). % one_power2
thf(fact_138_one__power2, axiom,
    (((power_power_nat @ one_one_nat @ (numeral_numeral_nat @ (bit0 @ one))) = one_one_nat))). % one_power2
thf(fact_139_power__even__eq, axiom,
    ((![A2 : complex, N : nat]: ((power_power_complex @ A2 @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N)) = (power_power_complex @ (power_power_complex @ A2 @ N) @ (numeral_numeral_nat @ (bit0 @ one))))))). % power_even_eq
thf(fact_140_power__even__eq, axiom,
    ((![A2 : nat, N : nat]: ((power_power_nat @ A2 @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N)) = (power_power_nat @ (power_power_nat @ A2 @ N) @ (numeral_numeral_nat @ (bit0 @ one))))))). % power_even_eq
thf(fact_141_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_142_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_143_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_144_semiring__norm_I2_J, axiom,
    (((plus_plus_num @ one @ one) = (bit0 @ one)))). % semiring_norm(2)
thf(fact_145_nat__induct2, axiom,
    ((![P : nat > $o, N : nat]: ((P @ zero_zero_nat) => ((P @ one_one_nat) => ((![N3 : nat]: ((P @ N3) => (P @ (plus_plus_nat @ N3 @ (numeral_numeral_nat @ (bit0 @ one)))))) => (P @ N))))))). % nat_induct2
thf(fact_146_exp__add__not__zero__imp__right, axiom,
    ((![M : nat, N : nat]: ((~ (((power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (plus_plus_nat @ M @ N)) = zero_zero_nat))) => (~ (((power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N) = zero_zero_nat))))))). % exp_add_not_zero_imp_right
thf(fact_147_exp__add__not__zero__imp__left, axiom,
    ((![M : nat, N : nat]: ((~ (((power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ (plus_plus_nat @ M @ N)) = zero_zero_nat))) => (~ (((power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ M) = zero_zero_nat))))))). % exp_add_not_zero_imp_left
thf(fact_148_sum_Oneutral__const, axiom,
    ((![A3 : set_nat]: ((groups59700922omplex @ (^[Uu : nat]: zero_zero_complex) @ A3) = zero_zero_complex)))). % sum.neutral_const
thf(fact_149_sum_Oneutral__const, axiom,
    ((![A3 : set_nat]: ((groups1842438620at_nat @ (^[Uu : nat]: zero_zero_nat) @ A3) = zero_zero_nat)))). % sum.neutral_const
thf(fact_150_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_151_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_152_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_153_mult__cancel2, axiom,
    ((![M : nat, K : nat, N : nat]: (((times_times_nat @ M @ K) = (times_times_nat @ N @ K)) = (((M = N)) | ((K = zero_zero_nat))))))). % mult_cancel2
thf(fact_154_mult__cancel1, axiom,
    ((![K : nat, M : nat, N : nat]: (((times_times_nat @ K @ M) = (times_times_nat @ K @ N)) = (((M = N)) | ((K = zero_zero_nat))))))). % mult_cancel1
thf(fact_155_semiring__norm_I87_J, axiom,
    ((![M : num, N : num]: (((bit0 @ M) = (bit0 @ N)) = (M = N))))). % semiring_norm(87)
thf(fact_156_semiring__norm_I85_J, axiom,
    ((![M : num]: (~ (((bit0 @ M) = one)))))). % semiring_norm(85)
thf(fact_157_semiring__norm_I83_J, axiom,
    ((![N : num]: (~ ((one = (bit0 @ N))))))). % semiring_norm(83)
thf(fact_158_add__is__0, axiom,
    ((![M : nat, N : nat]: (((plus_plus_nat @ M @ N) = zero_zero_nat) = (((M = zero_zero_nat)) & ((N = zero_zero_nat))))))). % add_is_0
thf(fact_159_Nat_Oadd__0__right, axiom,
    ((![M : nat]: ((plus_plus_nat @ M @ zero_zero_nat) = M)))). % Nat.add_0_right
thf(fact_160_mult__is__0, axiom,
    ((![M : nat, N : nat]: (((times_times_nat @ M @ N) = zero_zero_nat) = (((M = zero_zero_nat)) | ((N = zero_zero_nat))))))). % mult_is_0
thf(fact_161_mult__0__right, axiom,
    ((![M : nat]: ((times_times_nat @ M @ zero_zero_nat) = zero_zero_nat)))). % mult_0_right
thf(fact_162_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_163_semiring__norm_I12_J, axiom,
    ((![N : num]: ((times_times_num @ one @ N) = N)))). % semiring_norm(12)
thf(fact_164_semiring__norm_I11_J, axiom,
    ((![M : num]: ((times_times_num @ M @ one) = M)))). % semiring_norm(11)
thf(fact_165_sum_Ocong, axiom,
    ((![A3 : set_nat, B2 : set_nat, G : nat > complex, H2 : nat > complex]: ((A3 = B2) => ((![X3 : nat]: ((member_nat @ X3 @ B2) => ((G @ X3) = (H2 @ X3)))) => ((groups59700922omplex @ G @ A3) = (groups59700922omplex @ H2 @ B2))))))). % sum.cong
thf(fact_166_sum_Ocong, axiom,
    ((![A3 : set_nat, B2 : set_nat, G : nat > nat, H2 : nat > nat]: ((A3 = B2) => ((![X3 : nat]: ((member_nat @ X3 @ B2) => ((G @ X3) = (H2 @ X3)))) => ((groups1842438620at_nat @ G @ A3) = (groups1842438620at_nat @ H2 @ B2))))))). % sum.cong
thf(fact_167_sum_Oeq__general, axiom,
    ((![B2 : set_nat, A3 : set_nat, H2 : nat > nat, Gamma : nat > complex, Phi : nat > complex]: ((![Y2 : nat]: ((member_nat @ Y2 @ B2) => (?[X4 : nat]: (((member_nat @ X4 @ A3) & ((H2 @ X4) = Y2)) & (![Ya : nat]: (((member_nat @ Ya @ A3) & ((H2 @ Ya) = Y2)) => (Ya = X4))))))) => ((![X3 : nat]: ((member_nat @ X3 @ A3) => ((member_nat @ (H2 @ X3) @ B2) & ((Gamma @ (H2 @ X3)) = (Phi @ X3))))) => ((groups59700922omplex @ Phi @ A3) = (groups59700922omplex @ Gamma @ B2))))))). % sum.eq_general
thf(fact_168_sum_Oeq__general, axiom,
    ((![B2 : set_nat, A3 : set_nat, H2 : nat > nat, Gamma : nat > nat, Phi : nat > nat]: ((![Y2 : nat]: ((member_nat @ Y2 @ B2) => (?[X4 : nat]: (((member_nat @ X4 @ A3) & ((H2 @ X4) = Y2)) & (![Ya : nat]: (((member_nat @ Ya @ A3) & ((H2 @ Ya) = Y2)) => (Ya = X4))))))) => ((![X3 : nat]: ((member_nat @ X3 @ A3) => ((member_nat @ (H2 @ X3) @ B2) & ((Gamma @ (H2 @ X3)) = (Phi @ X3))))) => ((groups1842438620at_nat @ Phi @ A3) = (groups1842438620at_nat @ Gamma @ B2))))))). % sum.eq_general
thf(fact_169_sum_Oeq__general__inverses, axiom,
    ((![B2 : set_nat, K : nat > nat, A3 : set_nat, H2 : nat > nat, Gamma : nat > complex, Phi : nat > complex]: ((![Y2 : nat]: ((member_nat @ Y2 @ B2) => ((member_nat @ (K @ Y2) @ A3) & ((H2 @ (K @ Y2)) = Y2)))) => ((![X3 : nat]: ((member_nat @ X3 @ A3) => ((member_nat @ (H2 @ X3) @ B2) & (((K @ (H2 @ X3)) = X3) & ((Gamma @ (H2 @ X3)) = (Phi @ X3)))))) => ((groups59700922omplex @ Phi @ A3) = (groups59700922omplex @ Gamma @ B2))))))). % sum.eq_general_inverses
thf(fact_170_sum_Oeq__general__inverses, axiom,
    ((![B2 : set_nat, K : nat > nat, A3 : set_nat, H2 : nat > nat, Gamma : nat > nat, Phi : nat > nat]: ((![Y2 : nat]: ((member_nat @ Y2 @ B2) => ((member_nat @ (K @ Y2) @ A3) & ((H2 @ (K @ Y2)) = Y2)))) => ((![X3 : nat]: ((member_nat @ X3 @ A3) => ((member_nat @ (H2 @ X3) @ B2) & (((K @ (H2 @ X3)) = X3) & ((Gamma @ (H2 @ X3)) = (Phi @ X3)))))) => ((groups1842438620at_nat @ Phi @ A3) = (groups1842438620at_nat @ Gamma @ B2))))))). % sum.eq_general_inverses
thf(fact_171_sum_Oreindex__bij__witness, axiom,
    ((![S : set_nat, I2 : nat > nat, J2 : nat > nat, T : set_nat, H2 : nat > complex, G : nat > complex]: ((![A4 : nat]: ((member_nat @ A4 @ S) => ((I2 @ (J2 @ A4)) = A4))) => ((![A4 : nat]: ((member_nat @ A4 @ S) => (member_nat @ (J2 @ A4) @ T))) => ((![B3 : nat]: ((member_nat @ B3 @ T) => ((J2 @ (I2 @ B3)) = B3))) => ((![B3 : nat]: ((member_nat @ B3 @ T) => (member_nat @ (I2 @ B3) @ S))) => ((![A4 : nat]: ((member_nat @ A4 @ S) => ((H2 @ (J2 @ A4)) = (G @ A4)))) => ((groups59700922omplex @ G @ S) = (groups59700922omplex @ H2 @ T)))))))))). % sum.reindex_bij_witness
thf(fact_172_sum_Oreindex__bij__witness, axiom,
    ((![S : set_nat, I2 : nat > nat, J2 : nat > nat, T : set_nat, H2 : nat > nat, G : nat > nat]: ((![A4 : nat]: ((member_nat @ A4 @ S) => ((I2 @ (J2 @ A4)) = A4))) => ((![A4 : nat]: ((member_nat @ A4 @ S) => (member_nat @ (J2 @ A4) @ T))) => ((![B3 : nat]: ((member_nat @ B3 @ T) => ((J2 @ (I2 @ B3)) = B3))) => ((![B3 : nat]: ((member_nat @ B3 @ T) => (member_nat @ (I2 @ B3) @ S))) => ((![A4 : nat]: ((member_nat @ A4 @ S) => ((H2 @ (J2 @ A4)) = (G @ A4)))) => ((groups1842438620at_nat @ G @ S) = (groups1842438620at_nat @ H2 @ T)))))))))). % sum.reindex_bij_witness
thf(fact_173_sum_Oswap, axiom,
    ((![G : nat > nat > complex, B2 : set_nat, A3 : set_nat]: ((groups59700922omplex @ (^[I : nat]: (groups59700922omplex @ (G @ I) @ B2)) @ A3) = (groups59700922omplex @ (^[J : nat]: (groups59700922omplex @ (^[I : nat]: (G @ I @ J)) @ A3)) @ B2))))). % sum.swap
thf(fact_174_sum_Oswap, axiom,
    ((![G : nat > nat > nat, B2 : set_nat, A3 : set_nat]: ((groups1842438620at_nat @ (^[I : nat]: (groups1842438620at_nat @ (G @ I) @ B2)) @ A3) = (groups1842438620at_nat @ (^[J : nat]: (groups1842438620at_nat @ (^[I : nat]: (G @ I @ J)) @ A3)) @ B2))))). % sum.swap
thf(fact_175_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_176_sum_Oneutral, axiom,
    ((![A3 : set_nat, G : nat > nat]: ((![X3 : nat]: ((member_nat @ X3 @ A3) => ((G @ X3) = zero_zero_nat))) => ((groups1842438620at_nat @ G @ A3) = zero_zero_nat))))). % sum.neutral
thf(fact_177_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_178_sum_Onot__neutral__contains__not__neutral, axiom,
    ((![G : nat > nat, A3 : set_nat]: ((~ (((groups1842438620at_nat @ G @ A3) = zero_zero_nat))) => (~ ((![A4 : nat]: ((member_nat @ A4 @ A3) => ((G @ A4) = zero_zero_nat))))))))). % sum.not_neutral_contains_not_neutral
thf(fact_179_plus__nat_Oadd__0, axiom,
    ((![N : nat]: ((plus_plus_nat @ zero_zero_nat @ N) = N)))). % plus_nat.add_0
thf(fact_180_add__eq__self__zero, axiom,
    ((![M : nat, N : nat]: (((plus_plus_nat @ M @ N) = M) => (N = zero_zero_nat))))). % add_eq_self_zero
thf(fact_181_mult__0, axiom,
    ((![N : nat]: ((times_times_nat @ zero_zero_nat @ N) = zero_zero_nat)))). % mult_0
thf(fact_182_nat__mult__eq__cancel__disj, axiom,
    ((![K : nat, M : nat, N : nat]: (((times_times_nat @ K @ M) = (times_times_nat @ K @ N)) = (((K = zero_zero_nat)) | ((M = N))))))). % nat_mult_eq_cancel_disj
thf(fact_183_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_184_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_185_left__add__mult__distrib, axiom,
    ((![I2 : nat, U : nat, J2 : nat, K : nat]: ((plus_plus_nat @ (times_times_nat @ I2 @ U) @ (plus_plus_nat @ (times_times_nat @ J2 @ U) @ K)) = (plus_plus_nat @ (times_times_nat @ (plus_plus_nat @ I2 @ J2) @ U) @ K))))). % left_add_mult_distrib
thf(fact_186_nat__mult__1, axiom,
    ((![N : nat]: ((times_times_nat @ one_one_nat @ N) = N)))). % nat_mult_1
thf(fact_187_nat__mult__1__right, axiom,
    ((![N : nat]: ((times_times_nat @ N @ one_one_nat) = N)))). % nat_mult_1_right
thf(fact_188_sum__product, axiom,
    ((![F : nat > complex, A3 : set_nat, G : nat > complex, B2 : set_nat]: ((times_times_complex @ (groups59700922omplex @ F @ A3) @ (groups59700922omplex @ G @ B2)) = (groups59700922omplex @ (^[I : nat]: (groups59700922omplex @ (^[J : nat]: (times_times_complex @ (F @ I) @ (G @ J))) @ B2)) @ A3))))). % sum_product
thf(fact_189_sum__product, axiom,
    ((![F : nat > nat, A3 : set_nat, G : nat > nat, B2 : set_nat]: ((times_times_nat @ (groups1842438620at_nat @ F @ A3) @ (groups1842438620at_nat @ G @ B2)) = (groups1842438620at_nat @ (^[I : nat]: (groups1842438620at_nat @ (^[J : nat]: (times_times_nat @ (F @ I) @ (G @ J))) @ B2)) @ A3))))). % sum_product
thf(fact_190_sum__distrib__left, axiom,
    ((![R : complex, F : nat > complex, A3 : set_nat]: ((times_times_complex @ R @ (groups59700922omplex @ F @ A3)) = (groups59700922omplex @ (^[N2 : nat]: (times_times_complex @ R @ (F @ N2))) @ A3))))). % sum_distrib_left
thf(fact_191_sum__distrib__left, axiom,
    ((![R : nat, F : nat > nat, A3 : set_nat]: ((times_times_nat @ R @ (groups1842438620at_nat @ F @ A3)) = (groups1842438620at_nat @ (^[N2 : nat]: (times_times_nat @ R @ (F @ N2))) @ A3))))). % sum_distrib_left
thf(fact_192_sum__distrib__right, axiom,
    ((![F : nat > complex, A3 : set_nat, R : complex]: ((times_times_complex @ (groups59700922omplex @ F @ A3) @ R) = (groups59700922omplex @ (^[N2 : nat]: (times_times_complex @ (F @ N2) @ R)) @ A3))))). % sum_distrib_right
thf(fact_193_sum__distrib__right, axiom,
    ((![F : nat > nat, A3 : set_nat, R : nat]: ((times_times_nat @ (groups1842438620at_nat @ F @ A3) @ R) = (groups1842438620at_nat @ (^[N2 : nat]: (times_times_nat @ (F @ N2) @ R)) @ A3))))). % sum_distrib_right
thf(fact_194_sum_Odistrib, axiom,
    ((![G : nat > complex, H2 : nat > complex, A3 : set_nat]: ((groups59700922omplex @ (^[X2 : nat]: (plus_plus_complex @ (G @ X2) @ (H2 @ X2))) @ A3) = (plus_plus_complex @ (groups59700922omplex @ G @ A3) @ (groups59700922omplex @ H2 @ A3)))))). % sum.distrib
thf(fact_195_sum_Odistrib, axiom,
    ((![G : nat > nat, H2 : nat > nat, A3 : set_nat]: ((groups1842438620at_nat @ (^[X2 : nat]: (plus_plus_nat @ (G @ X2) @ (H2 @ X2))) @ A3) = (plus_plus_nat @ (groups1842438620at_nat @ G @ A3) @ (groups1842438620at_nat @ H2 @ A3)))))). % sum.distrib
thf(fact_196_mult__eq__self__implies__10, axiom,
    ((![M : nat, N : nat]: ((M = (times_times_nat @ M @ N)) => ((N = one_one_nat) | (M = zero_zero_nat)))))). % mult_eq_self_implies_10
thf(fact_197_mult_Oleft__neutral, axiom,
    ((![A2 : complex]: ((times_times_complex @ one_one_complex @ A2) = A2)))). % mult.left_neutral
thf(fact_198_mult_Oleft__neutral, axiom,
    ((![A2 : nat]: ((times_times_nat @ one_one_nat @ A2) = A2)))). % mult.left_neutral
thf(fact_199_mult_Oright__neutral, axiom,
    ((![A2 : complex]: ((times_times_complex @ A2 @ one_one_complex) = A2)))). % mult.right_neutral
thf(fact_200_mult_Oright__neutral, axiom,
    ((![A2 : nat]: ((times_times_nat @ A2 @ one_one_nat) = A2)))). % mult.right_neutral
thf(fact_201_add_Oleft__neutral, axiom,
    ((![A2 : nat]: ((plus_plus_nat @ zero_zero_nat @ A2) = A2)))). % add.left_neutral
thf(fact_202_add_Oleft__neutral, axiom,
    ((![A2 : complex]: ((plus_plus_complex @ zero_zero_complex @ A2) = A2)))). % add.left_neutral
thf(fact_203_add__right__cancel, axiom,
    ((![B : complex, A2 : complex, C : complex]: (((plus_plus_complex @ B @ A2) = (plus_plus_complex @ C @ A2)) = (B = C))))). % add_right_cancel
thf(fact_204_add__right__cancel, axiom,
    ((![B : nat, A2 : nat, C : nat]: (((plus_plus_nat @ B @ A2) = (plus_plus_nat @ C @ A2)) = (B = C))))). % add_right_cancel
thf(fact_205_add__left__cancel, axiom,
    ((![A2 : complex, B : complex, C : complex]: (((plus_plus_complex @ A2 @ B) = (plus_plus_complex @ A2 @ C)) = (B = C))))). % add_left_cancel
thf(fact_206_add__left__cancel, axiom,
    ((![A2 : nat, B : nat, C : nat]: (((plus_plus_nat @ A2 @ B) = (plus_plus_nat @ A2 @ C)) = (B = C))))). % add_left_cancel
thf(fact_207_zero__eq__add__iff__both__eq__0, axiom,
    ((![X : nat, Y : nat]: ((zero_zero_nat = (plus_plus_nat @ X @ Y)) = (((X = zero_zero_nat)) & ((Y = zero_zero_nat))))))). % zero_eq_add_iff_both_eq_0
thf(fact_208_add__eq__0__iff__both__eq__0, axiom,
    ((![X : nat, Y : nat]: (((plus_plus_nat @ X @ Y) = zero_zero_nat) = (((X = zero_zero_nat)) & ((Y = zero_zero_nat))))))). % add_eq_0_iff_both_eq_0
thf(fact_209_add__cancel__right__right, axiom,
    ((![A2 : nat, B : nat]: ((A2 = (plus_plus_nat @ A2 @ B)) = (B = zero_zero_nat))))). % add_cancel_right_right
thf(fact_210_add__cancel__right__right, axiom,
    ((![A2 : complex, B : complex]: ((A2 = (plus_plus_complex @ A2 @ B)) = (B = zero_zero_complex))))). % add_cancel_right_right
thf(fact_211_add__cancel__right__left, axiom,
    ((![A2 : nat, B : nat]: ((A2 = (plus_plus_nat @ B @ A2)) = (B = zero_zero_nat))))). % add_cancel_right_left
thf(fact_212_add__cancel__right__left, axiom,
    ((![A2 : complex, B : complex]: ((A2 = (plus_plus_complex @ B @ A2)) = (B = zero_zero_complex))))). % add_cancel_right_left
thf(fact_213_add__cancel__left__right, axiom,
    ((![A2 : nat, B : nat]: (((plus_plus_nat @ A2 @ B) = A2) = (B = zero_zero_nat))))). % add_cancel_left_right
thf(fact_214_add__cancel__left__right, axiom,
    ((![A2 : complex, B : complex]: (((plus_plus_complex @ A2 @ B) = A2) = (B = zero_zero_complex))))). % add_cancel_left_right
thf(fact_215_add__cancel__left__left, axiom,
    ((![B : nat, A2 : nat]: (((plus_plus_nat @ B @ A2) = A2) = (B = zero_zero_nat))))). % add_cancel_left_left
thf(fact_216_add__cancel__left__left, axiom,
    ((![B : complex, A2 : complex]: (((plus_plus_complex @ B @ A2) = A2) = (B = zero_zero_complex))))). % add_cancel_left_left
thf(fact_217_add_Oright__neutral, axiom,
    ((![A2 : nat]: ((plus_plus_nat @ A2 @ zero_zero_nat) = A2)))). % add.right_neutral
thf(fact_218_add_Oright__neutral, axiom,
    ((![A2 : complex]: ((plus_plus_complex @ A2 @ zero_zero_complex) = A2)))). % add.right_neutral
thf(fact_219_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_220_zero__reorient, axiom,
    ((![X : complex]: ((zero_zero_complex = X) = (X = zero_zero_complex))))). % zero_reorient
thf(fact_221_mult_Oleft__commute, axiom,
    ((![B : complex, A2 : complex, C : complex]: ((times_times_complex @ B @ (times_times_complex @ A2 @ C)) = (times_times_complex @ A2 @ (times_times_complex @ B @ C)))))). % mult.left_commute
thf(fact_222_mult_Oleft__commute, axiom,
    ((![B : nat, A2 : nat, C : nat]: ((times_times_nat @ B @ (times_times_nat @ A2 @ C)) = (times_times_nat @ A2 @ (times_times_nat @ B @ C)))))). % mult.left_commute
thf(fact_223_mult_Ocommute, axiom,
    ((times_times_complex = (^[A : complex]: (^[B4 : complex]: (times_times_complex @ B4 @ A)))))). % mult.commute
thf(fact_224_mult_Ocommute, axiom,
    ((times_times_nat = (^[A : nat]: (^[B4 : nat]: (times_times_nat @ B4 @ A)))))). % mult.commute

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