% 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_432__3227476_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:11:16.066

% Could-be-implicit typings (3)
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__Nat__Onat, type,
    nat : $tType).

% Explicit typings (21)
thf(sy_c_FFT__Mirabelle__ulikgskiun_Oroot, type,
    fFT_Mirabelle_root : nat > complex).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat, type,
    minus_minus_nat : nat > nat > nat).
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_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_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_If_001t__Complex__Ocomplex, type,
    if_complex : $o > complex > complex > complex).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat, type,
    ord_less_nat : nat > nat > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat, type,
    ord_less_eq_nat : nat > nat > $o).
thf(sy_c_Power_Opower__class_Opower_001t__Complex__Ocomplex, type,
    power_power_complex : complex > nat > complex).
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_n, type,
    n : nat).

% Relevant facts (98)
thf(fact_0_i__less, axiom,
    ((ord_less_nat @ i @ n))). % i_less
thf(fact_1_i__diff, axiom,
    ((![K : nat]: (ord_less_nat @ (minus_minus_nat @ i @ K) @ n)))). % i_diff
thf(fact_2_diff__i, axiom,
    ((![K : nat]: ((ord_less_nat @ K @ n) => (ord_less_nat @ (minus_minus_nat @ K @ i) @ n))))). % diff_i
thf(fact_3_root0, axiom,
    (((fFT_Mirabelle_root @ zero_zero_nat) = one_one_complex))). % root0
thf(fact_4_root__unity, axiom,
    ((![N : nat]: ((power_power_complex @ (fFT_Mirabelle_root @ N) @ N) = one_one_complex)))). % root_unity
thf(fact_5_calculation, axiom,
    (((groups59700922omplex @ (^[J : nat]: (times_times_complex @ (groups59700922omplex @ (power_power_complex @ (if_complex @ (ord_less_eq_nat @ i @ J) @ (power_power_complex @ (fFT_Mirabelle_root @ n) @ (minus_minus_nat @ J @ i)) @ (power_power_complex @ (divide1210191872omplex @ one_one_complex @ (fFT_Mirabelle_root @ n)) @ (minus_minus_nat @ i @ J)))) @ (set_or562006527an_nat @ zero_zero_nat @ n)) @ (a @ J))) @ (set_or562006527an_nat @ zero_zero_nat @ n)) = (plus_plus_complex @ (groups59700922omplex @ (^[J : nat]: (times_times_complex @ (groups59700922omplex @ (power_power_complex @ (power_power_complex @ (divide1210191872omplex @ one_one_complex @ (fFT_Mirabelle_root @ n)) @ (minus_minus_nat @ i @ J))) @ (set_or562006527an_nat @ zero_zero_nat @ n)) @ (a @ J))) @ (set_or562006527an_nat @ zero_zero_nat @ i)) @ (groups59700922omplex @ (^[J : nat]: (times_times_complex @ (groups59700922omplex @ (power_power_complex @ (power_power_complex @ (fFT_Mirabelle_root @ n) @ (minus_minus_nat @ J @ i))) @ (set_or562006527an_nat @ zero_zero_nat @ n)) @ (a @ J))) @ (set_or562006527an_nat @ i @ n)))))). % calculation
thf(fact_6_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_7_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_8_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_9_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_10_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_11_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_12_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_13_div__self, axiom,
    ((![A : complex]: ((~ ((A = zero_zero_complex))) => ((divide1210191872omplex @ A @ A) = one_one_complex))))). % div_self
thf(fact_14_div__self, axiom,
    ((![A : nat]: ((~ ((A = zero_zero_nat))) => ((divide_divide_nat @ A @ A) = one_one_nat))))). % div_self
thf(fact_15_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_16_divide__self, axiom,
    ((![A : complex]: ((~ ((A = zero_zero_complex))) => ((divide1210191872omplex @ A @ A) = one_one_complex))))). % divide_self
thf(fact_17_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_18_mult__cancel__right, axiom,
    ((![A : complex, C : complex, B : complex]: (((times_times_complex @ A @ C) = (times_times_complex @ B @ C)) = (((C = zero_zero_complex)) | ((A = B))))))). % mult_cancel_right
thf(fact_19_mult__cancel__right, axiom,
    ((![A : nat, C : nat, B : nat]: (((times_times_nat @ A @ C) = (times_times_nat @ B @ C)) = (((C = zero_zero_nat)) | ((A = B))))))). % mult_cancel_right
thf(fact_20_mult__cancel__left, axiom,
    ((![C : complex, A : complex, B : complex]: (((times_times_complex @ C @ A) = (times_times_complex @ C @ B)) = (((C = zero_zero_complex)) | ((A = B))))))). % mult_cancel_left
thf(fact_21_mult__cancel__left, axiom,
    ((![C : nat, A : nat, B : nat]: (((times_times_nat @ C @ A) = (times_times_nat @ C @ B)) = (((C = zero_zero_nat)) | ((A = B))))))). % mult_cancel_left
thf(fact_22_mult__eq__0__iff, axiom,
    ((![A : complex, B : complex]: (((times_times_complex @ A @ B) = zero_zero_complex) = (((A = zero_zero_complex)) | ((B = zero_zero_complex))))))). % mult_eq_0_iff
thf(fact_23_mult__eq__0__iff, axiom,
    ((![A : nat, B : nat]: (((times_times_nat @ A @ B) = zero_zero_nat) = (((A = zero_zero_nat)) | ((B = zero_zero_nat))))))). % mult_eq_0_iff
thf(fact_24_mult__zero__right, axiom,
    ((![A : complex]: ((times_times_complex @ A @ zero_zero_complex) = zero_zero_complex)))). % mult_zero_right
thf(fact_25_mult__zero__right, axiom,
    ((![A : nat]: ((times_times_nat @ A @ zero_zero_nat) = zero_zero_nat)))). % mult_zero_right
thf(fact_26_mult__zero__left, axiom,
    ((![A : complex]: ((times_times_complex @ zero_zero_complex @ A) = zero_zero_complex)))). % mult_zero_left
thf(fact_27_mult__zero__left, axiom,
    ((![A : nat]: ((times_times_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % mult_zero_left
thf(fact_28_division__ring__divide__zero, axiom,
    ((![A : complex]: ((divide1210191872omplex @ A @ zero_zero_complex) = zero_zero_complex)))). % division_ring_divide_zero
thf(fact_29_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_30_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_31_div__by__0, axiom,
    ((![A : complex]: ((divide1210191872omplex @ A @ zero_zero_complex) = zero_zero_complex)))). % div_by_0
thf(fact_32_div__by__0, axiom,
    ((![A : nat]: ((divide_divide_nat @ A @ zero_zero_nat) = zero_zero_nat)))). % div_by_0
thf(fact_33_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_34_div__0, axiom,
    ((![A : complex]: ((divide1210191872omplex @ zero_zero_complex @ A) = zero_zero_complex)))). % div_0
thf(fact_35_div__0, axiom,
    ((![A : nat]: ((divide_divide_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % div_0
thf(fact_36_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_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_div__by__1, axiom,
    ((![A : complex]: ((divide1210191872omplex @ A @ one_one_complex) = A)))). % div_by_1
thf(fact_39_div__by__1, axiom,
    ((![A : nat]: ((divide_divide_nat @ A @ one_one_nat) = A)))). % div_by_1
thf(fact_40_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_41_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_42_div__less, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => ((divide_divide_nat @ M @ N) = zero_zero_nat))))). % div_less
thf(fact_43_root1, axiom,
    (((fFT_Mirabelle_root @ one_one_nat) = one_one_complex))). % root1
thf(fact_44_mult__cancel__right2, axiom,
    ((![A : complex, C : complex]: (((times_times_complex @ A @ C) = C) = (((C = zero_zero_complex)) | ((A = one_one_complex))))))). % mult_cancel_right2
thf(fact_45_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_46_mult__cancel__left2, axiom,
    ((![C : complex, A : complex]: (((times_times_complex @ C @ A) = C) = (((C = zero_zero_complex)) | ((A = one_one_complex))))))). % mult_cancel_left2
thf(fact_47_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_48_le__add__diff__inverse2, axiom,
    ((![B : nat, A : nat]: ((ord_less_eq_nat @ B @ A) => ((plus_plus_nat @ (minus_minus_nat @ A @ B) @ B) = A))))). % le_add_diff_inverse2
thf(fact_49_le__add__diff__inverse, axiom,
    ((![B : nat, A : nat]: ((ord_less_eq_nat @ B @ A) => ((plus_plus_nat @ B @ (minus_minus_nat @ A @ B)) = A))))). % le_add_diff_inverse
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__div__cancel__right, axiom,
    ((![B : complex, A : complex]: ((~ ((B = zero_zero_complex))) => ((divide1210191872omplex @ (times_times_complex @ A @ B) @ B) = A))))). % nonzero_mult_div_cancel_right
thf(fact_55_nonzero__mult__div__cancel__right, axiom,
    ((![B : nat, A : nat]: ((~ ((B = zero_zero_nat))) => ((divide_divide_nat @ (times_times_nat @ A @ B) @ B) = A))))). % nonzero_mult_div_cancel_right
thf(fact_56_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_57_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_58_nonzero__mult__div__cancel__left, axiom,
    ((![A : complex, B : complex]: ((~ ((A = zero_zero_complex))) => ((divide1210191872omplex @ (times_times_complex @ A @ B) @ A) = B))))). % nonzero_mult_div_cancel_left
thf(fact_59_nonzero__mult__div__cancel__left, axiom,
    ((![A : nat, B : nat]: ((~ ((A = zero_zero_nat))) => ((divide_divide_nat @ (times_times_nat @ A @ B) @ A) = B))))). % nonzero_mult_div_cancel_left
thf(fact_60_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_61_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_62_split__div, axiom,
    ((![P : nat > $o, M : nat, N : nat]: ((P @ (divide_divide_nat @ M @ N)) = (((((N = zero_zero_nat)) => ((P @ zero_zero_nat)))) & ((((~ ((N = zero_zero_nat)))) => ((![I : nat]: (![J : nat]: (((ord_less_nat @ J @ N)) => ((((M = (plus_plus_nat @ (times_times_nat @ N @ I) @ J))) => ((P @ I))))))))))))))). % split_div
thf(fact_63_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_64_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_65_div__le__dividend, axiom,
    ((![M : nat, N : nat]: (ord_less_eq_nat @ (divide_divide_nat @ M @ N) @ M)))). % div_le_dividend
thf(fact_66_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_67_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_68_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_69_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_70_dividend__less__div__times, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ zero_zero_nat @ N) => (ord_less_nat @ M @ (plus_plus_nat @ N @ (times_times_nat @ (divide_divide_nat @ M @ N) @ N))))))). % dividend_less_div_times
thf(fact_71_dividend__less__times__div, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ zero_zero_nat @ N) => (ord_less_nat @ M @ (plus_plus_nat @ N @ (times_times_nat @ N @ (divide_divide_nat @ M @ N)))))))). % dividend_less_times_div
thf(fact_72_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_73_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_74_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_75_mult__less__le__imp__less, axiom,
    ((![A : nat, B : nat, C : nat, D : nat]: ((ord_less_nat @ A @ B) => ((ord_less_eq_nat @ C @ D) => ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_nat @ zero_zero_nat @ C) => (ord_less_nat @ (times_times_nat @ A @ C) @ (times_times_nat @ B @ D))))))))). % mult_less_le_imp_less
thf(fact_76_mult__le__less__imp__less, axiom,
    ((![A : nat, B : nat, C : nat, D : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_nat @ C @ D) => ((ord_less_nat @ zero_zero_nat @ A) => ((ord_less_eq_nat @ zero_zero_nat @ C) => (ord_less_nat @ (times_times_nat @ A @ C) @ (times_times_nat @ B @ D))))))))). % mult_le_less_imp_less
thf(fact_77_mult__right__le__imp__le, axiom,
    ((![A : nat, C : nat, B : nat]: ((ord_less_eq_nat @ (times_times_nat @ A @ C) @ (times_times_nat @ B @ C)) => ((ord_less_nat @ zero_zero_nat @ C) => (ord_less_eq_nat @ A @ B)))))). % mult_right_le_imp_le
thf(fact_78_mult__left__le__imp__le, axiom,
    ((![C : nat, A : nat, B : nat]: ((ord_less_eq_nat @ (times_times_nat @ C @ A) @ (times_times_nat @ C @ B)) => ((ord_less_nat @ zero_zero_nat @ C) => (ord_less_eq_nat @ A @ B)))))). % mult_left_le_imp_le
thf(fact_79_mult__strict__mono_H, axiom,
    ((![A : nat, B : nat, C : nat, D : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ C @ D) => ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_eq_nat @ zero_zero_nat @ C) => (ord_less_nat @ (times_times_nat @ A @ C) @ (times_times_nat @ B @ D))))))))). % mult_strict_mono'
thf(fact_80_mult__right__less__imp__less, axiom,
    ((![A : nat, C : nat, B : nat]: ((ord_less_nat @ (times_times_nat @ A @ C) @ (times_times_nat @ B @ C)) => ((ord_less_eq_nat @ zero_zero_nat @ C) => (ord_less_nat @ A @ B)))))). % mult_right_less_imp_less
thf(fact_81_mult__strict__mono, axiom,
    ((![A : nat, B : nat, C : nat, D : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ C @ D) => ((ord_less_nat @ zero_zero_nat @ B) => ((ord_less_eq_nat @ zero_zero_nat @ C) => (ord_less_nat @ (times_times_nat @ A @ C) @ (times_times_nat @ B @ D))))))))). % mult_strict_mono
thf(fact_82_mult__left__less__imp__less, axiom,
    ((![C : nat, A : nat, B : nat]: ((ord_less_nat @ (times_times_nat @ C @ A) @ (times_times_nat @ C @ B)) => ((ord_less_eq_nat @ zero_zero_nat @ C) => (ord_less_nat @ A @ B)))))). % mult_left_less_imp_less
thf(fact_83_ordered__comm__semiring__class_Ocomm__mult__left__mono, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ zero_zero_nat @ C) => (ord_less_eq_nat @ (times_times_nat @ C @ A) @ (times_times_nat @ C @ B))))))). % ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_84_mult__nonneg__nonpos2, axiom,
    ((![A : nat, B : nat]: ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_eq_nat @ B @ zero_zero_nat) => (ord_less_eq_nat @ (times_times_nat @ B @ A) @ zero_zero_nat)))))). % mult_nonneg_nonpos2
thf(fact_85_mult__nonpos__nonneg, axiom,
    ((![A : nat, B : nat]: ((ord_less_eq_nat @ A @ zero_zero_nat) => ((ord_less_eq_nat @ zero_zero_nat @ B) => (ord_less_eq_nat @ (times_times_nat @ A @ B) @ zero_zero_nat)))))). % mult_nonpos_nonneg
thf(fact_86_mult__nonneg__nonpos, axiom,
    ((![A : nat, B : nat]: ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_eq_nat @ B @ zero_zero_nat) => (ord_less_eq_nat @ (times_times_nat @ A @ B) @ zero_zero_nat)))))). % mult_nonneg_nonpos
thf(fact_87_mult__nonneg__nonneg, axiom,
    ((![A : nat, B : nat]: ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_eq_nat @ zero_zero_nat @ B) => (ord_less_eq_nat @ zero_zero_nat @ (times_times_nat @ A @ B))))))). % mult_nonneg_nonneg
thf(fact_88_split__mult__neg__le, axiom,
    ((![A : nat, B : nat]: ((((ord_less_eq_nat @ zero_zero_nat @ A) & (ord_less_eq_nat @ B @ zero_zero_nat)) | ((ord_less_eq_nat @ A @ zero_zero_nat) & (ord_less_eq_nat @ zero_zero_nat @ B))) => (ord_less_eq_nat @ (times_times_nat @ A @ B) @ zero_zero_nat))))). % split_mult_neg_le
thf(fact_89_mult__right__mono, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ zero_zero_nat @ C) => (ord_less_eq_nat @ (times_times_nat @ A @ C) @ (times_times_nat @ B @ C))))))). % mult_right_mono
thf(fact_90_mult__left__mono, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ zero_zero_nat @ C) => (ord_less_eq_nat @ (times_times_nat @ C @ A) @ (times_times_nat @ C @ B))))))). % mult_left_mono
thf(fact_91_mult__mono_H, axiom,
    ((![A : nat, B : nat, C : nat, D : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ C @ D) => ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_eq_nat @ zero_zero_nat @ C) => (ord_less_eq_nat @ (times_times_nat @ A @ C) @ (times_times_nat @ B @ D))))))))). % mult_mono'
thf(fact_92_mult__mono, axiom,
    ((![A : nat, B : nat, C : nat, D : nat]: ((ord_less_eq_nat @ A @ B) => ((ord_less_eq_nat @ C @ D) => ((ord_less_eq_nat @ zero_zero_nat @ B) => ((ord_less_eq_nat @ zero_zero_nat @ C) => (ord_less_eq_nat @ (times_times_nat @ A @ C) @ (times_times_nat @ B @ D))))))))). % mult_mono
thf(fact_93_not__one__le__zero, axiom,
    ((~ ((ord_less_eq_nat @ one_one_nat @ zero_zero_nat))))). % not_one_le_zero
thf(fact_94_zero__le__one, axiom,
    ((ord_less_eq_nat @ zero_zero_nat @ one_one_nat))). % zero_le_one
thf(fact_95_add__le__add__imp__diff__le, axiom,
    ((![I2 : nat, K : nat, N : nat, J2 : nat]: ((ord_less_eq_nat @ (plus_plus_nat @ I2 @ K) @ N) => ((ord_less_eq_nat @ N @ (plus_plus_nat @ J2 @ K)) => ((ord_less_eq_nat @ (plus_plus_nat @ I2 @ K) @ N) => ((ord_less_eq_nat @ N @ (plus_plus_nat @ J2 @ K)) => (ord_less_eq_nat @ (minus_minus_nat @ N @ K) @ J2)))))))). % add_le_add_imp_diff_le
thf(fact_96_add__le__imp__le__diff, axiom,
    ((![I2 : nat, K : nat, N : nat]: ((ord_less_eq_nat @ (plus_plus_nat @ I2 @ K) @ N) => (ord_less_eq_nat @ I2 @ (minus_minus_nat @ N @ K)))))). % add_le_imp_le_diff
thf(fact_97_root__nonzero, axiom,
    ((![N : nat]: (~ (((fFT_Mirabelle_root @ N) = zero_zero_complex)))))). % root_nonzero

% Helper facts (3)
thf(help_If_3_1_If_001t__Complex__Ocomplex_T, axiom,
    ((![P : $o]: ((P = $true) | (P = $false))))).
thf(help_If_2_1_If_001t__Complex__Ocomplex_T, axiom,
    ((![X : complex, Y : complex]: ((if_complex @ $false @ X @ Y) = Y)))).
thf(help_If_1_1_If_001t__Complex__Ocomplex_T, axiom,
    ((![X : complex, Y : complex]: ((if_complex @ $true @ X @ Y) = X)))).

% Conjectures (1)
thf(conj_0, conjecture,
    (((plus_plus_complex @ (groups59700922omplex @ (^[J : nat]: (times_times_complex @ (groups59700922omplex @ (power_power_complex @ (power_power_complex @ (divide1210191872omplex @ one_one_complex @ (fFT_Mirabelle_root @ n)) @ (minus_minus_nat @ i @ J))) @ (set_or562006527an_nat @ zero_zero_nat @ n)) @ (a @ J))) @ (set_or562006527an_nat @ zero_zero_nat @ i)) @ (groups59700922omplex @ (^[J : nat]: (times_times_complex @ (groups59700922omplex @ (power_power_complex @ (power_power_complex @ (fFT_Mirabelle_root @ n) @ (minus_minus_nat @ J @ i))) @ (set_or562006527an_nat @ zero_zero_nat @ n)) @ (a @ J))) @ (set_or562006527an_nat @ i @ n))) = (groups59700922omplex @ (^[J : nat]: (times_times_complex @ (groups59700922omplex @ (power_power_complex @ (power_power_complex @ (fFT_Mirabelle_root @ n) @ (minus_minus_nat @ J @ i))) @ (set_or562006527an_nat @ zero_zero_nat @ n)) @ (a @ J))) @ (set_or562006527an_nat @ i @ n))))).
