% 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_440__3227748_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:11:19.767

% 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 (20)
thf(sy_c_FFT__Mirabelle__ulikgskiun_Oroot, type,
    fFT_Mirabelle_root : nat > complex).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Complex__Ocomplex, type,
    minus_minus_complex : complex > complex > complex).
thf(sy_c_Groups_Ominus__class_Ominus_001t__Nat__Onat, type,
    minus_minus_nat : nat > 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_Groups__Big_Ocomm__monoid__add__class_Osum_001t__Nat__Onat_001t__Nat__Onat, type,
    groups1842438620at_nat : (nat > nat) > set_nat > nat).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat, type,
    ord_less_nat : nat > nat > $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_Set__Interval_Oord__class_OatLeastLessThan_001t__Nat__Onat, type,
    set_or562006527an_nat : nat > nat > set_nat).
thf(sy_c_Set__Interval_Oord__class_OgreaterThanLessThan_001t__Nat__Onat, type,
    set_or1544565540an_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_n, type,
    n : nat).

% Relevant facts (164)
thf(fact_0_i__less, axiom,
    ((ord_less_nat @ i @ n))). % i_less
thf(fact_1_diff__i, axiom,
    ((![K : nat]: ((ord_less_nat @ K @ n) => (ord_less_nat @ (minus_minus_nat @ K @ i) @ n))))). % diff_i
thf(fact_2_i__diff, axiom,
    ((![K : nat]: (ord_less_nat @ (minus_minus_nat @ i @ K) @ n)))). % i_diff
thf(fact_3_diff__add__zero, axiom,
    ((![A : nat, B : nat]: ((minus_minus_nat @ A @ (plus_plus_nat @ A @ B)) = zero_zero_nat)))). % diff_add_zero
thf(fact_4_sum_Oneutral__const, axiom,
    ((![A2 : set_nat]: ((groups1842438620at_nat @ (^[Uu : nat]: zero_zero_nat) @ A2) = zero_zero_nat)))). % sum.neutral_const
thf(fact_5_sum_Oneutral__const, axiom,
    ((![A2 : set_nat]: ((groups59700922omplex @ (^[Uu : nat]: zero_zero_complex) @ A2) = zero_zero_complex)))). % sum.neutral_const
thf(fact_6_sum__add__nat__ivl__singleton, axiom,
    ((![M : nat, N : nat, F : nat > nat]: ((ord_less_nat @ M @ N) => ((plus_plus_nat @ (F @ M) @ (groups1842438620at_nat @ F @ (set_or1544565540an_nat @ M @ N))) = (groups1842438620at_nat @ F @ (set_or562006527an_nat @ M @ N))))))). % sum_add_nat_ivl_singleton
thf(fact_7_sum__add__nat__ivl__singleton, axiom,
    ((![M : nat, N : nat, F : nat > complex]: ((ord_less_nat @ M @ N) => ((plus_plus_complex @ (F @ M) @ (groups59700922omplex @ F @ (set_or1544565540an_nat @ M @ N))) = (groups59700922omplex @ F @ (set_or562006527an_nat @ M @ N))))))). % sum_add_nat_ivl_singleton
thf(fact_8_sum__add__split__nat__ivl__singleton, axiom,
    ((![M : nat, N : nat, G : nat > nat, F : nat > nat]: ((ord_less_nat @ M @ N) => ((![I : nat]: ((ord_less_nat @ M @ I) => ((ord_less_nat @ I @ N) => ((G @ I) = (F @ I))))) => ((plus_plus_nat @ (F @ M) @ (groups1842438620at_nat @ G @ (set_or1544565540an_nat @ M @ N))) = (groups1842438620at_nat @ F @ (set_or562006527an_nat @ M @ N)))))))). % sum_add_split_nat_ivl_singleton
thf(fact_9_sum__add__split__nat__ivl__singleton, axiom,
    ((![M : nat, N : nat, G : nat > complex, F : nat > complex]: ((ord_less_nat @ M @ N) => ((![I : nat]: ((ord_less_nat @ M @ I) => ((ord_less_nat @ I @ N) => ((G @ I) = (F @ I))))) => ((plus_plus_complex @ (F @ M) @ (groups59700922omplex @ G @ (set_or1544565540an_nat @ M @ N))) = (groups59700922omplex @ F @ (set_or562006527an_nat @ M @ N)))))))). % sum_add_split_nat_ivl_singleton
thf(fact_10_diff__0__eq__0, axiom,
    ((![N : nat]: ((minus_minus_nat @ zero_zero_nat @ N) = zero_zero_nat)))). % diff_0_eq_0
thf(fact_11_diff__self__eq__0, axiom,
    ((![M : nat]: ((minus_minus_nat @ M @ M) = zero_zero_nat)))). % diff_self_eq_0
thf(fact_12_add__diff__cancel, axiom,
    ((![A : complex, B : complex]: ((minus_minus_complex @ (plus_plus_complex @ A @ B) @ B) = A)))). % add_diff_cancel
thf(fact_13_diff__add__cancel, axiom,
    ((![A : complex, B : complex]: ((plus_plus_complex @ (minus_minus_complex @ A @ B) @ B) = A)))). % diff_add_cancel
thf(fact_14_add__diff__cancel__left, axiom,
    ((![C : complex, A : complex, B : complex]: ((minus_minus_complex @ (plus_plus_complex @ C @ A) @ (plus_plus_complex @ C @ B)) = (minus_minus_complex @ A @ B))))). % add_diff_cancel_left
thf(fact_15_add__diff__cancel__left, axiom,
    ((![C : nat, A : nat, B : nat]: ((minus_minus_nat @ (plus_plus_nat @ C @ A) @ (plus_plus_nat @ C @ B)) = (minus_minus_nat @ A @ B))))). % add_diff_cancel_left
thf(fact_16_add__diff__cancel__left_H, axiom,
    ((![A : complex, B : complex]: ((minus_minus_complex @ (plus_plus_complex @ A @ B) @ A) = B)))). % add_diff_cancel_left'
thf(fact_17_add__diff__cancel__left_H, axiom,
    ((![A : nat, B : nat]: ((minus_minus_nat @ (plus_plus_nat @ A @ B) @ A) = B)))). % add_diff_cancel_left'
thf(fact_18_add__diff__cancel__right, axiom,
    ((![A : complex, C : complex, B : complex]: ((minus_minus_complex @ (plus_plus_complex @ A @ C) @ (plus_plus_complex @ B @ C)) = (minus_minus_complex @ A @ B))))). % add_diff_cancel_right
thf(fact_19_add__diff__cancel__right, axiom,
    ((![A : nat, C : nat, B : nat]: ((minus_minus_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ C)) = (minus_minus_nat @ A @ B))))). % add_diff_cancel_right
thf(fact_20_add__right__cancel, axiom,
    ((![B : complex, A : complex, C : complex]: (((plus_plus_complex @ B @ A) = (plus_plus_complex @ C @ A)) = (B = C))))). % add_right_cancel
thf(fact_21_add__right__cancel, axiom,
    ((![B : nat, A : nat, C : nat]: (((plus_plus_nat @ B @ A) = (plus_plus_nat @ C @ A)) = (B = C))))). % add_right_cancel
thf(fact_22_add__left__cancel, axiom,
    ((![A : complex, B : complex, C : complex]: (((plus_plus_complex @ A @ B) = (plus_plus_complex @ A @ C)) = (B = C))))). % add_left_cancel
thf(fact_23_add__left__cancel, axiom,
    ((![A : nat, B : nat, C : nat]: (((plus_plus_nat @ A @ B) = (plus_plus_nat @ A @ C)) = (B = C))))). % add_left_cancel
thf(fact_24_Nat_Oadd__0__right, axiom,
    ((![M : nat]: ((plus_plus_nat @ M @ zero_zero_nat) = M)))). % Nat.add_0_right
thf(fact_25_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_26_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_27_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_28_mult__0__right, axiom,
    ((![M : nat]: ((times_times_nat @ M @ zero_zero_nat) = zero_zero_nat)))). % mult_0_right
thf(fact_29_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_30_nat__add__left__cancel__less, axiom,
    ((![K : nat, M : nat, N : nat]: ((ord_less_nat @ (plus_plus_nat @ K @ M) @ (plus_plus_nat @ K @ N)) = (ord_less_nat @ M @ N))))). % nat_add_left_cancel_less
thf(fact_31_diff__diff__left, axiom,
    ((![I2 : nat, J : nat, K : nat]: ((minus_minus_nat @ (minus_minus_nat @ I2 @ J) @ K) = (minus_minus_nat @ I2 @ (plus_plus_nat @ J @ K)))))). % diff_diff_left
thf(fact_32_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_33_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_34_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_35_add__cancel__right__right, axiom,
    ((![A : nat, B : nat]: ((A = (plus_plus_nat @ A @ B)) = (B = zero_zero_nat))))). % add_cancel_right_right
thf(fact_36_add__cancel__right__right, axiom,
    ((![A : complex, B : complex]: ((A = (plus_plus_complex @ A @ B)) = (B = zero_zero_complex))))). % add_cancel_right_right
thf(fact_37_add__cancel__right__left, axiom,
    ((![A : nat, B : nat]: ((A = (plus_plus_nat @ B @ A)) = (B = zero_zero_nat))))). % add_cancel_right_left
thf(fact_38_add__cancel__right__left, axiom,
    ((![A : complex, B : complex]: ((A = (plus_plus_complex @ B @ A)) = (B = zero_zero_complex))))). % add_cancel_right_left
thf(fact_39_add__cancel__left__right, axiom,
    ((![A : nat, B : nat]: (((plus_plus_nat @ A @ B) = A) = (B = zero_zero_nat))))). % add_cancel_left_right
thf(fact_40_add__cancel__left__right, axiom,
    ((![A : complex, B : complex]: (((plus_plus_complex @ A @ B) = A) = (B = zero_zero_complex))))). % add_cancel_left_right
thf(fact_41_add__cancel__left__left, axiom,
    ((![B : nat, A : nat]: (((plus_plus_nat @ B @ A) = A) = (B = zero_zero_nat))))). % add_cancel_left_left
thf(fact_42_add__cancel__left__left, axiom,
    ((![B : complex, A : complex]: (((plus_plus_complex @ B @ A) = A) = (B = zero_zero_complex))))). % add_cancel_left_left
thf(fact_43_add_Oright__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ A @ zero_zero_nat) = A)))). % add.right_neutral
thf(fact_44_add_Oright__neutral, axiom,
    ((![A : complex]: ((plus_plus_complex @ A @ zero_zero_complex) = A)))). % add.right_neutral
thf(fact_45_add_Oleft__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ zero_zero_nat @ A) = A)))). % add.left_neutral
thf(fact_46_add_Oleft__neutral, axiom,
    ((![A : complex]: ((plus_plus_complex @ zero_zero_complex @ A) = A)))). % add.left_neutral
thf(fact_47_cancel__comm__monoid__add__class_Odiff__cancel, axiom,
    ((![A : complex]: ((minus_minus_complex @ A @ A) = zero_zero_complex)))). % cancel_comm_monoid_add_class.diff_cancel
thf(fact_48_cancel__comm__monoid__add__class_Odiff__cancel, axiom,
    ((![A : nat]: ((minus_minus_nat @ A @ A) = zero_zero_nat)))). % cancel_comm_monoid_add_class.diff_cancel
thf(fact_49_diff__zero, axiom,
    ((![A : complex]: ((minus_minus_complex @ A @ zero_zero_complex) = A)))). % diff_zero
thf(fact_50_diff__zero, axiom,
    ((![A : nat]: ((minus_minus_nat @ A @ zero_zero_nat) = A)))). % diff_zero
thf(fact_51_zero__diff, axiom,
    ((![A : nat]: ((minus_minus_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % zero_diff
thf(fact_52_diff__0__right, axiom,
    ((![A : complex]: ((minus_minus_complex @ A @ zero_zero_complex) = A)))). % diff_0_right
thf(fact_53_diff__self, axiom,
    ((![A : complex]: ((minus_minus_complex @ A @ A) = zero_zero_complex)))). % diff_self
thf(fact_54_add__less__cancel__right, axiom,
    ((![A : nat, C : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ C)) = (ord_less_nat @ A @ B))))). % add_less_cancel_right
thf(fact_55_add__less__cancel__left, axiom,
    ((![C : nat, A : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ C @ A) @ (plus_plus_nat @ C @ B)) = (ord_less_nat @ A @ B))))). % add_less_cancel_left
thf(fact_56_add__diff__cancel__right_H, axiom,
    ((![A : complex, B : complex]: ((minus_minus_complex @ (plus_plus_complex @ A @ B) @ B) = A)))). % add_diff_cancel_right'
thf(fact_57_add__diff__cancel__right_H, axiom,
    ((![A : nat, B : nat]: ((minus_minus_nat @ (plus_plus_nat @ A @ B) @ B) = A)))). % add_diff_cancel_right'
thf(fact_58_add__gr__0, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ (plus_plus_nat @ M @ N)) = (((ord_less_nat @ zero_zero_nat @ M)) | ((ord_less_nat @ zero_zero_nat @ N))))))). % add_gr_0
thf(fact_59_nat__0__less__mult__iff, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ (times_times_nat @ M @ N)) = (((ord_less_nat @ zero_zero_nat @ M)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % nat_0_less_mult_iff
thf(fact_60_mult__less__cancel2, axiom,
    ((![M : nat, K : nat, N : nat]: ((ord_less_nat @ (times_times_nat @ M @ K) @ (times_times_nat @ N @ K)) = (((ord_less_nat @ zero_zero_nat @ K)) & ((ord_less_nat @ M @ N))))))). % mult_less_cancel2
thf(fact_61_bot__nat__0_Onot__eq__extremum, axiom,
    ((![A : nat]: ((~ ((A = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ A))))). % bot_nat_0.not_eq_extremum
thf(fact_62_less__nat__zero__code, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_63_neq0__conv, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N))))). % neq0_conv
thf(fact_64_less__add__same__cancel2, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ (plus_plus_nat @ B @ A)) = (ord_less_nat @ zero_zero_nat @ B))))). % less_add_same_cancel2
thf(fact_65_less__add__same__cancel1, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ A @ (plus_plus_nat @ A @ B)) = (ord_less_nat @ zero_zero_nat @ B))))). % less_add_same_cancel1
thf(fact_66_add__less__same__cancel2, axiom,
    ((![A : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ A @ B) @ B) = (ord_less_nat @ A @ zero_zero_nat))))). % add_less_same_cancel2
thf(fact_67_add__less__same__cancel1, axiom,
    ((![B : nat, A : nat]: ((ord_less_nat @ (plus_plus_nat @ B @ A) @ B) = (ord_less_nat @ A @ zero_zero_nat))))). % add_less_same_cancel1
thf(fact_68_zero__less__diff, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ zero_zero_nat @ (minus_minus_nat @ N @ M)) = (ord_less_nat @ M @ N))))). % zero_less_diff
thf(fact_69_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_70_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_71_add__lessD1, axiom,
    ((![I2 : nat, J : nat, K : nat]: ((ord_less_nat @ (plus_plus_nat @ I2 @ J) @ K) => (ord_less_nat @ I2 @ K))))). % add_lessD1
thf(fact_72_nat__neq__iff, axiom,
    ((![M : nat, N : nat]: ((~ ((M = N))) = (((ord_less_nat @ M @ N)) | ((ord_less_nat @ N @ M))))))). % nat_neq_iff
thf(fact_73_add__less__mono, axiom,
    ((![I2 : nat, J : nat, K : nat, L : nat]: ((ord_less_nat @ I2 @ J) => ((ord_less_nat @ K @ L) => (ord_less_nat @ (plus_plus_nat @ I2 @ K) @ (plus_plus_nat @ J @ L))))))). % add_less_mono
thf(fact_74_less__not__refl, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_not_refl
thf(fact_75_not__add__less1, axiom,
    ((![I2 : nat, J : nat]: (~ ((ord_less_nat @ (plus_plus_nat @ I2 @ J) @ I2)))))). % not_add_less1
thf(fact_76_not__add__less2, axiom,
    ((![J : nat, I2 : nat]: (~ ((ord_less_nat @ (plus_plus_nat @ J @ I2) @ I2)))))). % not_add_less2
thf(fact_77_add__less__mono1, axiom,
    ((![I2 : nat, J : nat, K : nat]: ((ord_less_nat @ I2 @ J) => (ord_less_nat @ (plus_plus_nat @ I2 @ K) @ (plus_plus_nat @ J @ K)))))). % add_less_mono1
thf(fact_78_less__not__refl2, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ N @ M) => (~ ((M = N))))))). % less_not_refl2
thf(fact_79_less__not__refl3, axiom,
    ((![S : nat, T : nat]: ((ord_less_nat @ S @ T) => (~ ((S = T))))))). % less_not_refl3
thf(fact_80_less__irrefl__nat, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_irrefl_nat
thf(fact_81_nat__less__induct, axiom,
    ((![P : nat > $o, N : nat]: ((![N2 : nat]: ((![M2 : nat]: ((ord_less_nat @ M2 @ N2) => (P @ M2))) => (P @ N2))) => (P @ N))))). % nat_less_induct
thf(fact_82_trans__less__add1, axiom,
    ((![I2 : nat, J : nat, M : nat]: ((ord_less_nat @ I2 @ J) => (ord_less_nat @ I2 @ (plus_plus_nat @ J @ M)))))). % trans_less_add1
thf(fact_83_trans__less__add2, axiom,
    ((![I2 : nat, J : nat, M : nat]: ((ord_less_nat @ I2 @ J) => (ord_less_nat @ I2 @ (plus_plus_nat @ M @ J)))))). % trans_less_add2
thf(fact_84_infinite__descent, axiom,
    ((![P : nat > $o, N : nat]: ((![N2 : nat]: ((~ ((P @ N2))) => (?[M2 : nat]: ((ord_less_nat @ M2 @ N2) & (~ ((P @ M2))))))) => (P @ N))))). % infinite_descent
thf(fact_85_less__add__eq__less, axiom,
    ((![K : nat, L : nat, M : nat, N : nat]: ((ord_less_nat @ K @ L) => (((plus_plus_nat @ M @ L) = (plus_plus_nat @ K @ N)) => (ord_less_nat @ M @ N)))))). % less_add_eq_less
thf(fact_86_linorder__neqE__nat, axiom,
    ((![X : nat, Y : nat]: ((~ ((X = Y))) => ((~ ((ord_less_nat @ X @ Y))) => (ord_less_nat @ Y @ X)))))). % linorder_neqE_nat
thf(fact_87_less__imp__add__positive, axiom,
    ((![I2 : nat, J : nat]: ((ord_less_nat @ I2 @ J) => (?[K2 : nat]: ((ord_less_nat @ zero_zero_nat @ K2) & ((plus_plus_nat @ I2 @ K2) = J))))))). % less_imp_add_positive
thf(fact_88_mult__less__mono2, axiom,
    ((![I2 : nat, J : nat, K : nat]: ((ord_less_nat @ I2 @ J) => ((ord_less_nat @ zero_zero_nat @ K) => (ord_less_nat @ (times_times_nat @ K @ I2) @ (times_times_nat @ K @ J))))))). % mult_less_mono2
thf(fact_89_mult__less__mono1, axiom,
    ((![I2 : nat, J : nat, K : nat]: ((ord_less_nat @ I2 @ J) => ((ord_less_nat @ zero_zero_nat @ K) => (ord_less_nat @ (times_times_nat @ I2 @ K) @ (times_times_nat @ J @ K))))))). % mult_less_mono1
thf(fact_90_add__diff__inverse__nat, axiom,
    ((![M : nat, N : nat]: ((~ ((ord_less_nat @ M @ N))) => ((plus_plus_nat @ N @ (minus_minus_nat @ M @ N)) = M))))). % add_diff_inverse_nat
thf(fact_91_less__diff__conv, axiom,
    ((![I2 : nat, J : nat, K : nat]: ((ord_less_nat @ I2 @ (minus_minus_nat @ J @ K)) = (ord_less_nat @ (plus_plus_nat @ I2 @ K) @ J))))). % less_diff_conv
thf(fact_92_power__mult, axiom,
    ((![A : complex, M : nat, N : nat]: ((power_power_complex @ A @ (times_times_nat @ M @ N)) = (power_power_complex @ (power_power_complex @ A @ M) @ N))))). % power_mult
thf(fact_93_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_94_plus__nat_Oadd__0, axiom,
    ((![N : nat]: ((plus_plus_nat @ zero_zero_nat @ N) = N)))). % plus_nat.add_0
thf(fact_95_mult__0, axiom,
    ((![N : nat]: ((times_times_nat @ zero_zero_nat @ N) = zero_zero_nat)))). % mult_0
thf(fact_96_diff__add__inverse2, axiom,
    ((![M : nat, N : nat]: ((minus_minus_nat @ (plus_plus_nat @ M @ N) @ N) = M)))). % diff_add_inverse2
thf(fact_97_diff__add__inverse, axiom,
    ((![N : nat, M : nat]: ((minus_minus_nat @ (plus_plus_nat @ N @ M) @ N) = M)))). % diff_add_inverse
thf(fact_98_diff__cancel2, axiom,
    ((![M : nat, K : nat, N : nat]: ((minus_minus_nat @ (plus_plus_nat @ M @ K) @ (plus_plus_nat @ N @ K)) = (minus_minus_nat @ M @ N))))). % diff_cancel2
thf(fact_99_Nat_Odiff__cancel, axiom,
    ((![K : nat, M : nat, N : nat]: ((minus_minus_nat @ (plus_plus_nat @ K @ M) @ (plus_plus_nat @ K @ N)) = (minus_minus_nat @ M @ N))))). % Nat.diff_cancel
thf(fact_100_diff__mult__distrib2, axiom,
    ((![K : nat, M : nat, N : nat]: ((times_times_nat @ K @ (minus_minus_nat @ M @ N)) = (minus_minus_nat @ (times_times_nat @ K @ M) @ (times_times_nat @ K @ N)))))). % diff_mult_distrib2
thf(fact_101_diff__mult__distrib, axiom,
    ((![M : nat, N : nat, K : nat]: ((times_times_nat @ (minus_minus_nat @ M @ N) @ K) = (minus_minus_nat @ (times_times_nat @ M @ K) @ (times_times_nat @ N @ K)))))). % diff_mult_distrib
thf(fact_102_nat__diff__split__asm, axiom,
    ((![P : nat > $o, A : nat, B : nat]: ((P @ (minus_minus_nat @ A @ B)) = (~ ((((((ord_less_nat @ A @ B)) & ((~ ((P @ zero_zero_nat)))))) | ((?[D : nat]: (((A = (plus_plus_nat @ B @ D))) & ((~ ((P @ D)))))))))))))). % nat_diff_split_asm
thf(fact_103_nat__diff__split, axiom,
    ((![P : nat > $o, A : nat, B : nat]: ((P @ (minus_minus_nat @ A @ B)) = (((((ord_less_nat @ A @ B)) => ((P @ zero_zero_nat)))) & ((![D : nat]: (((A = (plus_plus_nat @ B @ D))) => ((P @ D)))))))))). % nat_diff_split
thf(fact_104_power__add, axiom,
    ((![A : complex, M : nat, N : nat]: ((power_power_complex @ A @ (plus_plus_nat @ M @ N)) = (times_times_complex @ (power_power_complex @ A @ M) @ (power_power_complex @ A @ N)))))). % power_add
thf(fact_105_power__add, axiom,
    ((![A : nat, M : nat, N : nat]: ((power_power_nat @ A @ (plus_plus_nat @ M @ N)) = (times_times_nat @ (power_power_nat @ A @ M) @ (power_power_nat @ A @ N)))))). % power_add
thf(fact_106_diff__add__0, axiom,
    ((![N : nat, M : nat]: ((minus_minus_nat @ N @ (plus_plus_nat @ N @ M)) = zero_zero_nat)))). % diff_add_0
thf(fact_107_root__nonzero, axiom,
    ((![N : nat]: (~ (((fFT_Mirabelle_root @ N) = zero_zero_complex)))))). % root_nonzero
thf(fact_108_zero__less__iff__neq__zero, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) = (~ ((N = zero_zero_nat))))))). % zero_less_iff_neq_zero
thf(fact_109_gr__implies__not__zero, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not_zero
thf(fact_110_not__less__zero, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less_zero
thf(fact_111_gr__zeroI, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr_zeroI
thf(fact_112_add__less__imp__less__right, axiom,
    ((![A : nat, C : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ C)) => (ord_less_nat @ A @ B))))). % add_less_imp_less_right
thf(fact_113_add__less__imp__less__left, axiom,
    ((![C : nat, A : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ C @ A) @ (plus_plus_nat @ C @ B)) => (ord_less_nat @ A @ B))))). % add_less_imp_less_left
thf(fact_114_add__strict__right__mono, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => (ord_less_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ C)))))). % add_strict_right_mono
thf(fact_115_add__strict__left__mono, axiom,
    ((![A : nat, B : nat, C : nat]: ((ord_less_nat @ A @ B) => (ord_less_nat @ (plus_plus_nat @ C @ A) @ (plus_plus_nat @ C @ B)))))). % add_strict_left_mono
thf(fact_116_add__strict__mono, axiom,
    ((![A : nat, B : nat, C : nat, D2 : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ C @ D2) => (ord_less_nat @ (plus_plus_nat @ A @ C) @ (plus_plus_nat @ B @ D2))))))). % add_strict_mono
thf(fact_117_add__mono__thms__linordered__field_I1_J, axiom,
    ((![I2 : nat, J : nat, K : nat, L : nat]: (((ord_less_nat @ I2 @ J) & (K = L)) => (ord_less_nat @ (plus_plus_nat @ I2 @ K) @ (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_field(1)
thf(fact_118_add__mono__thms__linordered__field_I2_J, axiom,
    ((![I2 : nat, J : nat, K : nat, L : nat]: (((I2 = J) & (ord_less_nat @ K @ L)) => (ord_less_nat @ (plus_plus_nat @ I2 @ K) @ (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_field(2)
thf(fact_119_add__mono__thms__linordered__field_I5_J, axiom,
    ((![I2 : nat, J : nat, K : nat, L : nat]: (((ord_less_nat @ I2 @ J) & (ord_less_nat @ K @ L)) => (ord_less_nat @ (plus_plus_nat @ I2 @ K) @ (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_field(5)
thf(fact_120_bot__nat__0_Oextremum__strict, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ zero_zero_nat)))))). % bot_nat_0.extremum_strict
thf(fact_121_infinite__descent0, axiom,
    ((![P : nat > $o, N : nat]: ((P @ zero_zero_nat) => ((![N2 : nat]: ((ord_less_nat @ zero_zero_nat @ N2) => ((~ ((P @ N2))) => (?[M2 : nat]: ((ord_less_nat @ M2 @ N2) & (~ ((P @ M2)))))))) => (P @ N)))))). % infinite_descent0
thf(fact_122_gr__implies__not0, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not0
thf(fact_123_less__zeroE, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_zeroE
thf(fact_124_not__less0, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less0
thf(fact_125_not__gr0, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr0
thf(fact_126_gr0I, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr0I
thf(fact_127_less__imp__diff__less, axiom,
    ((![J : nat, K : nat, N : nat]: ((ord_less_nat @ J @ K) => (ord_less_nat @ (minus_minus_nat @ J @ N) @ K))))). % less_imp_diff_less
thf(fact_128_diff__less__mono2, axiom,
    ((![M : nat, N : nat, L : nat]: ((ord_less_nat @ M @ N) => ((ord_less_nat @ M @ L) => (ord_less_nat @ (minus_minus_nat @ L @ N) @ (minus_minus_nat @ L @ M))))))). % diff_less_mono2
thf(fact_129_root__cancel, axiom,
    ((![D2 : nat, N : nat, K : nat]: ((ord_less_nat @ zero_zero_nat @ D2) => ((power_power_complex @ (fFT_Mirabelle_root @ (times_times_nat @ D2 @ N)) @ (times_times_nat @ D2 @ K)) = (power_power_complex @ (fFT_Mirabelle_root @ N) @ K)))))). % root_cancel
thf(fact_130_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_131_zero__reorient, axiom,
    ((![X : complex]: ((zero_zero_complex = X) = (X = zero_zero_complex))))). % zero_reorient
thf(fact_132_mult_Oleft__commute, axiom,
    ((![B : complex, A : complex, C : complex]: ((times_times_complex @ B @ (times_times_complex @ A @ C)) = (times_times_complex @ A @ (times_times_complex @ B @ C)))))). % mult.left_commute
thf(fact_133_mult_Oleft__commute, axiom,
    ((![B : nat, A : nat, C : nat]: ((times_times_nat @ B @ (times_times_nat @ A @ C)) = (times_times_nat @ A @ (times_times_nat @ B @ C)))))). % mult.left_commute
thf(fact_134_mult_Ocommute, axiom,
    ((times_times_complex = (^[A3 : complex]: (^[B2 : complex]: (times_times_complex @ B2 @ A3)))))). % mult.commute
thf(fact_135_mult_Ocommute, axiom,
    ((times_times_nat = (^[A3 : nat]: (^[B2 : nat]: (times_times_nat @ B2 @ A3)))))). % mult.commute
thf(fact_136_mult_Oassoc, axiom,
    ((![A : complex, B : complex, C : complex]: ((times_times_complex @ (times_times_complex @ A @ B) @ C) = (times_times_complex @ A @ (times_times_complex @ B @ C)))))). % mult.assoc
thf(fact_137_mult_Oassoc, axiom,
    ((![A : nat, B : nat, C : nat]: ((times_times_nat @ (times_times_nat @ A @ B) @ C) = (times_times_nat @ A @ (times_times_nat @ B @ C)))))). % mult.assoc
thf(fact_138_ab__semigroup__mult__class_Omult__ac_I1_J, axiom,
    ((![A : complex, B : complex, C : complex]: ((times_times_complex @ (times_times_complex @ A @ B) @ C) = (times_times_complex @ A @ (times_times_complex @ B @ C)))))). % ab_semigroup_mult_class.mult_ac(1)
thf(fact_139_ab__semigroup__mult__class_Omult__ac_I1_J, axiom,
    ((![A : nat, B : nat, C : nat]: ((times_times_nat @ (times_times_nat @ A @ B) @ C) = (times_times_nat @ A @ (times_times_nat @ B @ C)))))). % ab_semigroup_mult_class.mult_ac(1)
thf(fact_140_add__right__imp__eq, axiom,
    ((![B : complex, A : complex, C : complex]: (((plus_plus_complex @ B @ A) = (plus_plus_complex @ C @ A)) => (B = C))))). % add_right_imp_eq
thf(fact_141_add__right__imp__eq, axiom,
    ((![B : nat, A : nat, C : nat]: (((plus_plus_nat @ B @ A) = (plus_plus_nat @ C @ A)) => (B = C))))). % add_right_imp_eq
thf(fact_142_add__left__imp__eq, axiom,
    ((![A : complex, B : complex, C : complex]: (((plus_plus_complex @ A @ B) = (plus_plus_complex @ A @ C)) => (B = C))))). % add_left_imp_eq
thf(fact_143_add__left__imp__eq, axiom,
    ((![A : nat, B : nat, C : nat]: (((plus_plus_nat @ A @ B) = (plus_plus_nat @ A @ C)) => (B = C))))). % add_left_imp_eq
thf(fact_144_add_Oleft__commute, axiom,
    ((![B : complex, A : complex, C : complex]: ((plus_plus_complex @ B @ (plus_plus_complex @ A @ C)) = (plus_plus_complex @ A @ (plus_plus_complex @ B @ C)))))). % add.left_commute
thf(fact_145_add_Oleft__commute, axiom,
    ((![B : nat, A : nat, C : nat]: ((plus_plus_nat @ B @ (plus_plus_nat @ A @ C)) = (plus_plus_nat @ A @ (plus_plus_nat @ B @ C)))))). % add.left_commute
thf(fact_146_add_Ocommute, axiom,
    ((plus_plus_complex = (^[A3 : complex]: (^[B2 : complex]: (plus_plus_complex @ B2 @ A3)))))). % add.commute
thf(fact_147_add_Ocommute, axiom,
    ((plus_plus_nat = (^[A3 : nat]: (^[B2 : nat]: (plus_plus_nat @ B2 @ A3)))))). % add.commute
thf(fact_148_add_Oright__cancel, axiom,
    ((![B : complex, A : complex, C : complex]: (((plus_plus_complex @ B @ A) = (plus_plus_complex @ C @ A)) = (B = C))))). % add.right_cancel
thf(fact_149_add_Oleft__cancel, axiom,
    ((![A : complex, B : complex, C : complex]: (((plus_plus_complex @ A @ B) = (plus_plus_complex @ A @ C)) = (B = C))))). % add.left_cancel
thf(fact_150_add_Oassoc, axiom,
    ((![A : complex, B : complex, C : complex]: ((plus_plus_complex @ (plus_plus_complex @ A @ B) @ C) = (plus_plus_complex @ A @ (plus_plus_complex @ B @ C)))))). % add.assoc
thf(fact_151_add_Oassoc, axiom,
    ((![A : nat, B : nat, C : nat]: ((plus_plus_nat @ (plus_plus_nat @ A @ B) @ C) = (plus_plus_nat @ A @ (plus_plus_nat @ B @ C)))))). % add.assoc
thf(fact_152_group__cancel_Oadd2, axiom,
    ((![B3 : complex, K : complex, B : complex, A : complex]: ((B3 = (plus_plus_complex @ K @ B)) => ((plus_plus_complex @ A @ B3) = (plus_plus_complex @ K @ (plus_plus_complex @ A @ B))))))). % group_cancel.add2
thf(fact_153_group__cancel_Oadd2, axiom,
    ((![B3 : nat, K : nat, B : nat, A : nat]: ((B3 = (plus_plus_nat @ K @ B)) => ((plus_plus_nat @ A @ B3) = (plus_plus_nat @ K @ (plus_plus_nat @ A @ B))))))). % group_cancel.add2
thf(fact_154_group__cancel_Oadd1, axiom,
    ((![A2 : complex, K : complex, A : complex, B : complex]: ((A2 = (plus_plus_complex @ K @ A)) => ((plus_plus_complex @ A2 @ B) = (plus_plus_complex @ K @ (plus_plus_complex @ A @ B))))))). % group_cancel.add1
thf(fact_155_group__cancel_Oadd1, axiom,
    ((![A2 : nat, K : nat, A : nat, B : nat]: ((A2 = (plus_plus_nat @ K @ A)) => ((plus_plus_nat @ A2 @ B) = (plus_plus_nat @ K @ (plus_plus_nat @ A @ B))))))). % group_cancel.add1
thf(fact_156_add__mono__thms__linordered__semiring_I4_J, axiom,
    ((![I2 : nat, J : nat, K : nat, L : nat]: (((I2 = J) & (K = L)) => ((plus_plus_nat @ I2 @ K) = (plus_plus_nat @ J @ L)))))). % add_mono_thms_linordered_semiring(4)
thf(fact_157_ab__semigroup__add__class_Oadd__ac_I1_J, axiom,
    ((![A : complex, B : complex, C : complex]: ((plus_plus_complex @ (plus_plus_complex @ A @ B) @ C) = (plus_plus_complex @ A @ (plus_plus_complex @ B @ C)))))). % ab_semigroup_add_class.add_ac(1)
thf(fact_158_ab__semigroup__add__class_Oadd__ac_I1_J, axiom,
    ((![A : nat, B : nat, C : nat]: ((plus_plus_nat @ (plus_plus_nat @ A @ B) @ C) = (plus_plus_nat @ A @ (plus_plus_nat @ B @ C)))))). % ab_semigroup_add_class.add_ac(1)
thf(fact_159_diff__right__commute, axiom,
    ((![A : nat, C : nat, B : nat]: ((minus_minus_nat @ (minus_minus_nat @ A @ C) @ B) = (minus_minus_nat @ (minus_minus_nat @ A @ B) @ C))))). % diff_right_commute
thf(fact_160_sum_Oreindex__bij__witness, axiom,
    ((![S2 : set_nat, I2 : nat > nat, J : nat > nat, T2 : set_nat, H : nat > complex, G : nat > complex]: ((![A4 : nat]: ((member_nat @ A4 @ S2) => ((I2 @ (J @ A4)) = A4))) => ((![A4 : nat]: ((member_nat @ A4 @ S2) => (member_nat @ (J @ A4) @ T2))) => ((![B4 : nat]: ((member_nat @ B4 @ T2) => ((J @ (I2 @ B4)) = B4))) => ((![B4 : nat]: ((member_nat @ B4 @ T2) => (member_nat @ (I2 @ B4) @ S2))) => ((![A4 : nat]: ((member_nat @ A4 @ S2) => ((H @ (J @ A4)) = (G @ A4)))) => ((groups59700922omplex @ G @ S2) = (groups59700922omplex @ H @ T2)))))))))). % sum.reindex_bij_witness
thf(fact_161_sum_Oreindex__bij__witness, axiom,
    ((![S2 : set_nat, I2 : nat > nat, J : nat > nat, T2 : set_nat, H : nat > nat, G : nat > nat]: ((![A4 : nat]: ((member_nat @ A4 @ S2) => ((I2 @ (J @ A4)) = A4))) => ((![A4 : nat]: ((member_nat @ A4 @ S2) => (member_nat @ (J @ A4) @ T2))) => ((![B4 : nat]: ((member_nat @ B4 @ T2) => ((J @ (I2 @ B4)) = B4))) => ((![B4 : nat]: ((member_nat @ B4 @ T2) => (member_nat @ (I2 @ B4) @ S2))) => ((![A4 : nat]: ((member_nat @ A4 @ S2) => ((H @ (J @ A4)) = (G @ A4)))) => ((groups1842438620at_nat @ G @ S2) = (groups1842438620at_nat @ H @ T2)))))))))). % sum.reindex_bij_witness
thf(fact_162_sum_Oeq__general__inverses, axiom,
    ((![B3 : set_nat, K : nat > nat, A2 : set_nat, H : nat > nat, Gamma : nat > complex, Phi : nat > complex]: ((![Y2 : nat]: ((member_nat @ Y2 @ B3) => ((member_nat @ (K @ Y2) @ A2) & ((H @ (K @ Y2)) = Y2)))) => ((![X2 : nat]: ((member_nat @ X2 @ A2) => ((member_nat @ (H @ X2) @ B3) & (((K @ (H @ X2)) = X2) & ((Gamma @ (H @ X2)) = (Phi @ X2)))))) => ((groups59700922omplex @ Phi @ A2) = (groups59700922omplex @ Gamma @ B3))))))). % sum.eq_general_inverses
thf(fact_163_sum_Oeq__general__inverses, axiom,
    ((![B3 : set_nat, K : nat > nat, A2 : set_nat, H : nat > nat, Gamma : nat > nat, Phi : nat > nat]: ((![Y2 : nat]: ((member_nat @ Y2 @ B3) => ((member_nat @ (K @ Y2) @ A2) & ((H @ (K @ Y2)) = Y2)))) => ((![X2 : nat]: ((member_nat @ X2 @ A2) => ((member_nat @ (H @ X2) @ B3) & (((K @ (H @ X2)) = X2) & ((Gamma @ (H @ X2)) = (Phi @ X2)))))) => ((groups1842438620at_nat @ Phi @ A2) = (groups1842438620at_nat @ Gamma @ B3))))))). % sum.eq_general_inverses

% Conjectures (1)
thf(conj_0, conjecture,
    (((plus_plus_complex @ (times_times_complex @ (groups59700922omplex @ (power_power_complex @ (power_power_complex @ (fFT_Mirabelle_root @ n) @ (minus_minus_nat @ i @ i))) @ (set_or562006527an_nat @ zero_zero_nat @ n)) @ (a @ i)) @ (groups59700922omplex @ (^[J2 : nat]: (times_times_complex @ (groups59700922omplex @ (power_power_complex @ (power_power_complex @ (fFT_Mirabelle_root @ n) @ (minus_minus_nat @ J2 @ i))) @ (set_or562006527an_nat @ zero_zero_nat @ n)) @ (a @ J2))) @ (set_or1544565540an_nat @ i @ n))) = (times_times_complex @ (groups59700922omplex @ (power_power_complex @ (power_power_complex @ (fFT_Mirabelle_root @ n) @ (minus_minus_nat @ i @ i))) @ (set_or562006527an_nat @ zero_zero_nat @ n)) @ (a @ i))))).
