% 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_438__3227588_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:11:18.540

% 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 (25)
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_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J, type,
    minus_minus_set_nat : set_nat > set_nat > set_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_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J, type,
    sup_sup_set_nat : set_nat > set_nat > set_nat).
thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J, type,
    bot_bot_set_nat : set_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_OCollect_001t__Nat__Onat, type,
    collect_nat : (nat > $o) > set_nat).
thf(sy_c_Set_Oinsert_001t__Nat__Onat, type,
    insert_nat : nat > set_nat > set_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 (152)
thf(fact_0_i__less, axiom,
    ((ord_less_nat @ i @ n))). % i_less
thf(fact_1__092_060open_062_I_092_060Sum_062j_A_061_Ai_O_O_060n_O_Asum_A_I_I_094_J_A_IFFT__Mirabelle__ulikgskiun_Oroot_An_A_094_A_Ij_A_N_Ai_J_J_J_A_1230_O_O_060n_125_A_K_Aa_Aj_J_A_061_A_I_092_060Sum_062j_092_060in_062_123i_125_A_092_060union_062_A_123i_060_O_O_060n_125_O_Asum_A_I_I_094_J_A_IFFT__Mirabelle__ulikgskiun_Oroot_An_A_094_A_Ij_A_N_Ai_J_J_J_A_1230_O_O_060n_125_A_K_Aa_Aj_J_092_060close_062, axiom,
    (((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))) @ (sup_sup_set_nat @ (insert_nat @ i @ bot_bot_set_nat) @ (set_or1544565540an_nat @ i @ n)))))). % \<open>(\<Sum>j = i..<n. sum ((^) (FFT_Mirabelle_ulikgskiun.root n ^ (j - i))) {0..<n} * a j) = (\<Sum>j\<in>{i} \<union> {i<..<n}. sum ((^) (FFT_Mirabelle_ulikgskiun.root n ^ (j - i))) {0..<n} * a j)\<close>
thf(fact_2_sum_Oempty, axiom,
    ((![G : nat > nat]: ((groups1842438620at_nat @ G @ bot_bot_set_nat) = zero_zero_nat)))). % sum.empty
thf(fact_3_sum_Oempty, axiom,
    ((![G : nat > complex]: ((groups59700922omplex @ G @ bot_bot_set_nat) = zero_zero_complex)))). % sum.empty
thf(fact_4_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_5_singleton__conv, axiom,
    ((![A : nat]: ((collect_nat @ (^[X : nat]: (X = A))) = (insert_nat @ A @ bot_bot_set_nat))))). % singleton_conv
thf(fact_6_singleton__conv2, axiom,
    ((![A : nat]: ((collect_nat @ ((^[Y : nat]: (^[Z : nat]: (Y = Z))) @ A)) = (insert_nat @ A @ bot_bot_set_nat))))). % singleton_conv2
thf(fact_7_sum_Oneutral__const, axiom,
    ((![A2 : set_nat]: ((groups1842438620at_nat @ (^[Uu : nat]: zero_zero_nat) @ A2) = zero_zero_nat)))). % sum.neutral_const
thf(fact_8_sum_Oneutral__const, axiom,
    ((![A2 : set_nat]: ((groups59700922omplex @ (^[Uu : nat]: zero_zero_complex) @ A2) = zero_zero_complex)))). % sum.neutral_const
thf(fact_9_Un__insert__left, axiom,
    ((![A : nat, B2 : set_nat, C : set_nat]: ((sup_sup_set_nat @ (insert_nat @ A @ B2) @ C) = (insert_nat @ A @ (sup_sup_set_nat @ B2 @ C)))))). % Un_insert_left
thf(fact_10_Un__insert__right, axiom,
    ((![A2 : set_nat, A : nat, B2 : set_nat]: ((sup_sup_set_nat @ A2 @ (insert_nat @ A @ B2)) = (insert_nat @ A @ (sup_sup_set_nat @ A2 @ B2)))))). % Un_insert_right
thf(fact_11_Un__empty, axiom,
    ((![A2 : set_nat, B2 : set_nat]: (((sup_sup_set_nat @ A2 @ B2) = bot_bot_set_nat) = (((A2 = bot_bot_set_nat)) & ((B2 = bot_bot_set_nat))))))). % Un_empty
thf(fact_12_diff__0__eq__0, axiom,
    ((![N : nat]: ((minus_minus_nat @ zero_zero_nat @ N) = zero_zero_nat)))). % diff_0_eq_0
thf(fact_13_diff__self__eq__0, axiom,
    ((![M : nat]: ((minus_minus_nat @ M @ M) = zero_zero_nat)))). % diff_self_eq_0
thf(fact_14_singletonI, axiom,
    ((![A : nat]: (member_nat @ A @ (insert_nat @ A @ bot_bot_set_nat))))). % singletonI
thf(fact_15_add__right__cancel, axiom,
    ((![B : complex, A : complex, C2 : complex]: (((plus_plus_complex @ B @ A) = (plus_plus_complex @ C2 @ A)) = (B = C2))))). % add_right_cancel
thf(fact_16_add__right__cancel, axiom,
    ((![B : nat, A : nat, C2 : nat]: (((plus_plus_nat @ B @ A) = (plus_plus_nat @ C2 @ A)) = (B = C2))))). % add_right_cancel
thf(fact_17_add__left__cancel, axiom,
    ((![A : complex, B : complex, C2 : complex]: (((plus_plus_complex @ A @ B) = (plus_plus_complex @ A @ C2)) = (B = C2))))). % add_left_cancel
thf(fact_18_add__left__cancel, axiom,
    ((![A : nat, B : nat, C2 : nat]: (((plus_plus_nat @ A @ B) = (plus_plus_nat @ A @ C2)) = (B = C2))))). % add_left_cancel
thf(fact_19_Diff__cancel, axiom,
    ((![A2 : set_nat]: ((minus_minus_set_nat @ A2 @ A2) = bot_bot_set_nat)))). % Diff_cancel
thf(fact_20_empty__Diff, axiom,
    ((![A2 : set_nat]: ((minus_minus_set_nat @ bot_bot_set_nat @ A2) = bot_bot_set_nat)))). % empty_Diff
thf(fact_21_Diff__empty, axiom,
    ((![A2 : set_nat]: ((minus_minus_set_nat @ A2 @ bot_bot_set_nat) = A2)))). % Diff_empty
thf(fact_22_empty__Collect__eq, axiom,
    ((![P : nat > $o]: ((bot_bot_set_nat = (collect_nat @ P)) = (![X : nat]: (~ ((P @ X)))))))). % empty_Collect_eq
thf(fact_23_Collect__empty__eq, axiom,
    ((![P : nat > $o]: (((collect_nat @ P) = bot_bot_set_nat) = (![X : nat]: (~ ((P @ X)))))))). % Collect_empty_eq
thf(fact_24_all__not__in__conv, axiom,
    ((![A2 : set_nat]: ((![X : nat]: (~ ((member_nat @ X @ A2)))) = (A2 = bot_bot_set_nat))))). % all_not_in_conv
thf(fact_25_empty__iff, axiom,
    ((![C2 : nat]: (~ ((member_nat @ C2 @ bot_bot_set_nat)))))). % empty_iff
thf(fact_26_Nat_Oadd__0__right, axiom,
    ((![M : nat]: ((plus_plus_nat @ M @ zero_zero_nat) = M)))). % Nat.add_0_right
thf(fact_27_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_28_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_29_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_30_mult__0__right, axiom,
    ((![M : nat]: ((times_times_nat @ M @ zero_zero_nat) = zero_zero_nat)))). % mult_0_right
thf(fact_31_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_32_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_33_insert__Diff1, axiom,
    ((![X2 : nat, B2 : set_nat, A2 : set_nat]: ((member_nat @ X2 @ B2) => ((minus_minus_set_nat @ (insert_nat @ X2 @ A2) @ B2) = (minus_minus_set_nat @ A2 @ B2)))))). % insert_Diff1
thf(fact_34_Diff__insert0, axiom,
    ((![X2 : nat, A2 : set_nat, B2 : set_nat]: ((~ ((member_nat @ X2 @ A2))) => ((minus_minus_set_nat @ A2 @ (insert_nat @ X2 @ B2)) = (minus_minus_set_nat @ A2 @ B2)))))). % Diff_insert0
thf(fact_35_insert__absorb2, axiom,
    ((![X2 : nat, A2 : set_nat]: ((insert_nat @ X2 @ (insert_nat @ X2 @ A2)) = (insert_nat @ X2 @ A2))))). % insert_absorb2
thf(fact_36_insert__iff, axiom,
    ((![A : nat, B : nat, A2 : set_nat]: ((member_nat @ A @ (insert_nat @ B @ A2)) = (((A = B)) | ((member_nat @ A @ A2))))))). % insert_iff
thf(fact_37_insertCI, axiom,
    ((![A : nat, B2 : set_nat, B : nat]: (((~ ((member_nat @ A @ B2))) => (A = B)) => (member_nat @ A @ (insert_nat @ B @ B2)))))). % insertCI
thf(fact_38_diff__diff__left, axiom,
    ((![I : nat, J2 : nat, K : nat]: ((minus_minus_nat @ (minus_minus_nat @ I @ J2) @ K) = (minus_minus_nat @ I @ (plus_plus_nat @ J2 @ K)))))). % diff_diff_left
thf(fact_39_Un__Diff__cancel2, axiom,
    ((![B2 : set_nat, A2 : set_nat]: ((sup_sup_set_nat @ (minus_minus_set_nat @ B2 @ A2) @ A2) = (sup_sup_set_nat @ B2 @ A2))))). % Un_Diff_cancel2
thf(fact_40_Un__Diff__cancel, axiom,
    ((![A2 : set_nat, B2 : set_nat]: ((sup_sup_set_nat @ A2 @ (minus_minus_set_nat @ B2 @ A2)) = (sup_sup_set_nat @ A2 @ B2))))). % Un_Diff_cancel
thf(fact_41_Un__iff, axiom,
    ((![C2 : nat, A2 : set_nat, B2 : set_nat]: ((member_nat @ C2 @ (sup_sup_set_nat @ A2 @ B2)) = (((member_nat @ C2 @ A2)) | ((member_nat @ C2 @ B2))))))). % Un_iff
thf(fact_42_UnCI, axiom,
    ((![C2 : nat, B2 : set_nat, A2 : set_nat]: (((~ ((member_nat @ C2 @ B2))) => (member_nat @ C2 @ A2)) => (member_nat @ C2 @ (sup_sup_set_nat @ A2 @ B2)))))). % UnCI
thf(fact_43_i__diff, axiom,
    ((![K : nat]: (ord_less_nat @ (minus_minus_nat @ i @ K) @ n)))). % i_diff
thf(fact_44_diff__i, axiom,
    ((![K : nat]: ((ord_less_nat @ K @ n) => (ord_less_nat @ (minus_minus_nat @ K @ i) @ n))))). % diff_i
thf(fact_45_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_46_zero__eq__add__iff__both__eq__0, axiom,
    ((![X2 : nat, Y2 : nat]: ((zero_zero_nat = (plus_plus_nat @ X2 @ Y2)) = (((X2 = zero_zero_nat)) & ((Y2 = zero_zero_nat))))))). % zero_eq_add_iff_both_eq_0
thf(fact_47_mem__Collect__eq, axiom,
    ((![A : nat, P : nat > $o]: ((member_nat @ A @ (collect_nat @ P)) = (P @ A))))). % mem_Collect_eq
thf(fact_48_Collect__mem__eq, axiom,
    ((![A2 : set_nat]: ((collect_nat @ (^[X : nat]: (member_nat @ X @ A2))) = A2)))). % Collect_mem_eq
thf(fact_49_Collect__cong, axiom,
    ((![P : nat > $o, Q : nat > $o]: ((![X3 : nat]: ((P @ X3) = (Q @ X3))) => ((collect_nat @ P) = (collect_nat @ Q)))))). % Collect_cong
thf(fact_50_add__eq__0__iff__both__eq__0, axiom,
    ((![X2 : nat, Y2 : nat]: (((plus_plus_nat @ X2 @ Y2) = zero_zero_nat) = (((X2 = zero_zero_nat)) & ((Y2 = zero_zero_nat))))))). % add_eq_0_iff_both_eq_0
thf(fact_51_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_52_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_53_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_54_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_55_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_56_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_57_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_58_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_59_add_Oright__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ A @ zero_zero_nat) = A)))). % add.right_neutral
thf(fact_60_add_Oright__neutral, axiom,
    ((![A : complex]: ((plus_plus_complex @ A @ zero_zero_complex) = A)))). % add.right_neutral
thf(fact_61_add_Oleft__neutral, axiom,
    ((![A : nat]: ((plus_plus_nat @ zero_zero_nat @ A) = A)))). % add.left_neutral
thf(fact_62_add_Oleft__neutral, axiom,
    ((![A : complex]: ((plus_plus_complex @ zero_zero_complex @ A) = A)))). % add.left_neutral
thf(fact_63_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_64_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_65_diff__zero, axiom,
    ((![A : complex]: ((minus_minus_complex @ A @ zero_zero_complex) = A)))). % diff_zero
thf(fact_66_diff__zero, axiom,
    ((![A : nat]: ((minus_minus_nat @ A @ zero_zero_nat) = A)))). % diff_zero
thf(fact_67_zero__diff, axiom,
    ((![A : nat]: ((minus_minus_nat @ zero_zero_nat @ A) = zero_zero_nat)))). % zero_diff
thf(fact_68_diff__0__right, axiom,
    ((![A : complex]: ((minus_minus_complex @ A @ zero_zero_complex) = A)))). % diff_0_right
thf(fact_69_diff__self, axiom,
    ((![A : complex]: ((minus_minus_complex @ A @ A) = zero_zero_complex)))). % diff_self
thf(fact_70_add__less__cancel__right, axiom,
    ((![A : nat, C2 : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ A @ C2) @ (plus_plus_nat @ B @ C2)) = (ord_less_nat @ A @ B))))). % add_less_cancel_right
thf(fact_71_add__less__cancel__left, axiom,
    ((![C2 : nat, A : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ C2 @ A) @ (plus_plus_nat @ C2 @ B)) = (ord_less_nat @ A @ B))))). % add_less_cancel_left
thf(fact_72_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_73_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_74_add__diff__cancel__right, axiom,
    ((![A : complex, C2 : complex, B : complex]: ((minus_minus_complex @ (plus_plus_complex @ A @ C2) @ (plus_plus_complex @ B @ C2)) = (minus_minus_complex @ A @ B))))). % add_diff_cancel_right
thf(fact_75_add__diff__cancel__right, axiom,
    ((![A : nat, C2 : nat, B : nat]: ((minus_minus_nat @ (plus_plus_nat @ A @ C2) @ (plus_plus_nat @ B @ C2)) = (minus_minus_nat @ A @ B))))). % add_diff_cancel_right
thf(fact_76_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_77_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_78_add__diff__cancel__left, axiom,
    ((![C2 : complex, A : complex, B : complex]: ((minus_minus_complex @ (plus_plus_complex @ C2 @ A) @ (plus_plus_complex @ C2 @ B)) = (minus_minus_complex @ A @ B))))). % add_diff_cancel_left
thf(fact_79_add__diff__cancel__left, axiom,
    ((![C2 : nat, A : nat, B : nat]: ((minus_minus_nat @ (plus_plus_nat @ C2 @ A) @ (plus_plus_nat @ C2 @ B)) = (minus_minus_nat @ A @ B))))). % add_diff_cancel_left
thf(fact_80_diff__add__cancel, axiom,
    ((![A : complex, B : complex]: ((plus_plus_complex @ (minus_minus_complex @ A @ B) @ B) = A)))). % diff_add_cancel
thf(fact_81_add__diff__cancel, axiom,
    ((![A : complex, B : complex]: ((minus_minus_complex @ (plus_plus_complex @ A @ B) @ B) = A)))). % add_diff_cancel
thf(fact_82_insert__Diff__single, axiom,
    ((![A : nat, A2 : set_nat]: ((insert_nat @ A @ (minus_minus_set_nat @ A2 @ (insert_nat @ A @ bot_bot_set_nat))) = (insert_nat @ A @ A2))))). % insert_Diff_single
thf(fact_83_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_84_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_85_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_86_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_87_less__nat__zero__code, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_88_neq0__conv, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N))))). % neq0_conv
thf(fact_89_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_90_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_91_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_92_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_93_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_94_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_95_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_96_add__lessD1, axiom,
    ((![I : nat, J2 : nat, K : nat]: ((ord_less_nat @ (plus_plus_nat @ I @ J2) @ K) => (ord_less_nat @ I @ K))))). % add_lessD1
thf(fact_97_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_98_add__less__mono, axiom,
    ((![I : nat, J2 : nat, K : nat, L : nat]: ((ord_less_nat @ I @ J2) => ((ord_less_nat @ K @ L) => (ord_less_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J2 @ L))))))). % add_less_mono
thf(fact_99_less__not__refl, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_not_refl
thf(fact_100_not__add__less1, axiom,
    ((![I : nat, J2 : nat]: (~ ((ord_less_nat @ (plus_plus_nat @ I @ J2) @ I)))))). % not_add_less1
thf(fact_101_not__add__less2, axiom,
    ((![J2 : nat, I : nat]: (~ ((ord_less_nat @ (plus_plus_nat @ J2 @ I) @ I)))))). % not_add_less2
thf(fact_102_add__less__mono1, axiom,
    ((![I : nat, J2 : nat, K : nat]: ((ord_less_nat @ I @ J2) => (ord_less_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J2 @ K)))))). % add_less_mono1
thf(fact_103_less__diff__conv, axiom,
    ((![I : nat, J2 : nat, K : nat]: ((ord_less_nat @ I @ (minus_minus_nat @ J2 @ K)) = (ord_less_nat @ (plus_plus_nat @ I @ K) @ J2))))). % less_diff_conv
thf(fact_104_less__not__refl2, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ N @ M) => (~ ((M = N))))))). % less_not_refl2
thf(fact_105_less__not__refl3, axiom,
    ((![S : nat, T : nat]: ((ord_less_nat @ S @ T) => (~ ((S = T))))))). % less_not_refl3
thf(fact_106_less__irrefl__nat, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_irrefl_nat
thf(fact_107_mult__less__mono1, axiom,
    ((![I : nat, J2 : nat, K : nat]: ((ord_less_nat @ I @ J2) => ((ord_less_nat @ zero_zero_nat @ K) => (ord_less_nat @ (times_times_nat @ I @ K) @ (times_times_nat @ J2 @ K))))))). % mult_less_mono1
thf(fact_108_mult__less__mono2, axiom,
    ((![I : nat, J2 : nat, K : nat]: ((ord_less_nat @ I @ J2) => ((ord_less_nat @ zero_zero_nat @ K) => (ord_less_nat @ (times_times_nat @ K @ I) @ (times_times_nat @ K @ J2))))))). % mult_less_mono2
thf(fact_109_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_110_trans__less__add1, axiom,
    ((![I : nat, J2 : nat, M : nat]: ((ord_less_nat @ I @ J2) => (ord_less_nat @ I @ (plus_plus_nat @ J2 @ M)))))). % trans_less_add1
thf(fact_111_trans__less__add2, axiom,
    ((![I : nat, J2 : nat, M : nat]: ((ord_less_nat @ I @ J2) => (ord_less_nat @ I @ (plus_plus_nat @ M @ J2)))))). % trans_less_add2
thf(fact_112_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_113_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_114_linorder__neqE__nat, axiom,
    ((![X2 : nat, Y2 : nat]: ((~ ((X2 = Y2))) => ((~ ((ord_less_nat @ X2 @ Y2))) => (ord_less_nat @ Y2 @ X2)))))). % linorder_neqE_nat
thf(fact_115_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_116_less__imp__add__positive, axiom,
    ((![I : nat, J2 : nat]: ((ord_less_nat @ I @ J2) => (?[K2 : nat]: ((ord_less_nat @ zero_zero_nat @ K2) & ((plus_plus_nat @ I @ K2) = J2))))))). % less_imp_add_positive
thf(fact_117_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_118_gr__implies__not__zero, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not_zero
thf(fact_119_not__less__zero, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less_zero
thf(fact_120_gr__zeroI, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr_zeroI
thf(fact_121_add__less__imp__less__right, axiom,
    ((![A : nat, C2 : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ A @ C2) @ (plus_plus_nat @ B @ C2)) => (ord_less_nat @ A @ B))))). % add_less_imp_less_right
thf(fact_122_add__less__imp__less__left, axiom,
    ((![C2 : nat, A : nat, B : nat]: ((ord_less_nat @ (plus_plus_nat @ C2 @ A) @ (plus_plus_nat @ C2 @ B)) => (ord_less_nat @ A @ B))))). % add_less_imp_less_left
thf(fact_123_add__strict__right__mono, axiom,
    ((![A : nat, B : nat, C2 : nat]: ((ord_less_nat @ A @ B) => (ord_less_nat @ (plus_plus_nat @ A @ C2) @ (plus_plus_nat @ B @ C2)))))). % add_strict_right_mono
thf(fact_124_add__strict__left__mono, axiom,
    ((![A : nat, B : nat, C2 : nat]: ((ord_less_nat @ A @ B) => (ord_less_nat @ (plus_plus_nat @ C2 @ A) @ (plus_plus_nat @ C2 @ B)))))). % add_strict_left_mono
thf(fact_125_add__strict__mono, axiom,
    ((![A : nat, B : nat, C2 : nat, D : nat]: ((ord_less_nat @ A @ B) => ((ord_less_nat @ C2 @ D) => (ord_less_nat @ (plus_plus_nat @ A @ C2) @ (plus_plus_nat @ B @ D))))))). % add_strict_mono
thf(fact_126_add__mono__thms__linordered__field_I1_J, axiom,
    ((![I : nat, J2 : nat, K : nat, L : nat]: (((ord_less_nat @ I @ J2) & (K = L)) => (ord_less_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J2 @ L)))))). % add_mono_thms_linordered_field(1)
thf(fact_127_add__mono__thms__linordered__field_I2_J, axiom,
    ((![I : nat, J2 : nat, K : nat, L : nat]: (((I = J2) & (ord_less_nat @ K @ L)) => (ord_less_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J2 @ L)))))). % add_mono_thms_linordered_field(2)
thf(fact_128_add__mono__thms__linordered__field_I5_J, axiom,
    ((![I : nat, J2 : nat, K : nat, L : nat]: (((ord_less_nat @ I @ J2) & (ord_less_nat @ K @ L)) => (ord_less_nat @ (plus_plus_nat @ I @ K) @ (plus_plus_nat @ J2 @ L)))))). % add_mono_thms_linordered_field(5)
thf(fact_129_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_130_insert__Diff__if, axiom,
    ((![X2 : nat, B2 : set_nat, A2 : set_nat]: (((member_nat @ X2 @ B2) => ((minus_minus_set_nat @ (insert_nat @ X2 @ A2) @ B2) = (minus_minus_set_nat @ A2 @ B2))) & ((~ ((member_nat @ X2 @ B2))) => ((minus_minus_set_nat @ (insert_nat @ X2 @ A2) @ B2) = (insert_nat @ X2 @ (minus_minus_set_nat @ A2 @ B2)))))))). % insert_Diff_if
thf(fact_131_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_132_plus__nat_Oadd__0, axiom,
    ((![N : nat]: ((plus_plus_nat @ zero_zero_nat @ N) = N)))). % plus_nat.add_0
thf(fact_133_mult__0, axiom,
    ((![N : nat]: ((times_times_nat @ zero_zero_nat @ N) = zero_zero_nat)))). % mult_0
thf(fact_134_bot__nat__0_Oextremum__strict, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ zero_zero_nat)))))). % bot_nat_0.extremum_strict
thf(fact_135_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_136_gr__implies__not0, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not0
thf(fact_137_less__zeroE, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_zeroE
thf(fact_138_not__less0, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less0
thf(fact_139_not__gr0, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr0
thf(fact_140_gr0I, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr0I
thf(fact_141_Un__Diff, axiom,
    ((![A2 : set_nat, B2 : set_nat, C : set_nat]: ((minus_minus_set_nat @ (sup_sup_set_nat @ A2 @ B2) @ C) = (sup_sup_set_nat @ (minus_minus_set_nat @ A2 @ C) @ (minus_minus_set_nat @ B2 @ C)))))). % Un_Diff
thf(fact_142_diff__add__inverse2, axiom,
    ((![M : nat, N : nat]: ((minus_minus_nat @ (plus_plus_nat @ M @ N) @ N) = M)))). % diff_add_inverse2
thf(fact_143_diff__add__inverse, axiom,
    ((![N : nat, M : nat]: ((minus_minus_nat @ (plus_plus_nat @ N @ M) @ N) = M)))). % diff_add_inverse
thf(fact_144_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_145_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_146_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_147_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_148_less__imp__diff__less, axiom,
    ((![J2 : nat, K : nat, N : nat]: ((ord_less_nat @ J2 @ K) => (ord_less_nat @ (minus_minus_nat @ J2 @ N) @ K))))). % less_imp_diff_less
thf(fact_149_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_150_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)))))) | ((?[D2 : nat]: (((A = (plus_plus_nat @ B @ D2))) & ((~ ((P @ D2)))))))))))))). % nat_diff_split_asm
thf(fact_151_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)))) & ((![D2 : nat]: (((A = (plus_plus_nat @ B @ D2))) => ((P @ D2)))))))))). % nat_diff_split

% Conjectures (1)
thf(conj_0, conjecture,
    (((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))) @ (sup_sup_set_nat @ (insert_nat @ i @ bot_bot_set_nat) @ (set_or1544565540an_nat @ i @ n))) = (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 @ (^[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_or1544565540an_nat @ i @ n)))))).
