% 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_236__3225012_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:10:03.896

% Could-be-implicit typings (3)
thf(ty_n_t__Complex__Ocomplex, type,
    complex : $tType).
thf(ty_n_t__Num__Onum, type,
    num : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).

% Explicit typings (18)
thf(sy_c_FFT__Mirabelle__ulikgskiun_Oroot, type,
    fFT_Mirabelle_root : nat > complex).
thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex, type,
    one_one_complex : complex).
thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat, type,
    one_one_nat : nat).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Complex__Ocomplex, type,
    times_times_complex : complex > complex > complex).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Nat__Onat, type,
    times_times_nat : nat > nat > nat).
thf(sy_c_Groups_Otimes__class_Otimes_001t__Num__Onum, type,
    times_times_num : num > num > num).
thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Complex__Ocomplex, type,
    uminus1204672759omplex : complex > complex).
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_Num_Onum_OBit0, type,
    bit0 : num > num).
thf(sy_c_Num_Onum_OOne, type,
    one : num).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Complex__Ocomplex, type,
    numera632737353omplex : num > complex).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat, type,
    numeral_numeral_nat : num > nat).
thf(sy_c_Orderings_Oord__class_Oless_001t__Nat__Onat, type,
    ord_less_nat : nat > nat > $o).
thf(sy_c_Orderings_Oord__class_Oless_001t__Num__Onum, type,
    ord_less_num : num > num > $o).
thf(sy_c_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_v_n, type,
    n : nat).

% Relevant facts (149)
thf(fact_0__092_060open_062_092_060And_062d_O_A0_A_060_Ad_A_092_060Longrightarrow_062_AFFT__Mirabelle__ulikgskiun_Oroot_A_Id_A_K_A2_J_A_094_A_Id_A_K_A1_J_A_061_AFFT__Mirabelle__ulikgskiun_Oroot_A2_A_094_A1_092_060close_062, axiom,
    ((![D : nat]: ((ord_less_nat @ zero_zero_nat @ D) => ((power_power_complex @ (fFT_Mirabelle_root @ (times_times_nat @ D @ (numeral_numeral_nat @ (bit0 @ one)))) @ (times_times_nat @ D @ one_one_nat)) = (power_power_complex @ (fFT_Mirabelle_root @ (numeral_numeral_nat @ (bit0 @ one))) @ one_one_nat)))))). % \<open>\<And>d. 0 < d \<Longrightarrow> FFT_Mirabelle_ulikgskiun.root (d * 2) ^ (d * 1) = FFT_Mirabelle_ulikgskiun.root 2 ^ 1\<close>
thf(fact_1_root0, axiom,
    (((fFT_Mirabelle_root @ zero_zero_nat) = one_one_complex))). % root0
thf(fact_2_root2, axiom,
    (((fFT_Mirabelle_root @ (numeral_numeral_nat @ (bit0 @ one))) = (uminus1204672759omplex @ one_one_complex)))). % root2
thf(fact_3_root__unity, axiom,
    ((![N : nat]: ((power_power_complex @ (fFT_Mirabelle_root @ N) @ N) = one_one_complex)))). % root_unity
thf(fact_4_root__cancel, axiom,
    ((![D : nat, N : nat, K : nat]: ((ord_less_nat @ zero_zero_nat @ D) => ((power_power_complex @ (fFT_Mirabelle_root @ (times_times_nat @ D @ N)) @ (times_times_nat @ D @ K)) = (power_power_complex @ (fFT_Mirabelle_root @ N) @ K)))))). % root_cancel
thf(fact_5_root__cancel1, axiom,
    ((![M : nat, I : nat, J : nat]: ((power_power_complex @ (fFT_Mirabelle_root @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ M)) @ (times_times_nat @ I @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ J))) = (power_power_complex @ (fFT_Mirabelle_root @ M) @ (times_times_nat @ I @ J)))))). % root_cancel1
thf(fact_6_power__minus1__even, axiom,
    ((![N : nat]: ((power_power_complex @ (uminus1204672759omplex @ one_one_complex) @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N)) = one_one_complex)))). % power_minus1_even
thf(fact_7_Power_Oring__1__class_Opower__minus__even, axiom,
    ((![A : complex, N : nat]: ((power_power_complex @ (uminus1204672759omplex @ A) @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N)) = (power_power_complex @ A @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N)))))). % Power.ring_1_class.power_minus_even
thf(fact_8_power2__minus, axiom,
    ((![A : complex]: ((power_power_complex @ (uminus1204672759omplex @ A) @ (numeral_numeral_nat @ (bit0 @ one))) = (power_power_complex @ A @ (numeral_numeral_nat @ (bit0 @ one))))))). % power2_minus
thf(fact_9_zero__eq__power2, axiom,
    ((![A : complex]: (((power_power_complex @ A @ (numeral_numeral_nat @ (bit0 @ one))) = zero_zero_complex) = (A = zero_zero_complex))))). % zero_eq_power2
thf(fact_10_zero__eq__power2, axiom,
    ((![A : nat]: (((power_power_nat @ A @ (numeral_numeral_nat @ (bit0 @ one))) = zero_zero_nat) = (A = zero_zero_nat))))). % zero_eq_power2
thf(fact_11_power__strict__decreasing__iff, axiom,
    ((![B : nat, M : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ B) => ((ord_less_nat @ B @ one_one_nat) => ((ord_less_nat @ (power_power_nat @ B @ M) @ (power_power_nat @ B @ N)) = (ord_less_nat @ N @ M))))))). % power_strict_decreasing_iff
thf(fact_12_root1, axiom,
    (((fFT_Mirabelle_root @ one_one_nat) = one_one_complex))). % root1
thf(fact_13_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_14_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_15_power__strict__increasing__iff, axiom,
    ((![B : nat, X : nat, Y : nat]: ((ord_less_nat @ one_one_nat @ B) => ((ord_less_nat @ (power_power_nat @ B @ X) @ (power_power_nat @ B @ Y)) = (ord_less_nat @ X @ Y)))))). % power_strict_increasing_iff
thf(fact_16_minus__one__mult__self, axiom,
    ((![N : nat]: ((times_times_complex @ (power_power_complex @ (uminus1204672759omplex @ one_one_complex) @ N) @ (power_power_complex @ (uminus1204672759omplex @ one_one_complex) @ N)) = one_one_complex)))). % minus_one_mult_self
thf(fact_17_left__minus__one__mult__self, axiom,
    ((![N : nat, A : complex]: ((times_times_complex @ (power_power_complex @ (uminus1204672759omplex @ one_one_complex) @ N) @ (times_times_complex @ (power_power_complex @ (uminus1204672759omplex @ one_one_complex) @ N) @ A)) = A)))). % left_minus_one_mult_self
thf(fact_18_numeral__eq__iff, axiom,
    ((![M : num, N : num]: (((numeral_numeral_nat @ M) = (numeral_numeral_nat @ N)) = (M = N))))). % numeral_eq_iff
thf(fact_19_numeral__less__iff, axiom,
    ((![M : num, N : num]: ((ord_less_nat @ (numeral_numeral_nat @ M) @ (numeral_numeral_nat @ N)) = (ord_less_num @ M @ N))))). % numeral_less_iff
thf(fact_20_numeral__times__numeral, axiom,
    ((![M : num, N : num]: ((times_times_nat @ (numeral_numeral_nat @ M) @ (numeral_numeral_nat @ N)) = (numeral_numeral_nat @ (times_times_num @ M @ N)))))). % numeral_times_numeral
thf(fact_21_mult__numeral__left__semiring__numeral, axiom,
    ((![V : num, W : num, Z : nat]: ((times_times_nat @ (numeral_numeral_nat @ V) @ (times_times_nat @ (numeral_numeral_nat @ W) @ Z)) = (times_times_nat @ (numeral_numeral_nat @ (times_times_num @ V @ W)) @ Z))))). % mult_numeral_left_semiring_numeral
thf(fact_22_neg__numeral__eq__iff, axiom,
    ((![M : num, N : num]: (((uminus1204672759omplex @ (numera632737353omplex @ M)) = (uminus1204672759omplex @ (numera632737353omplex @ N))) = (M = N))))). % neg_numeral_eq_iff
thf(fact_23_power__one, axiom,
    ((![N : nat]: ((power_power_complex @ one_one_complex @ N) = one_one_complex)))). % power_one
thf(fact_24_power__one, axiom,
    ((![N : nat]: ((power_power_nat @ one_one_nat @ N) = one_one_nat)))). % power_one
thf(fact_25_num__double, axiom,
    ((![N : num]: ((times_times_num @ (bit0 @ one) @ N) = (bit0 @ N))))). % num_double
thf(fact_26_power__mult__numeral, axiom,
    ((![A : complex, M : num, N : num]: ((power_power_complex @ (power_power_complex @ A @ (numeral_numeral_nat @ M)) @ (numeral_numeral_nat @ N)) = (power_power_complex @ A @ (numeral_numeral_nat @ (times_times_num @ M @ N))))))). % power_mult_numeral
thf(fact_27_power__mult__numeral, axiom,
    ((![A : nat, M : num, N : num]: ((power_power_nat @ (power_power_nat @ A @ (numeral_numeral_nat @ M)) @ (numeral_numeral_nat @ N)) = (power_power_nat @ A @ (numeral_numeral_nat @ (times_times_num @ M @ N))))))). % power_mult_numeral
thf(fact_28_nat__zero__less__power__iff, axiom,
    ((![X : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ (power_power_nat @ X @ N)) = (((ord_less_nat @ zero_zero_nat @ X)) | ((N = zero_zero_nat))))))). % nat_zero_less_power_iff
thf(fact_29_power__one__right, axiom,
    ((![A : complex]: ((power_power_complex @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_30_power__one__right, axiom,
    ((![A : nat]: ((power_power_nat @ A @ one_one_nat) = A)))). % power_one_right
thf(fact_31_power__inject__exp, axiom,
    ((![A : nat, M : nat, N : nat]: ((ord_less_nat @ one_one_nat @ A) => (((power_power_nat @ A @ M) = (power_power_nat @ A @ N)) = (M = N)))))). % power_inject_exp
thf(fact_32_mult__neg__numeral__simps_I1_J, axiom,
    ((![M : num, N : num]: ((times_times_complex @ (uminus1204672759omplex @ (numera632737353omplex @ M)) @ (uminus1204672759omplex @ (numera632737353omplex @ N))) = (numera632737353omplex @ (times_times_num @ M @ N)))))). % mult_neg_numeral_simps(1)
thf(fact_33_mult__neg__numeral__simps_I2_J, axiom,
    ((![M : num, N : num]: ((times_times_complex @ (uminus1204672759omplex @ (numera632737353omplex @ M)) @ (numera632737353omplex @ N)) = (uminus1204672759omplex @ (numera632737353omplex @ (times_times_num @ M @ N))))))). % mult_neg_numeral_simps(2)
thf(fact_34_mult__neg__numeral__simps_I3_J, axiom,
    ((![M : num, N : num]: ((times_times_complex @ (numera632737353omplex @ M) @ (uminus1204672759omplex @ (numera632737353omplex @ N))) = (uminus1204672759omplex @ (numera632737353omplex @ (times_times_num @ M @ N))))))). % mult_neg_numeral_simps(3)
thf(fact_35_mult__minus1__right, axiom,
    ((![Z : complex]: ((times_times_complex @ Z @ (uminus1204672759omplex @ one_one_complex)) = (uminus1204672759omplex @ Z))))). % mult_minus1_right
thf(fact_36_mult__minus1, axiom,
    ((![Z : complex]: ((times_times_complex @ (uminus1204672759omplex @ one_one_complex) @ Z) = (uminus1204672759omplex @ Z))))). % mult_minus1
thf(fact_37_one__eq__numeral__iff, axiom,
    ((![N : num]: ((one_one_complex = (numera632737353omplex @ N)) = (one = N))))). % one_eq_numeral_iff
thf(fact_38_one__eq__numeral__iff, axiom,
    ((![N : num]: ((one_one_nat = (numeral_numeral_nat @ N)) = (one = N))))). % one_eq_numeral_iff
thf(fact_39_numeral__eq__one__iff, axiom,
    ((![N : num]: (((numera632737353omplex @ N) = one_one_complex) = (N = one))))). % numeral_eq_one_iff
thf(fact_40_numeral__eq__one__iff, axiom,
    ((![N : num]: (((numeral_numeral_nat @ N) = one_one_nat) = (N = one))))). % numeral_eq_one_iff
thf(fact_41_power__zero__numeral, axiom,
    ((![K : num]: ((power_power_complex @ zero_zero_complex @ (numeral_numeral_nat @ K)) = zero_zero_complex)))). % power_zero_numeral
thf(fact_42_power__zero__numeral, axiom,
    ((![K : num]: ((power_power_nat @ zero_zero_nat @ (numeral_numeral_nat @ K)) = zero_zero_nat)))). % power_zero_numeral
thf(fact_43_one__less__numeral__iff, axiom,
    ((![N : num]: ((ord_less_nat @ one_one_nat @ (numeral_numeral_nat @ N)) = (ord_less_num @ one @ N))))). % one_less_numeral_iff
thf(fact_44_numeral__eq__neg__one__iff, axiom,
    ((![N : num]: (((uminus1204672759omplex @ (numera632737353omplex @ N)) = (uminus1204672759omplex @ one_one_complex)) = (N = one))))). % numeral_eq_neg_one_iff
thf(fact_45_neg__one__eq__numeral__iff, axiom,
    ((![N : num]: (((uminus1204672759omplex @ one_one_complex) = (uminus1204672759omplex @ (numera632737353omplex @ N))) = (N = one))))). % neg_one_eq_numeral_iff
thf(fact_46_nat__power__less__imp__less, axiom,
    ((![I : nat, M : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ I) => ((ord_less_nat @ (power_power_nat @ I @ M) @ (power_power_nat @ I @ N)) => (ord_less_nat @ M @ N)))))). % nat_power_less_imp_less
thf(fact_47_root__nonzero, axiom,
    ((![N : nat]: (~ (((fFT_Mirabelle_root @ N) = zero_zero_complex)))))). % root_nonzero
thf(fact_48_numerals_I1_J, axiom,
    (((numeral_numeral_nat @ one) = one_one_nat))). % numerals(1)
thf(fact_49_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_nat @ zero_zero_nat @ zero_zero_nat))))). % less_numeral_extra(3)
thf(fact_50_zero__neq__numeral, axiom,
    ((![N : num]: (~ ((zero_zero_complex = (numera632737353omplex @ N))))))). % zero_neq_numeral
thf(fact_51_zero__neq__numeral, axiom,
    ((![N : num]: (~ ((zero_zero_nat = (numeral_numeral_nat @ N))))))). % zero_neq_numeral
thf(fact_52_less__numeral__extra_I4_J, axiom,
    ((~ ((ord_less_nat @ one_one_nat @ one_one_nat))))). % less_numeral_extra(4)
thf(fact_53_power__not__zero, axiom,
    ((![A : complex, N : nat]: ((~ ((A = zero_zero_complex))) => (~ (((power_power_complex @ A @ N) = zero_zero_complex))))))). % power_not_zero
thf(fact_54_power__not__zero, axiom,
    ((![A : nat, N : nat]: ((~ ((A = zero_zero_nat))) => (~ (((power_power_nat @ A @ N) = zero_zero_nat))))))). % power_not_zero
thf(fact_55_numeral__neq__neg__numeral, axiom,
    ((![M : num, N : num]: (~ (((numera632737353omplex @ M) = (uminus1204672759omplex @ (numera632737353omplex @ N)))))))). % numeral_neq_neg_numeral
thf(fact_56_neg__numeral__neq__numeral, axiom,
    ((![M : num, N : num]: (~ (((uminus1204672759omplex @ (numera632737353omplex @ M)) = (numera632737353omplex @ N))))))). % neg_numeral_neq_numeral
thf(fact_57_one__neq__neg__one, axiom,
    ((~ ((one_one_complex = (uminus1204672759omplex @ one_one_complex)))))). % one_neq_neg_one
thf(fact_58_power__commuting__commutes, axiom,
    ((![X : complex, Y : complex, N : nat]: (((times_times_complex @ X @ Y) = (times_times_complex @ Y @ X)) => ((times_times_complex @ (power_power_complex @ X @ N) @ Y) = (times_times_complex @ Y @ (power_power_complex @ X @ N))))))). % power_commuting_commutes
thf(fact_59_power__commuting__commutes, axiom,
    ((![X : nat, Y : nat, N : nat]: (((times_times_nat @ X @ Y) = (times_times_nat @ Y @ X)) => ((times_times_nat @ (power_power_nat @ X @ N) @ Y) = (times_times_nat @ Y @ (power_power_nat @ X @ N))))))). % power_commuting_commutes
thf(fact_60_power__mult__distrib, axiom,
    ((![A : complex, B : complex, N : nat]: ((power_power_complex @ (times_times_complex @ A @ B) @ N) = (times_times_complex @ (power_power_complex @ A @ N) @ (power_power_complex @ B @ N)))))). % power_mult_distrib
thf(fact_61_power__mult__distrib, axiom,
    ((![A : nat, B : nat, N : nat]: ((power_power_nat @ (times_times_nat @ A @ B) @ N) = (times_times_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ B @ N)))))). % power_mult_distrib
thf(fact_62_power__commutes, axiom,
    ((![A : complex, N : nat]: ((times_times_complex @ (power_power_complex @ A @ N) @ A) = (times_times_complex @ A @ (power_power_complex @ A @ N)))))). % power_commutes
thf(fact_63_power__commutes, axiom,
    ((![A : nat, N : nat]: ((times_times_nat @ (power_power_nat @ A @ N) @ A) = (times_times_nat @ A @ (power_power_nat @ A @ N)))))). % power_commutes
thf(fact_64_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_65_power__mult, axiom,
    ((![A : nat, M : nat, N : nat]: ((power_power_nat @ A @ (times_times_nat @ M @ N)) = (power_power_nat @ (power_power_nat @ A @ M) @ N))))). % power_mult
thf(fact_66_not__numeral__less__zero, axiom,
    ((![N : num]: (~ ((ord_less_nat @ (numeral_numeral_nat @ N) @ zero_zero_nat)))))). % not_numeral_less_zero
thf(fact_67_zero__less__numeral, axiom,
    ((![N : num]: (ord_less_nat @ zero_zero_nat @ (numeral_numeral_nat @ N))))). % zero_less_numeral
thf(fact_68_less__numeral__extra_I1_J, axiom,
    ((ord_less_nat @ zero_zero_nat @ one_one_nat))). % less_numeral_extra(1)
thf(fact_69_zero__less__power, axiom,
    ((![A : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ A) => (ord_less_nat @ zero_zero_nat @ (power_power_nat @ A @ N)))))). % zero_less_power
thf(fact_70_not__numeral__less__one, axiom,
    ((![N : num]: (~ ((ord_less_nat @ (numeral_numeral_nat @ N) @ one_one_nat)))))). % not_numeral_less_one
thf(fact_71_zero__neq__neg__numeral, axiom,
    ((![N : num]: (~ ((zero_zero_complex = (uminus1204672759omplex @ (numera632737353omplex @ N)))))))). % zero_neq_neg_numeral
thf(fact_72_zero__neq__neg__one, axiom,
    ((~ ((zero_zero_complex = (uminus1204672759omplex @ one_one_complex)))))). % zero_neq_neg_one
thf(fact_73_mult__numeral__1__right, axiom,
    ((![A : nat]: ((times_times_nat @ A @ (numeral_numeral_nat @ one)) = A)))). % mult_numeral_1_right
thf(fact_74_mult__numeral__1, axiom,
    ((![A : nat]: ((times_times_nat @ (numeral_numeral_nat @ one) @ A) = A)))). % mult_numeral_1
thf(fact_75_numeral__One, axiom,
    (((numera632737353omplex @ one) = one_one_complex))). % numeral_One
thf(fact_76_numeral__One, axiom,
    (((numeral_numeral_nat @ one) = one_one_nat))). % numeral_One
thf(fact_77_one__neq__neg__numeral, axiom,
    ((![N : num]: (~ ((one_one_complex = (uminus1204672759omplex @ (numera632737353omplex @ N)))))))). % one_neq_neg_numeral
thf(fact_78_numeral__neq__neg__one, axiom,
    ((![N : num]: (~ (((numera632737353omplex @ N) = (uminus1204672759omplex @ one_one_complex))))))). % numeral_neq_neg_one
thf(fact_79_left__right__inverse__power, axiom,
    ((![X : complex, Y : complex, N : nat]: (((times_times_complex @ X @ Y) = one_one_complex) => ((times_times_complex @ (power_power_complex @ X @ N) @ (power_power_complex @ Y @ N)) = one_one_complex))))). % left_right_inverse_power
thf(fact_80_left__right__inverse__power, axiom,
    ((![X : nat, Y : nat, N : nat]: (((times_times_nat @ X @ Y) = one_one_nat) => ((times_times_nat @ (power_power_nat @ X @ N) @ (power_power_nat @ Y @ N)) = one_one_nat))))). % left_right_inverse_power
thf(fact_81_power__0, axiom,
    ((![A : complex]: ((power_power_complex @ A @ zero_zero_nat) = one_one_complex)))). % power_0
thf(fact_82_power__0, axiom,
    ((![A : nat]: ((power_power_nat @ A @ zero_zero_nat) = one_one_nat)))). % power_0
thf(fact_83_power__less__power__Suc, axiom,
    ((![A : nat, N : nat]: ((ord_less_nat @ one_one_nat @ A) => (ord_less_nat @ (power_power_nat @ A @ N) @ (times_times_nat @ A @ (power_power_nat @ A @ N))))))). % power_less_power_Suc
thf(fact_84_power__gt1__lemma, axiom,
    ((![A : nat, N : nat]: ((ord_less_nat @ one_one_nat @ A) => (ord_less_nat @ one_one_nat @ (times_times_nat @ A @ (power_power_nat @ A @ N))))))). % power_gt1_lemma
thf(fact_85_mult__1s__ring__1_I1_J, axiom,
    ((![B : complex]: ((times_times_complex @ (uminus1204672759omplex @ (numera632737353omplex @ one)) @ B) = (uminus1204672759omplex @ B))))). % mult_1s_ring_1(1)
thf(fact_86_mult__1s__ring__1_I2_J, axiom,
    ((![B : complex]: ((times_times_complex @ B @ (uminus1204672759omplex @ (numera632737353omplex @ one))) = (uminus1204672759omplex @ B))))). % mult_1s_ring_1(2)
thf(fact_87_power__0__left, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((power_power_complex @ zero_zero_complex @ N) = one_one_complex)) & ((~ ((N = zero_zero_nat))) => ((power_power_complex @ zero_zero_complex @ N) = zero_zero_complex)))))). % power_0_left
thf(fact_88_power__0__left, axiom,
    ((![N : nat]: (((N = zero_zero_nat) => ((power_power_nat @ zero_zero_nat @ N) = one_one_nat)) & ((~ ((N = zero_zero_nat))) => ((power_power_nat @ zero_zero_nat @ N) = zero_zero_nat)))))). % power_0_left
thf(fact_89_uminus__numeral__One, axiom,
    (((uminus1204672759omplex @ (numera632737353omplex @ one)) = (uminus1204672759omplex @ one_one_complex)))). % uminus_numeral_One
thf(fact_90_power__minus, axiom,
    ((![A : complex, N : nat]: ((power_power_complex @ (uminus1204672759omplex @ A) @ N) = (times_times_complex @ (power_power_complex @ (uminus1204672759omplex @ one_one_complex) @ N) @ (power_power_complex @ A @ N)))))). % power_minus
thf(fact_91_power__strict__increasing, axiom,
    ((![N : nat, N2 : nat, A : nat]: ((ord_less_nat @ N @ N2) => ((ord_less_nat @ one_one_nat @ A) => (ord_less_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ A @ N2))))))). % power_strict_increasing
thf(fact_92_power__less__imp__less__exp, axiom,
    ((![A : nat, M : nat, N : nat]: ((ord_less_nat @ one_one_nat @ A) => ((ord_less_nat @ (power_power_nat @ A @ M) @ (power_power_nat @ A @ N)) => (ord_less_nat @ M @ N)))))). % power_less_imp_less_exp
thf(fact_93_zero__power, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((power_power_complex @ zero_zero_complex @ N) = zero_zero_complex))))). % zero_power
thf(fact_94_zero__power, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((power_power_nat @ zero_zero_nat @ N) = zero_zero_nat))))). % zero_power
thf(fact_95_power__minus__Bit0, axiom,
    ((![X : complex, K : num]: ((power_power_complex @ (uminus1204672759omplex @ X) @ (numeral_numeral_nat @ (bit0 @ K))) = (power_power_complex @ X @ (numeral_numeral_nat @ (bit0 @ K))))))). % power_minus_Bit0
thf(fact_96_power__Suc__less, axiom,
    ((![A : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ A) => ((ord_less_nat @ A @ one_one_nat) => (ord_less_nat @ (times_times_nat @ A @ (power_power_nat @ A @ N)) @ (power_power_nat @ A @ N))))))). % power_Suc_less
thf(fact_97_power__strict__decreasing, axiom,
    ((![N : nat, N2 : nat, A : nat]: ((ord_less_nat @ N @ N2) => ((ord_less_nat @ zero_zero_nat @ A) => ((ord_less_nat @ A @ one_one_nat) => (ord_less_nat @ (power_power_nat @ A @ N2) @ (power_power_nat @ A @ N)))))))). % power_strict_decreasing
thf(fact_98_zero__power2, axiom,
    (((power_power_complex @ zero_zero_complex @ (numeral_numeral_nat @ (bit0 @ one))) = zero_zero_complex))). % zero_power2
thf(fact_99_zero__power2, axiom,
    (((power_power_nat @ zero_zero_nat @ (numeral_numeral_nat @ (bit0 @ one))) = zero_zero_nat))). % zero_power2
thf(fact_100_one__less__power, axiom,
    ((![A : nat, N : nat]: ((ord_less_nat @ one_one_nat @ A) => ((ord_less_nat @ zero_zero_nat @ N) => (ord_less_nat @ one_one_nat @ (power_power_nat @ A @ N))))))). % one_less_power
thf(fact_101_power2__eq__square, axiom,
    ((![A : complex]: ((power_power_complex @ A @ (numeral_numeral_nat @ (bit0 @ one))) = (times_times_complex @ A @ A))))). % power2_eq_square
thf(fact_102_power2__eq__square, axiom,
    ((![A : nat]: ((power_power_nat @ A @ (numeral_numeral_nat @ (bit0 @ one))) = (times_times_nat @ A @ A))))). % power2_eq_square
thf(fact_103_power4__eq__xxxx, axiom,
    ((![X : complex]: ((power_power_complex @ X @ (numeral_numeral_nat @ (bit0 @ (bit0 @ one)))) = (times_times_complex @ (times_times_complex @ (times_times_complex @ X @ X) @ X) @ X))))). % power4_eq_xxxx
thf(fact_104_power4__eq__xxxx, axiom,
    ((![X : nat]: ((power_power_nat @ X @ (numeral_numeral_nat @ (bit0 @ (bit0 @ one)))) = (times_times_nat @ (times_times_nat @ (times_times_nat @ X @ X) @ X) @ X))))). % power4_eq_xxxx
thf(fact_105_one__power2, axiom,
    (((power_power_complex @ one_one_complex @ (numeral_numeral_nat @ (bit0 @ one))) = one_one_complex))). % one_power2
thf(fact_106_one__power2, axiom,
    (((power_power_nat @ one_one_nat @ (numeral_numeral_nat @ (bit0 @ one))) = one_one_nat))). % one_power2
thf(fact_107_power2__eq__iff, axiom,
    ((![X : complex, Y : complex]: (((power_power_complex @ X @ (numeral_numeral_nat @ (bit0 @ one))) = (power_power_complex @ Y @ (numeral_numeral_nat @ (bit0 @ one)))) = (((X = Y)) | ((X = (uminus1204672759omplex @ Y)))))))). % power2_eq_iff
thf(fact_108_power__even__eq, axiom,
    ((![A : complex, N : nat]: ((power_power_complex @ A @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N)) = (power_power_complex @ (power_power_complex @ A @ N) @ (numeral_numeral_nat @ (bit0 @ one))))))). % power_even_eq
thf(fact_109_power__even__eq, axiom,
    ((![A : nat, N : nat]: ((power_power_nat @ A @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N)) = (power_power_nat @ (power_power_nat @ A @ N) @ (numeral_numeral_nat @ (bit0 @ one))))))). % power_even_eq
thf(fact_110_power2__eq__1__iff, axiom,
    ((![A : complex]: (((power_power_complex @ A @ (numeral_numeral_nat @ (bit0 @ one))) = one_one_complex) = (((A = one_one_complex)) | ((A = (uminus1204672759omplex @ one_one_complex)))))))). % power2_eq_1_iff
thf(fact_111_minus__power__mult__self, axiom,
    ((![A : complex, N : nat]: ((times_times_complex @ (power_power_complex @ (uminus1204672759omplex @ A) @ N) @ (power_power_complex @ (uminus1204672759omplex @ A) @ N)) = (power_power_complex @ A @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N)))))). % minus_power_mult_self
thf(fact_112_less__one, axiom,
    ((![N : nat]: ((ord_less_nat @ N @ one_one_nat) = (N = zero_zero_nat))))). % less_one
thf(fact_113_nat__mult__less__cancel__disj, axiom,
    ((![K : nat, M : nat, N : nat]: ((ord_less_nat @ (times_times_nat @ K @ M) @ (times_times_nat @ K @ N)) = (((ord_less_nat @ zero_zero_nat @ K)) & ((ord_less_nat @ M @ N))))))). % nat_mult_less_cancel_disj
thf(fact_114_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_115_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_116_semiring__norm_I170_J, axiom,
    ((![V : num, W : num, Y : complex]: ((times_times_complex @ (uminus1204672759omplex @ (numera632737353omplex @ V)) @ (times_times_complex @ (numera632737353omplex @ W) @ Y)) = (times_times_complex @ (uminus1204672759omplex @ (numera632737353omplex @ (times_times_num @ V @ W))) @ Y))))). % semiring_norm(170)
thf(fact_117_semiring__norm_I171_J, axiom,
    ((![V : num, W : num, Y : complex]: ((times_times_complex @ (numera632737353omplex @ V) @ (times_times_complex @ (uminus1204672759omplex @ (numera632737353omplex @ W)) @ Y)) = (times_times_complex @ (uminus1204672759omplex @ (numera632737353omplex @ (times_times_num @ V @ W))) @ Y))))). % semiring_norm(171)
thf(fact_118_semiring__norm_I172_J, axiom,
    ((![V : num, W : num, Y : complex]: ((times_times_complex @ (uminus1204672759omplex @ (numera632737353omplex @ V)) @ (times_times_complex @ (uminus1204672759omplex @ (numera632737353omplex @ W)) @ Y)) = (times_times_complex @ (numera632737353omplex @ (times_times_num @ V @ W)) @ Y))))). % semiring_norm(172)
thf(fact_119_semiring__norm_I87_J, axiom,
    ((![M : num, N : num]: (((bit0 @ M) = (bit0 @ N)) = (M = N))))). % semiring_norm(87)
thf(fact_120_add_Oinverse__inverse, axiom,
    ((![A : complex]: ((uminus1204672759omplex @ (uminus1204672759omplex @ A)) = A)))). % add.inverse_inverse
thf(fact_121_neg__equal__iff__equal, axiom,
    ((![A : complex, B : complex]: (((uminus1204672759omplex @ A) = (uminus1204672759omplex @ B)) = (A = B))))). % neg_equal_iff_equal
thf(fact_122_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_123_add_Oinverse__neutral, axiom,
    (((uminus1204672759omplex @ zero_zero_complex) = zero_zero_complex))). % add.inverse_neutral
thf(fact_124_neg__0__equal__iff__equal, axiom,
    ((![A : complex]: ((zero_zero_complex = (uminus1204672759omplex @ A)) = (zero_zero_complex = A))))). % neg_0_equal_iff_equal
thf(fact_125_neg__equal__0__iff__equal, axiom,
    ((![A : complex]: (((uminus1204672759omplex @ A) = zero_zero_complex) = (A = zero_zero_complex))))). % neg_equal_0_iff_equal
thf(fact_126_mult_Oleft__neutral, axiom,
    ((![A : complex]: ((times_times_complex @ one_one_complex @ A) = A)))). % mult.left_neutral
thf(fact_127_mult_Oleft__neutral, axiom,
    ((![A : nat]: ((times_times_nat @ one_one_nat @ A) = A)))). % mult.left_neutral
thf(fact_128_mult_Oright__neutral, axiom,
    ((![A : complex]: ((times_times_complex @ A @ one_one_complex) = A)))). % mult.right_neutral
thf(fact_129_mult_Oright__neutral, axiom,
    ((![A : nat]: ((times_times_nat @ A @ one_one_nat) = A)))). % mult.right_neutral
thf(fact_130_semiring__norm_I85_J, axiom,
    ((![M : num]: (~ (((bit0 @ M) = one)))))). % semiring_norm(85)
thf(fact_131_semiring__norm_I83_J, axiom,
    ((![N : num]: (~ ((one = (bit0 @ N))))))). % semiring_norm(83)
thf(fact_132_neq0__conv, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N))))). % neq0_conv
thf(fact_133_less__nat__zero__code, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_134_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_135_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_136_mult__0__right, axiom,
    ((![M : nat]: ((times_times_nat @ M @ zero_zero_nat) = zero_zero_nat)))). % mult_0_right
thf(fact_137_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_138_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_139_nat__1__eq__mult__iff, axiom,
    ((![M : nat, N : nat]: ((one_one_nat = (times_times_nat @ M @ N)) = (((M = one_one_nat)) & ((N = one_one_nat))))))). % nat_1_eq_mult_iff
thf(fact_140_nat__mult__eq__1__iff, axiom,
    ((![M : nat, N : nat]: (((times_times_nat @ M @ N) = one_one_nat) = (((M = one_one_nat)) & ((N = one_one_nat))))))). % nat_mult_eq_1_iff
thf(fact_141_semiring__norm_I13_J, axiom,
    ((![M : num, N : num]: ((times_times_num @ (bit0 @ M) @ (bit0 @ N)) = (bit0 @ (bit0 @ (times_times_num @ M @ N))))))). % semiring_norm(13)
thf(fact_142_semiring__norm_I12_J, axiom,
    ((![N : num]: ((times_times_num @ one @ N) = N)))). % semiring_norm(12)
thf(fact_143_semiring__norm_I11_J, axiom,
    ((![M : num]: ((times_times_num @ M @ one) = M)))). % semiring_norm(11)
thf(fact_144_semiring__norm_I78_J, axiom,
    ((![M : num, N : num]: ((ord_less_num @ (bit0 @ M) @ (bit0 @ N)) = (ord_less_num @ M @ N))))). % semiring_norm(78)
thf(fact_145_semiring__norm_I75_J, axiom,
    ((![M : num]: (~ ((ord_less_num @ M @ one)))))). % semiring_norm(75)
thf(fact_146_semiring__norm_I76_J, axiom,
    ((![N : num]: (ord_less_num @ one @ (bit0 @ N))))). % semiring_norm(76)
thf(fact_147_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_148_zero__reorient, axiom,
    ((![X : complex]: ((zero_zero_complex = X) = (X = zero_zero_complex))))). % zero_reorient

% Conjectures (2)
thf(conj_0, hypothesis,
    ((ord_less_nat @ zero_zero_nat @ n))).
thf(conj_1, conjecture,
    (((power_power_complex @ (fFT_Mirabelle_root @ (times_times_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ n)) @ n) = (uminus1204672759omplex @ one_one_complex)))).
