% 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_495__3228606_1 ) ; }
% This file was generated by Isabelle (most likely Sledgehammer)
% 2020-12-16 14:11:36.869

% Could-be-implicit typings (5)
thf(ty_n_t__Complex__Ocomplex, type,
    complex : $tType).
thf(ty_n_t__Real__Oreal, type,
    real : $tType).
thf(ty_n_t__Num__Onum, type,
    num : $tType).
thf(ty_n_t__Nat__Onat, type,
    nat : $tType).
thf(ty_n_t__Int__Oint, type,
    int : $tType).

% Explicit typings (32)
thf(sy_c_FFT__Mirabelle__ulikgskiun_ODFT, type,
    fFT_Mirabelle_DFT : nat > (nat > complex) > nat > complex).
thf(sy_c_FFT__Mirabelle__ulikgskiun_OFFT, type,
    fFT_Mirabelle_FFT : nat > (nat > complex) > nat > complex).
thf(sy_c_FFT__Mirabelle__ulikgskiun_OIDFT, type,
    fFT_Mirabelle_IDFT : nat > (nat > complex) > nat > complex).
thf(sy_c_FFT__Mirabelle__ulikgskiun_OIFFT, type,
    fFT_Mirabelle_IFFT : nat > (nat > complex) > nat > complex).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Int__Oint, type,
    zero_zero_int : int).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat, type,
    zero_zero_nat : nat).
thf(sy_c_Groups_Ozero__class_Ozero_001t__Real__Oreal, type,
    zero_zero_real : real).
thf(sy_c_If_001t__Nat__Onat, type,
    if_nat : $o > nat > nat > nat).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Int__Oint, type,
    semiri2019852685at_int : nat > int).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Nat__Onat, type,
    semiri1382578993at_nat : nat > nat).
thf(sy_c_Nat_Osemiring__1__class_Oof__nat_001t__Real__Oreal, type,
    semiri2110766477t_real : nat > real).
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__Int__Oint, type,
    numeral_numeral_int : num > int).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Nat__Onat, type,
    numeral_numeral_nat : num > nat).
thf(sy_c_Num_Onumeral__class_Onumeral_001t__Real__Oreal, type,
    numeral_numeral_real : num > real).
thf(sy_c_Num_Opow, type,
    pow : num > num > num).
thf(sy_c_Orderings_Oord__class_Oless_001t__Int__Oint, type,
    ord_less_int : int > int > $o).
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_Orderings_Oord__class_Oless_001t__Real__Oreal, type,
    ord_less_real : real > real > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Int__Oint, type,
    ord_less_eq_int : int > int > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Nat__Onat, type,
    ord_less_eq_nat : nat > nat > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Num__Onum, type,
    ord_less_eq_num : num > num > $o).
thf(sy_c_Orderings_Oord__class_Oless__eq_001t__Real__Oreal, type,
    ord_less_eq_real : real > real > $o).
thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint, type,
    power_power_int : int > nat > int).
thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat, type,
    power_power_nat : nat > nat > nat).
thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal, type,
    power_power_real : real > nat > real).
thf(sy_c_Transcendental_Olog, type,
    log : real > real > real).
thf(sy_v_a, type,
    a : nat > complex).
thf(sy_v_i, type,
    i : nat).
thf(sy_v_k, type,
    k : nat).

% Relevant facts (241)
thf(fact_0_semiring__norm_I85_J, axiom,
    ((![M : num]: (~ (((bit0 @ M) = one)))))). % semiring_norm(85)
thf(fact_1_semiring__norm_I83_J, axiom,
    ((![N : num]: (~ ((one = (bit0 @ N))))))). % semiring_norm(83)
thf(fact_2_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_3_numeral__less__iff, axiom,
    ((![M : num, N : num]: ((ord_less_int @ (numeral_numeral_int @ M) @ (numeral_numeral_int @ N)) = (ord_less_num @ M @ N))))). % numeral_less_iff
thf(fact_4_numeral__less__iff, axiom,
    ((![M : num, N : num]: ((ord_less_real @ (numeral_numeral_real @ M) @ (numeral_numeral_real @ N)) = (ord_less_num @ M @ N))))). % numeral_less_iff
thf(fact_5_verit__eq__simplify_I8_J, axiom,
    ((![X2 : num, Y2 : num]: (((bit0 @ X2) = (bit0 @ Y2)) = (X2 = Y2))))). % verit_eq_simplify(8)
thf(fact_6_semiring__norm_I87_J, axiom,
    ((![M : num, N : num]: (((bit0 @ M) = (bit0 @ N)) = (M = N))))). % semiring_norm(87)
thf(fact_7_numeral__eq__iff, axiom,
    ((![M : num, N : num]: (((numeral_numeral_nat @ M) = (numeral_numeral_nat @ N)) = (M = N))))). % numeral_eq_iff
thf(fact_8_numeral__eq__iff, axiom,
    ((![M : num, N : num]: (((numeral_numeral_int @ M) = (numeral_numeral_int @ N)) = (M = N))))). % numeral_eq_iff
thf(fact_9_numeral__eq__iff, axiom,
    ((![M : num, N : num]: (((numeral_numeral_real @ M) = (numeral_numeral_real @ N)) = (M = N))))). % numeral_eq_iff
thf(fact_10_verit__eq__simplify_I10_J, axiom,
    ((![X2 : num]: (~ ((one = (bit0 @ X2))))))). % verit_eq_simplify(10)
thf(fact_11_DFT__FFT, axiom,
    ((![I : nat, K : nat, A : nat > complex]: ((ord_less_nat @ I @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ K)) => ((fFT_Mirabelle_DFT @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ K) @ A @ I) = (fFT_Mirabelle_FFT @ K @ A @ I)))))). % DFT_FFT
thf(fact_12_numeral__power__less__of__nat__cancel__iff, axiom,
    ((![I : num, N : nat, X : nat]: ((ord_less_nat @ (power_power_nat @ (numeral_numeral_nat @ I) @ N) @ (semiri1382578993at_nat @ X)) = (ord_less_nat @ (power_power_nat @ (numeral_numeral_nat @ I) @ N) @ X))))). % numeral_power_less_of_nat_cancel_iff
thf(fact_13_numeral__power__less__of__nat__cancel__iff, axiom,
    ((![I : num, N : nat, X : nat]: ((ord_less_int @ (power_power_int @ (numeral_numeral_int @ I) @ N) @ (semiri2019852685at_int @ X)) = (ord_less_nat @ (power_power_nat @ (numeral_numeral_nat @ I) @ N) @ X))))). % numeral_power_less_of_nat_cancel_iff
thf(fact_14_numeral__power__less__of__nat__cancel__iff, axiom,
    ((![I : num, N : nat, X : nat]: ((ord_less_real @ (power_power_real @ (numeral_numeral_real @ I) @ N) @ (semiri2110766477t_real @ X)) = (ord_less_nat @ (power_power_nat @ (numeral_numeral_nat @ I) @ N) @ X))))). % numeral_power_less_of_nat_cancel_iff
thf(fact_15_of__nat__less__numeral__power__cancel__iff, axiom,
    ((![X : nat, I : num, N : nat]: ((ord_less_int @ (semiri2019852685at_int @ X) @ (power_power_int @ (numeral_numeral_int @ I) @ N)) = (ord_less_nat @ X @ (power_power_nat @ (numeral_numeral_nat @ I) @ N)))))). % of_nat_less_numeral_power_cancel_iff
thf(fact_16_of__nat__less__numeral__power__cancel__iff, axiom,
    ((![X : nat, I : num, N : nat]: ((ord_less_real @ (semiri2110766477t_real @ X) @ (power_power_real @ (numeral_numeral_real @ I) @ N)) = (ord_less_nat @ X @ (power_power_nat @ (numeral_numeral_nat @ I) @ N)))))). % of_nat_less_numeral_power_cancel_iff
thf(fact_17_of__nat__less__numeral__power__cancel__iff, axiom,
    ((![X : nat, I : num, N : nat]: ((ord_less_nat @ (semiri1382578993at_nat @ X) @ (power_power_nat @ (numeral_numeral_nat @ I) @ N)) = (ord_less_nat @ X @ (power_power_nat @ (numeral_numeral_nat @ I) @ N)))))). % of_nat_less_numeral_power_cancel_iff
thf(fact_18_power__numeral, axiom,
    ((![K : num, L : num]: ((power_power_nat @ (numeral_numeral_nat @ K) @ (numeral_numeral_nat @ L)) = (numeral_numeral_nat @ (pow @ K @ L)))))). % power_numeral
thf(fact_19_power__numeral, axiom,
    ((![K : num, L : num]: ((power_power_int @ (numeral_numeral_int @ K) @ (numeral_numeral_nat @ L)) = (numeral_numeral_int @ (pow @ K @ L)))))). % power_numeral
thf(fact_20_power__numeral, axiom,
    ((![K : num, L : num]: ((power_power_real @ (numeral_numeral_real @ K) @ (numeral_numeral_nat @ L)) = (numeral_numeral_real @ (pow @ K @ L)))))). % power_numeral
thf(fact_21_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_22_semiring__norm_I75_J, axiom,
    ((![M : num]: (~ ((ord_less_num @ M @ one)))))). % semiring_norm(75)
thf(fact_23_of__nat__numeral, axiom,
    ((![N : num]: ((semiri2019852685at_int @ (numeral_numeral_nat @ N)) = (numeral_numeral_int @ N))))). % of_nat_numeral
thf(fact_24_of__nat__numeral, axiom,
    ((![N : num]: ((semiri2110766477t_real @ (numeral_numeral_nat @ N)) = (numeral_numeral_real @ N))))). % of_nat_numeral
thf(fact_25_of__nat__numeral, axiom,
    ((![N : num]: ((semiri1382578993at_nat @ (numeral_numeral_nat @ N)) = (numeral_numeral_nat @ N))))). % of_nat_numeral
thf(fact_26_of__nat__power__eq__of__nat__cancel__iff, axiom,
    ((![X : nat, B : nat, W : nat]: (((semiri2019852685at_int @ X) = (power_power_int @ (semiri2019852685at_int @ B) @ W)) = (X = (power_power_nat @ B @ W)))))). % of_nat_power_eq_of_nat_cancel_iff
thf(fact_27_of__nat__power__eq__of__nat__cancel__iff, axiom,
    ((![X : nat, B : nat, W : nat]: (((semiri2110766477t_real @ X) = (power_power_real @ (semiri2110766477t_real @ B) @ W)) = (X = (power_power_nat @ B @ W)))))). % of_nat_power_eq_of_nat_cancel_iff
thf(fact_28_of__nat__power__eq__of__nat__cancel__iff, axiom,
    ((![X : nat, B : nat, W : nat]: (((semiri1382578993at_nat @ X) = (power_power_nat @ (semiri1382578993at_nat @ B) @ W)) = (X = (power_power_nat @ B @ W)))))). % of_nat_power_eq_of_nat_cancel_iff
thf(fact_29_of__nat__eq__of__nat__power__cancel__iff, axiom,
    ((![B : nat, W : nat, X : nat]: (((power_power_int @ (semiri2019852685at_int @ B) @ W) = (semiri2019852685at_int @ X)) = ((power_power_nat @ B @ W) = X))))). % of_nat_eq_of_nat_power_cancel_iff
thf(fact_30_of__nat__eq__of__nat__power__cancel__iff, axiom,
    ((![B : nat, W : nat, X : nat]: (((power_power_real @ (semiri2110766477t_real @ B) @ W) = (semiri2110766477t_real @ X)) = ((power_power_nat @ B @ W) = X))))). % of_nat_eq_of_nat_power_cancel_iff
thf(fact_31_of__nat__eq__of__nat__power__cancel__iff, axiom,
    ((![B : nat, W : nat, X : nat]: (((power_power_nat @ (semiri1382578993at_nat @ B) @ W) = (semiri1382578993at_nat @ X)) = ((power_power_nat @ B @ W) = X))))). % of_nat_eq_of_nat_power_cancel_iff
thf(fact_32_of__nat__power, axiom,
    ((![M : nat, N : nat]: ((semiri2019852685at_int @ (power_power_nat @ M @ N)) = (power_power_int @ (semiri2019852685at_int @ M) @ N))))). % of_nat_power
thf(fact_33_of__nat__power, axiom,
    ((![M : nat, N : nat]: ((semiri2110766477t_real @ (power_power_nat @ M @ N)) = (power_power_real @ (semiri2110766477t_real @ M) @ N))))). % of_nat_power
thf(fact_34_of__nat__power, axiom,
    ((![M : nat, N : nat]: ((semiri1382578993at_nat @ (power_power_nat @ M @ N)) = (power_power_nat @ (semiri1382578993at_nat @ M) @ N))))). % of_nat_power
thf(fact_35_semiring__norm_I76_J, axiom,
    ((![N : num]: (ord_less_num @ one @ (bit0 @ N))))). % semiring_norm(76)
thf(fact_36_of__nat__power__less__of__nat__cancel__iff, axiom,
    ((![X : nat, B : nat, W : nat]: ((ord_less_int @ (semiri2019852685at_int @ X) @ (power_power_int @ (semiri2019852685at_int @ B) @ W)) = (ord_less_nat @ X @ (power_power_nat @ B @ W)))))). % of_nat_power_less_of_nat_cancel_iff
thf(fact_37_of__nat__power__less__of__nat__cancel__iff, axiom,
    ((![X : nat, B : nat, W : nat]: ((ord_less_real @ (semiri2110766477t_real @ X) @ (power_power_real @ (semiri2110766477t_real @ B) @ W)) = (ord_less_nat @ X @ (power_power_nat @ B @ W)))))). % of_nat_power_less_of_nat_cancel_iff
thf(fact_38_of__nat__power__less__of__nat__cancel__iff, axiom,
    ((![X : nat, B : nat, W : nat]: ((ord_less_nat @ (semiri1382578993at_nat @ X) @ (power_power_nat @ (semiri1382578993at_nat @ B) @ W)) = (ord_less_nat @ X @ (power_power_nat @ B @ W)))))). % of_nat_power_less_of_nat_cancel_iff
thf(fact_39_of__nat__less__of__nat__power__cancel__iff, axiom,
    ((![B : nat, W : nat, X : nat]: ((ord_less_int @ (power_power_int @ (semiri2019852685at_int @ B) @ W) @ (semiri2019852685at_int @ X)) = (ord_less_nat @ (power_power_nat @ B @ W) @ X))))). % of_nat_less_of_nat_power_cancel_iff
thf(fact_40_of__nat__less__of__nat__power__cancel__iff, axiom,
    ((![B : nat, W : nat, X : nat]: ((ord_less_real @ (power_power_real @ (semiri2110766477t_real @ B) @ W) @ (semiri2110766477t_real @ X)) = (ord_less_nat @ (power_power_nat @ B @ W) @ X))))). % of_nat_less_of_nat_power_cancel_iff
thf(fact_41_of__nat__less__of__nat__power__cancel__iff, axiom,
    ((![B : nat, W : nat, X : nat]: ((ord_less_nat @ (power_power_nat @ (semiri1382578993at_nat @ B) @ W) @ (semiri1382578993at_nat @ X)) = (ord_less_nat @ (power_power_nat @ B @ W) @ X))))). % of_nat_less_of_nat_power_cancel_iff
thf(fact_42_real__of__nat__eq__numeral__power__cancel__iff, axiom,
    ((![Y : nat, X : num, N : nat]: (((semiri2019852685at_int @ Y) = (power_power_int @ (numeral_numeral_int @ X) @ N)) = (Y = (power_power_nat @ (numeral_numeral_nat @ X) @ N)))))). % real_of_nat_eq_numeral_power_cancel_iff
thf(fact_43_real__of__nat__eq__numeral__power__cancel__iff, axiom,
    ((![Y : nat, X : num, N : nat]: (((semiri2110766477t_real @ Y) = (power_power_real @ (numeral_numeral_real @ X) @ N)) = (Y = (power_power_nat @ (numeral_numeral_nat @ X) @ N)))))). % real_of_nat_eq_numeral_power_cancel_iff
thf(fact_44_real__of__nat__eq__numeral__power__cancel__iff, axiom,
    ((![Y : nat, X : num, N : nat]: (((semiri1382578993at_nat @ Y) = (power_power_nat @ (numeral_numeral_nat @ X) @ N)) = (Y = (power_power_nat @ (numeral_numeral_nat @ X) @ N)))))). % real_of_nat_eq_numeral_power_cancel_iff
thf(fact_45_numeral__power__eq__of__nat__cancel__iff, axiom,
    ((![X : num, N : nat, Y : nat]: (((power_power_int @ (numeral_numeral_int @ X) @ N) = (semiri2019852685at_int @ Y)) = ((power_power_nat @ (numeral_numeral_nat @ X) @ N) = Y))))). % numeral_power_eq_of_nat_cancel_iff
thf(fact_46_numeral__power__eq__of__nat__cancel__iff, axiom,
    ((![X : num, N : nat, Y : nat]: (((power_power_real @ (numeral_numeral_real @ X) @ N) = (semiri2110766477t_real @ Y)) = ((power_power_nat @ (numeral_numeral_nat @ X) @ N) = Y))))). % numeral_power_eq_of_nat_cancel_iff
thf(fact_47_numeral__power__eq__of__nat__cancel__iff, axiom,
    ((![X : num, N : nat, Y : nat]: (((power_power_nat @ (numeral_numeral_nat @ X) @ N) = (semiri1382578993at_nat @ Y)) = ((power_power_nat @ (numeral_numeral_nat @ X) @ N) = Y))))). % numeral_power_eq_of_nat_cancel_iff
thf(fact_48_int__ops_I3_J, axiom,
    ((![N : num]: ((semiri2019852685at_int @ (numeral_numeral_nat @ N)) = (numeral_numeral_int @ N))))). % int_ops(3)
thf(fact_49_nat__int__comparison_I2_J, axiom,
    ((ord_less_nat = (^[A2 : nat]: (^[B2 : nat]: (ord_less_int @ (semiri2019852685at_int @ A2) @ (semiri2019852685at_int @ B2))))))). % nat_int_comparison(2)
thf(fact_50_pow_Osimps_I1_J, axiom,
    ((![X : num]: ((pow @ X @ one) = X)))). % pow.simps(1)
thf(fact_51_verit__comp__simplify1_I1_J, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ A)))))). % verit_comp_simplify1(1)
thf(fact_52_verit__comp__simplify1_I1_J, axiom,
    ((![A : num]: (~ ((ord_less_num @ A @ A)))))). % verit_comp_simplify1(1)
thf(fact_53_verit__comp__simplify1_I1_J, axiom,
    ((![A : int]: (~ ((ord_less_int @ A @ A)))))). % verit_comp_simplify1(1)
thf(fact_54_verit__comp__simplify1_I1_J, axiom,
    ((![A : real]: (~ ((ord_less_real @ A @ A)))))). % verit_comp_simplify1(1)
thf(fact_55_of__nat__less__iff, axiom,
    ((![M : nat, N : nat]: ((ord_less_int @ (semiri2019852685at_int @ M) @ (semiri2019852685at_int @ N)) = (ord_less_nat @ M @ N))))). % of_nat_less_iff
thf(fact_56_of__nat__less__iff, axiom,
    ((![M : nat, N : nat]: ((ord_less_real @ (semiri2110766477t_real @ M) @ (semiri2110766477t_real @ N)) = (ord_less_nat @ M @ N))))). % of_nat_less_iff
thf(fact_57_of__nat__less__iff, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ (semiri1382578993at_nat @ M) @ (semiri1382578993at_nat @ N)) = (ord_less_nat @ M @ N))))). % of_nat_less_iff
thf(fact_58_of__nat__less__two__power, axiom,
    ((![N : nat]: (ord_less_int @ (semiri2019852685at_int @ N) @ (power_power_int @ (numeral_numeral_int @ (bit0 @ one)) @ N))))). % of_nat_less_two_power
thf(fact_59_of__nat__less__two__power, axiom,
    ((![N : nat]: (ord_less_real @ (semiri2110766477t_real @ N) @ (power_power_real @ (numeral_numeral_real @ (bit0 @ one)) @ N))))). % of_nat_less_two_power
thf(fact_60_real__of__nat__less__numeral__iff, axiom,
    ((![N : nat, W : num]: ((ord_less_real @ (semiri2110766477t_real @ N) @ (numeral_numeral_real @ W)) = (ord_less_nat @ N @ (numeral_numeral_nat @ W)))))). % real_of_nat_less_numeral_iff
thf(fact_61_numeral__less__real__of__nat__iff, axiom,
    ((![W : num, N : nat]: ((ord_less_real @ (numeral_numeral_real @ W) @ (semiri2110766477t_real @ N)) = (ord_less_nat @ (numeral_numeral_nat @ W) @ N))))). % numeral_less_real_of_nat_iff
thf(fact_62_of__nat__less__imp__less, axiom,
    ((![M : nat, N : nat]: ((ord_less_int @ (semiri2019852685at_int @ M) @ (semiri2019852685at_int @ N)) => (ord_less_nat @ M @ N))))). % of_nat_less_imp_less
thf(fact_63_of__nat__less__imp__less, axiom,
    ((![M : nat, N : nat]: ((ord_less_real @ (semiri2110766477t_real @ M) @ (semiri2110766477t_real @ N)) => (ord_less_nat @ M @ N))))). % of_nat_less_imp_less
thf(fact_64_of__nat__less__imp__less, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ (semiri1382578993at_nat @ M) @ (semiri1382578993at_nat @ N)) => (ord_less_nat @ M @ N))))). % of_nat_less_imp_less
thf(fact_65_less__imp__of__nat__less, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (ord_less_int @ (semiri2019852685at_int @ M) @ (semiri2019852685at_int @ N)))))). % less_imp_of_nat_less
thf(fact_66_less__imp__of__nat__less, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (ord_less_real @ (semiri2110766477t_real @ M) @ (semiri2110766477t_real @ N)))))). % less_imp_of_nat_less
thf(fact_67_less__imp__of__nat__less, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (ord_less_nat @ (semiri1382578993at_nat @ M) @ (semiri1382578993at_nat @ N)))))). % less_imp_of_nat_less
thf(fact_68_int__eq__iff__numeral, axiom,
    ((![M : nat, V : num]: (((semiri2019852685at_int @ M) = (numeral_numeral_int @ V)) = (M = (numeral_numeral_nat @ V)))))). % int_eq_iff_numeral
thf(fact_69_of__nat__eq__iff, axiom,
    ((![M : nat, N : nat]: (((semiri2019852685at_int @ M) = (semiri2019852685at_int @ N)) = (M = N))))). % of_nat_eq_iff
thf(fact_70_of__nat__eq__iff, axiom,
    ((![M : nat, N : nat]: (((semiri2110766477t_real @ M) = (semiri2110766477t_real @ N)) = (M = N))))). % of_nat_eq_iff
thf(fact_71_of__nat__eq__iff, axiom,
    ((![M : nat, N : nat]: (((semiri1382578993at_nat @ M) = (semiri1382578993at_nat @ N)) = (M = N))))). % of_nat_eq_iff
thf(fact_72_reals__Archimedean2, axiom,
    ((![X : real]: (?[N2 : nat]: (ord_less_real @ X @ (semiri2110766477t_real @ N2)))))). % reals_Archimedean2
thf(fact_73_zero__less__power2, axiom,
    ((![A : int]: ((ord_less_int @ zero_zero_int @ (power_power_int @ A @ (numeral_numeral_nat @ (bit0 @ one)))) = (~ ((A = zero_zero_int))))))). % zero_less_power2
thf(fact_74_zero__less__power2, axiom,
    ((![A : real]: ((ord_less_real @ zero_zero_real @ (power_power_real @ A @ (numeral_numeral_nat @ (bit0 @ one)))) = (~ ((A = zero_zero_real))))))). % zero_less_power2
thf(fact_75_neq0__conv, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) = (ord_less_nat @ zero_zero_nat @ N))))). % neq0_conv
thf(fact_76_less__nat__zero__code, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_nat_zero_code
thf(fact_77_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_78_of__nat__0, axiom,
    (((semiri2019852685at_int @ zero_zero_nat) = zero_zero_int))). % of_nat_0
thf(fact_79_of__nat__0, axiom,
    (((semiri2110766477t_real @ zero_zero_nat) = zero_zero_real))). % of_nat_0
thf(fact_80_of__nat__0, axiom,
    (((semiri1382578993at_nat @ zero_zero_nat) = zero_zero_nat))). % of_nat_0
thf(fact_81_of__nat__0__eq__iff, axiom,
    ((![N : nat]: ((zero_zero_int = (semiri2019852685at_int @ N)) = (zero_zero_nat = N))))). % of_nat_0_eq_iff
thf(fact_82_of__nat__0__eq__iff, axiom,
    ((![N : nat]: ((zero_zero_real = (semiri2110766477t_real @ N)) = (zero_zero_nat = N))))). % of_nat_0_eq_iff
thf(fact_83_of__nat__0__eq__iff, axiom,
    ((![N : nat]: ((zero_zero_nat = (semiri1382578993at_nat @ N)) = (zero_zero_nat = N))))). % of_nat_0_eq_iff
thf(fact_84_of__nat__eq__0__iff, axiom,
    ((![M : nat]: (((semiri2019852685at_int @ M) = zero_zero_int) = (M = zero_zero_nat))))). % of_nat_eq_0_iff
thf(fact_85_of__nat__eq__0__iff, axiom,
    ((![M : nat]: (((semiri2110766477t_real @ M) = zero_zero_real) = (M = zero_zero_nat))))). % of_nat_eq_0_iff
thf(fact_86_of__nat__eq__0__iff, axiom,
    ((![M : nat]: (((semiri1382578993at_nat @ M) = zero_zero_nat) = (M = zero_zero_nat))))). % of_nat_eq_0_iff
thf(fact_87_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_88_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_89_power__eq__0__iff, axiom,
    ((![A : real, N : nat]: (((power_power_real @ A @ N) = zero_zero_real) = (((A = zero_zero_real)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % power_eq_0_iff
thf(fact_90_power__eq__0__iff, axiom,
    ((![A : int, N : nat]: (((power_power_int @ A @ N) = zero_zero_int) = (((A = zero_zero_int)) & ((ord_less_nat @ zero_zero_nat @ N))))))). % power_eq_0_iff
thf(fact_91_power__zero__numeral, axiom,
    ((![K : num]: ((power_power_nat @ zero_zero_nat @ (numeral_numeral_nat @ K)) = zero_zero_nat)))). % power_zero_numeral
thf(fact_92_power__zero__numeral, axiom,
    ((![K : num]: ((power_power_real @ zero_zero_real @ (numeral_numeral_nat @ K)) = zero_zero_real)))). % power_zero_numeral
thf(fact_93_power__zero__numeral, axiom,
    ((![K : num]: ((power_power_int @ zero_zero_int @ (numeral_numeral_nat @ K)) = zero_zero_int)))). % power_zero_numeral
thf(fact_94_of__nat__0__less__iff, axiom,
    ((![N : nat]: ((ord_less_int @ zero_zero_int @ (semiri2019852685at_int @ N)) = (ord_less_nat @ zero_zero_nat @ N))))). % of_nat_0_less_iff
thf(fact_95_of__nat__0__less__iff, axiom,
    ((![N : nat]: ((ord_less_real @ zero_zero_real @ (semiri2110766477t_real @ N)) = (ord_less_nat @ zero_zero_nat @ N))))). % of_nat_0_less_iff
thf(fact_96_of__nat__0__less__iff, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ (semiri1382578993at_nat @ N)) = (ord_less_nat @ zero_zero_nat @ N))))). % of_nat_0_less_iff
thf(fact_97_of__nat__zero__less__power__iff, axiom,
    ((![X : nat, N : nat]: ((ord_less_int @ zero_zero_int @ (power_power_int @ (semiri2019852685at_int @ X) @ N)) = (((ord_less_nat @ zero_zero_nat @ X)) | ((N = zero_zero_nat))))))). % of_nat_zero_less_power_iff
thf(fact_98_of__nat__zero__less__power__iff, axiom,
    ((![X : nat, N : nat]: ((ord_less_real @ zero_zero_real @ (power_power_real @ (semiri2110766477t_real @ X) @ N)) = (((ord_less_nat @ zero_zero_nat @ X)) | ((N = zero_zero_nat))))))). % of_nat_zero_less_power_iff
thf(fact_99_of__nat__zero__less__power__iff, axiom,
    ((![X : nat, N : nat]: ((ord_less_nat @ zero_zero_nat @ (power_power_nat @ (semiri1382578993at_nat @ X) @ N)) = (((ord_less_nat @ zero_zero_nat @ X)) | ((N = zero_zero_nat))))))). % of_nat_zero_less_power_iff
thf(fact_100_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_101_zero__eq__power2, axiom,
    ((![A : real]: (((power_power_real @ A @ (numeral_numeral_nat @ (bit0 @ one))) = zero_zero_real) = (A = zero_zero_real))))). % zero_eq_power2
thf(fact_102_zero__eq__power2, axiom,
    ((![A : int]: (((power_power_int @ A @ (numeral_numeral_nat @ (bit0 @ one))) = zero_zero_int) = (A = zero_zero_int))))). % zero_eq_power2
thf(fact_103_int__ops_I1_J, axiom,
    (((semiri2019852685at_int @ zero_zero_nat) = zero_zero_int))). % int_ops(1)
thf(fact_104_less__int__code_I1_J, axiom,
    ((~ ((ord_less_int @ zero_zero_int @ zero_zero_int))))). % less_int_code(1)
thf(fact_105_nat__int__comparison_I1_J, axiom,
    (((^[Y3 : nat]: (^[Z : nat]: (Y3 = Z))) = (^[A2 : nat]: (^[B2 : nat]: ((semiri2019852685at_int @ A2) = (semiri2019852685at_int @ B2))))))). % nat_int_comparison(1)
thf(fact_106_int__if, axiom,
    ((![P : $o, A : nat, B : nat]: ((P => ((semiri2019852685at_int @ (if_nat @ P @ A @ B)) = (semiri2019852685at_int @ A))) & ((~ (P)) => ((semiri2019852685at_int @ (if_nat @ P @ A @ B)) = (semiri2019852685at_int @ B))))))). % int_if
thf(fact_107_int__int__eq, axiom,
    ((![M : nat, N : nat]: (((semiri2019852685at_int @ M) = (semiri2019852685at_int @ N)) = (M = N))))). % int_int_eq
thf(fact_108_pos__int__cases, axiom,
    ((![K : int]: ((ord_less_int @ zero_zero_int @ K) => (~ ((![N2 : nat]: ((K = (semiri2019852685at_int @ N2)) => (~ ((ord_less_nat @ zero_zero_nat @ N2))))))))))). % pos_int_cases
thf(fact_109_zero__less__imp__eq__int, axiom,
    ((![K : int]: ((ord_less_int @ zero_zero_int @ K) => (?[N2 : nat]: ((ord_less_nat @ zero_zero_nat @ N2) & (K = (semiri2019852685at_int @ N2)))))))). % zero_less_imp_eq_int
thf(fact_110_field__lbound__gt__zero, axiom,
    ((![D1 : real, D2 : real]: ((ord_less_real @ zero_zero_real @ D1) => ((ord_less_real @ zero_zero_real @ D2) => (?[E : real]: ((ord_less_real @ zero_zero_real @ E) & ((ord_less_real @ E @ D1) & (ord_less_real @ E @ D2))))))))). % field_lbound_gt_zero
thf(fact_111_gr0I, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr0I
thf(fact_112_not__gr0, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr0
thf(fact_113_not__less0, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less0
thf(fact_114_less__zeroE, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % less_zeroE
thf(fact_115_gr__implies__not0, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not0
thf(fact_116_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_117_bot__nat__0_Oextremum__strict, axiom,
    ((![A : nat]: (~ ((ord_less_nat @ A @ zero_zero_nat)))))). % bot_nat_0.extremum_strict
thf(fact_118_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_119_zero__power, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((power_power_real @ zero_zero_real @ N) = zero_zero_real))))). % zero_power
thf(fact_120_zero__power, axiom,
    ((![N : nat]: ((ord_less_nat @ zero_zero_nat @ N) => ((power_power_int @ zero_zero_int @ N) = zero_zero_int))))). % zero_power
thf(fact_121_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_nat @ zero_zero_nat @ zero_zero_nat))))). % less_numeral_extra(3)
thf(fact_122_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_int @ zero_zero_int @ zero_zero_int))))). % less_numeral_extra(3)
thf(fact_123_less__numeral__extra_I3_J, axiom,
    ((~ ((ord_less_real @ zero_zero_real @ zero_zero_real))))). % less_numeral_extra(3)
thf(fact_124_zero__neq__numeral, axiom,
    ((![N : num]: (~ ((zero_zero_nat = (numeral_numeral_nat @ N))))))). % zero_neq_numeral
thf(fact_125_zero__neq__numeral, axiom,
    ((![N : num]: (~ ((zero_zero_int = (numeral_numeral_int @ N))))))). % zero_neq_numeral
thf(fact_126_zero__neq__numeral, axiom,
    ((![N : num]: (~ ((zero_zero_real = (numeral_numeral_real @ N))))))). % zero_neq_numeral
thf(fact_127_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_128_power__not__zero, axiom,
    ((![A : real, N : nat]: ((~ ((A = zero_zero_real))) => (~ (((power_power_real @ A @ N) = zero_zero_real))))))). % power_not_zero
thf(fact_129_power__not__zero, axiom,
    ((![A : int, N : nat]: ((~ ((A = zero_zero_int))) => (~ (((power_power_int @ A @ N) = zero_zero_int))))))). % power_not_zero
thf(fact_130_of__nat__less__0__iff, axiom,
    ((![M : nat]: (~ ((ord_less_int @ (semiri2019852685at_int @ M) @ zero_zero_int)))))). % of_nat_less_0_iff
thf(fact_131_of__nat__less__0__iff, axiom,
    ((![M : nat]: (~ ((ord_less_real @ (semiri2110766477t_real @ M) @ zero_zero_real)))))). % of_nat_less_0_iff
thf(fact_132_of__nat__less__0__iff, axiom,
    ((![M : nat]: (~ ((ord_less_nat @ (semiri1382578993at_nat @ M) @ zero_zero_nat)))))). % of_nat_less_0_iff
thf(fact_133_FFT_Osimps_I1_J, axiom,
    ((![A : nat > complex]: ((fFT_Mirabelle_FFT @ zero_zero_nat @ A) = A)))). % FFT.simps(1)
thf(fact_134_IFFT_Osimps_I1_J, axiom,
    ((![A : nat > complex]: ((fFT_Mirabelle_IFFT @ zero_zero_nat @ A) = A)))). % IFFT.simps(1)
thf(fact_135_zero__less__numeral, axiom,
    ((![N : num]: (ord_less_nat @ zero_zero_nat @ (numeral_numeral_nat @ N))))). % zero_less_numeral
thf(fact_136_zero__less__numeral, axiom,
    ((![N : num]: (ord_less_int @ zero_zero_int @ (numeral_numeral_int @ N))))). % zero_less_numeral
thf(fact_137_zero__less__numeral, axiom,
    ((![N : num]: (ord_less_real @ zero_zero_real @ (numeral_numeral_real @ N))))). % zero_less_numeral
thf(fact_138_not__numeral__less__zero, axiom,
    ((![N : num]: (~ ((ord_less_nat @ (numeral_numeral_nat @ N) @ zero_zero_nat)))))). % not_numeral_less_zero
thf(fact_139_not__numeral__less__zero, axiom,
    ((![N : num]: (~ ((ord_less_int @ (numeral_numeral_int @ N) @ zero_zero_int)))))). % not_numeral_less_zero
thf(fact_140_not__numeral__less__zero, axiom,
    ((![N : num]: (~ ((ord_less_real @ (numeral_numeral_real @ N) @ zero_zero_real)))))). % not_numeral_less_zero
thf(fact_141_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_142_zero__less__power, axiom,
    ((![A : int, N : nat]: ((ord_less_int @ zero_zero_int @ A) => (ord_less_int @ zero_zero_int @ (power_power_int @ A @ N)))))). % zero_less_power
thf(fact_143_zero__less__power, axiom,
    ((![A : real, N : nat]: ((ord_less_real @ zero_zero_real @ A) => (ord_less_real @ zero_zero_real @ (power_power_real @ A @ N)))))). % zero_less_power
thf(fact_144_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_145_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_146_less__not__refl, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_not_refl
thf(fact_147_less__not__refl2, axiom,
    ((![N : nat, M : nat]: ((ord_less_nat @ N @ M) => (~ ((M = N))))))). % less_not_refl2
thf(fact_148_less__not__refl3, axiom,
    ((![S : nat, T : nat]: ((ord_less_nat @ S @ T) => (~ ((S = T))))))). % less_not_refl3
thf(fact_149_less__irrefl__nat, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ N)))))). % less_irrefl_nat
thf(fact_150_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_151_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_152_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_153_zero__power2, axiom,
    (((power_power_nat @ zero_zero_nat @ (numeral_numeral_nat @ (bit0 @ one))) = zero_zero_nat))). % zero_power2
thf(fact_154_zero__power2, axiom,
    (((power_power_real @ zero_zero_real @ (numeral_numeral_nat @ (bit0 @ one))) = zero_zero_real))). % zero_power2
thf(fact_155_zero__power2, axiom,
    (((power_power_int @ zero_zero_int @ (numeral_numeral_nat @ (bit0 @ one))) = zero_zero_int))). % zero_power2
thf(fact_156_power2__less__0, axiom,
    ((![A : int]: (~ ((ord_less_int @ (power_power_int @ A @ (numeral_numeral_nat @ (bit0 @ one))) @ zero_zero_int)))))). % power2_less_0
thf(fact_157_power2__less__0, axiom,
    ((![A : real]: (~ ((ord_less_real @ (power_power_real @ A @ (numeral_numeral_nat @ (bit0 @ one))) @ zero_zero_real)))))). % power2_less_0
thf(fact_158_not__gr__zero, axiom,
    ((![N : nat]: ((~ ((ord_less_nat @ zero_zero_nat @ N))) = (N = zero_zero_nat))))). % not_gr_zero
thf(fact_159_pos2, axiom,
    ((ord_less_nat @ zero_zero_nat @ (numeral_numeral_nat @ (bit0 @ one))))). % pos2
thf(fact_160_realpow__pos__nth, axiom,
    ((![N : nat, A : real]: ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_real @ zero_zero_real @ A) => (?[R : real]: ((ord_less_real @ zero_zero_real @ R) & ((power_power_real @ R @ N) = A)))))))). % realpow_pos_nth
thf(fact_161_realpow__pos__nth__unique, axiom,
    ((![N : nat, A : real]: ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_real @ zero_zero_real @ A) => (?[X3 : real]: (((ord_less_real @ zero_zero_real @ X3) & ((power_power_real @ X3 @ N) = A)) & (![Y4 : real]: (((ord_less_real @ zero_zero_real @ Y4) & ((power_power_real @ Y4 @ N) = A)) => (Y4 = X3)))))))))). % realpow_pos_nth_unique
thf(fact_162_gr__zeroI, axiom,
    ((![N : nat]: ((~ ((N = zero_zero_nat))) => (ord_less_nat @ zero_zero_nat @ N))))). % gr_zeroI
thf(fact_163_not__less__zero, axiom,
    ((![N : nat]: (~ ((ord_less_nat @ N @ zero_zero_nat)))))). % not_less_zero
thf(fact_164_zero__reorient, axiom,
    ((![X : nat]: ((zero_zero_nat = X) = (X = zero_zero_nat))))). % zero_reorient
thf(fact_165_zero__reorient, axiom,
    ((![X : int]: ((zero_zero_int = X) = (X = zero_zero_int))))). % zero_reorient
thf(fact_166_zero__reorient, axiom,
    ((![X : real]: ((zero_zero_real = X) = (X = zero_zero_real))))). % zero_reorient
thf(fact_167_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_168_gr__implies__not__zero, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (~ ((N = zero_zero_nat))))))). % gr_implies_not_zero
thf(fact_169_log2__of__power__less, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N)) => ((ord_less_nat @ zero_zero_nat @ M) => (ord_less_real @ (log @ (numeral_numeral_real @ (bit0 @ one)) @ (semiri2110766477t_real @ M)) @ (semiri2110766477t_real @ N))))))). % log2_of_power_less
thf(fact_170_power2__eq__iff__nonneg, axiom,
    ((![X : nat, Y : nat]: ((ord_less_eq_nat @ zero_zero_nat @ X) => ((ord_less_eq_nat @ zero_zero_nat @ Y) => (((power_power_nat @ X @ (numeral_numeral_nat @ (bit0 @ one))) = (power_power_nat @ Y @ (numeral_numeral_nat @ (bit0 @ one)))) = (X = Y))))))). % power2_eq_iff_nonneg
thf(fact_171_power2__eq__iff__nonneg, axiom,
    ((![X : real, Y : real]: ((ord_less_eq_real @ zero_zero_real @ X) => ((ord_less_eq_real @ zero_zero_real @ Y) => (((power_power_real @ X @ (numeral_numeral_nat @ (bit0 @ one))) = (power_power_real @ Y @ (numeral_numeral_nat @ (bit0 @ one)))) = (X = Y))))))). % power2_eq_iff_nonneg
thf(fact_172_power2__eq__iff__nonneg, axiom,
    ((![X : int, Y : int]: ((ord_less_eq_int @ zero_zero_int @ X) => ((ord_less_eq_int @ zero_zero_int @ Y) => (((power_power_int @ X @ (numeral_numeral_nat @ (bit0 @ one))) = (power_power_int @ Y @ (numeral_numeral_nat @ (bit0 @ one)))) = (X = Y))))))). % power2_eq_iff_nonneg
thf(fact_173_power2__less__eq__zero__iff, axiom,
    ((![A : real]: ((ord_less_eq_real @ (power_power_real @ A @ (numeral_numeral_nat @ (bit0 @ one))) @ zero_zero_real) = (A = zero_zero_real))))). % power2_less_eq_zero_iff
thf(fact_174_power2__less__eq__zero__iff, axiom,
    ((![A : int]: ((ord_less_eq_int @ (power_power_int @ A @ (numeral_numeral_nat @ (bit0 @ one))) @ zero_zero_int) = (A = zero_zero_int))))). % power2_less_eq_zero_iff
thf(fact_175_semiring__norm_I71_J, axiom,
    ((![M : num, N : num]: ((ord_less_eq_num @ (bit0 @ M) @ (bit0 @ N)) = (ord_less_eq_num @ M @ N))))). % semiring_norm(71)
thf(fact_176_semiring__norm_I68_J, axiom,
    ((![N : num]: (ord_less_eq_num @ one @ N)))). % semiring_norm(68)
thf(fact_177_bot__nat__0_Oextremum, axiom,
    ((![A : nat]: (ord_less_eq_nat @ zero_zero_nat @ A)))). % bot_nat_0.extremum
thf(fact_178_le0, axiom,
    ((![N : nat]: (ord_less_eq_nat @ zero_zero_nat @ N)))). % le0
thf(fact_179_le__zero__eq, axiom,
    ((![N : nat]: ((ord_less_eq_nat @ N @ zero_zero_nat) = (N = zero_zero_nat))))). % le_zero_eq
thf(fact_180_numeral__le__iff, axiom,
    ((![M : num, N : num]: ((ord_less_eq_nat @ (numeral_numeral_nat @ M) @ (numeral_numeral_nat @ N)) = (ord_less_eq_num @ M @ N))))). % numeral_le_iff
thf(fact_181_numeral__le__iff, axiom,
    ((![M : num, N : num]: ((ord_less_eq_real @ (numeral_numeral_real @ M) @ (numeral_numeral_real @ N)) = (ord_less_eq_num @ M @ N))))). % numeral_le_iff
thf(fact_182_numeral__le__iff, axiom,
    ((![M : num, N : num]: ((ord_less_eq_int @ (numeral_numeral_int @ M) @ (numeral_numeral_int @ N)) = (ord_less_eq_num @ M @ N))))). % numeral_le_iff
thf(fact_183_of__nat__le__iff, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ (semiri1382578993at_nat @ M) @ (semiri1382578993at_nat @ N)) = (ord_less_eq_nat @ M @ N))))). % of_nat_le_iff
thf(fact_184_of__nat__le__iff, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_real @ (semiri2110766477t_real @ M) @ (semiri2110766477t_real @ N)) = (ord_less_eq_nat @ M @ N))))). % of_nat_le_iff
thf(fact_185_of__nat__le__iff, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_int @ (semiri2019852685at_int @ M) @ (semiri2019852685at_int @ N)) = (ord_less_eq_nat @ M @ N))))). % of_nat_le_iff
thf(fact_186_semiring__norm_I69_J, axiom,
    ((![M : num]: (~ ((ord_less_eq_num @ (bit0 @ M) @ one)))))). % semiring_norm(69)
thf(fact_187_numeral__le__real__of__nat__iff, axiom,
    ((![N : num, M : nat]: ((ord_less_eq_real @ (numeral_numeral_real @ N) @ (semiri2110766477t_real @ M)) = (ord_less_eq_nat @ (numeral_numeral_nat @ N) @ M))))). % numeral_le_real_of_nat_iff
thf(fact_188_of__nat__le__0__iff, axiom,
    ((![M : nat]: ((ord_less_eq_nat @ (semiri1382578993at_nat @ M) @ zero_zero_nat) = (M = zero_zero_nat))))). % of_nat_le_0_iff
thf(fact_189_of__nat__le__0__iff, axiom,
    ((![M : nat]: ((ord_less_eq_real @ (semiri2110766477t_real @ M) @ zero_zero_real) = (M = zero_zero_nat))))). % of_nat_le_0_iff
thf(fact_190_of__nat__le__0__iff, axiom,
    ((![M : nat]: ((ord_less_eq_int @ (semiri2019852685at_int @ M) @ zero_zero_int) = (M = zero_zero_nat))))). % of_nat_le_0_iff
thf(fact_191_of__nat__power__le__of__nat__cancel__iff, axiom,
    ((![X : nat, B : nat, W : nat]: ((ord_less_eq_nat @ (semiri1382578993at_nat @ X) @ (power_power_nat @ (semiri1382578993at_nat @ B) @ W)) = (ord_less_eq_nat @ X @ (power_power_nat @ B @ W)))))). % of_nat_power_le_of_nat_cancel_iff
thf(fact_192_of__nat__power__le__of__nat__cancel__iff, axiom,
    ((![X : nat, B : nat, W : nat]: ((ord_less_eq_real @ (semiri2110766477t_real @ X) @ (power_power_real @ (semiri2110766477t_real @ B) @ W)) = (ord_less_eq_nat @ X @ (power_power_nat @ B @ W)))))). % of_nat_power_le_of_nat_cancel_iff
thf(fact_193_of__nat__power__le__of__nat__cancel__iff, axiom,
    ((![X : nat, B : nat, W : nat]: ((ord_less_eq_int @ (semiri2019852685at_int @ X) @ (power_power_int @ (semiri2019852685at_int @ B) @ W)) = (ord_less_eq_nat @ X @ (power_power_nat @ B @ W)))))). % of_nat_power_le_of_nat_cancel_iff
thf(fact_194_of__nat__le__of__nat__power__cancel__iff, axiom,
    ((![B : nat, W : nat, X : nat]: ((ord_less_eq_nat @ (power_power_nat @ (semiri1382578993at_nat @ B) @ W) @ (semiri1382578993at_nat @ X)) = (ord_less_eq_nat @ (power_power_nat @ B @ W) @ X))))). % of_nat_le_of_nat_power_cancel_iff
thf(fact_195_of__nat__le__of__nat__power__cancel__iff, axiom,
    ((![B : nat, W : nat, X : nat]: ((ord_less_eq_real @ (power_power_real @ (semiri2110766477t_real @ B) @ W) @ (semiri2110766477t_real @ X)) = (ord_less_eq_nat @ (power_power_nat @ B @ W) @ X))))). % of_nat_le_of_nat_power_cancel_iff
thf(fact_196_of__nat__le__of__nat__power__cancel__iff, axiom,
    ((![B : nat, W : nat, X : nat]: ((ord_less_eq_int @ (power_power_int @ (semiri2019852685at_int @ B) @ W) @ (semiri2019852685at_int @ X)) = (ord_less_eq_nat @ (power_power_nat @ B @ W) @ X))))). % of_nat_le_of_nat_power_cancel_iff
thf(fact_197_power__mono__iff, axiom,
    ((![A : nat, B : nat, N : nat]: ((ord_less_eq_nat @ zero_zero_nat @ A) => ((ord_less_eq_nat @ zero_zero_nat @ B) => ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_eq_nat @ (power_power_nat @ A @ N) @ (power_power_nat @ B @ N)) = (ord_less_eq_nat @ A @ B)))))))). % power_mono_iff
thf(fact_198_power__mono__iff, axiom,
    ((![A : real, B : real, N : nat]: ((ord_less_eq_real @ zero_zero_real @ A) => ((ord_less_eq_real @ zero_zero_real @ B) => ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_eq_real @ (power_power_real @ A @ N) @ (power_power_real @ B @ N)) = (ord_less_eq_real @ A @ B)))))))). % power_mono_iff
thf(fact_199_power__mono__iff, axiom,
    ((![A : int, B : int, N : nat]: ((ord_less_eq_int @ zero_zero_int @ A) => ((ord_less_eq_int @ zero_zero_int @ B) => ((ord_less_nat @ zero_zero_nat @ N) => ((ord_less_eq_int @ (power_power_int @ A @ N) @ (power_power_int @ B @ N)) = (ord_less_eq_int @ A @ B)))))))). % power_mono_iff
thf(fact_200_numeral__power__le__of__nat__cancel__iff, axiom,
    ((![I : num, N : nat, X : nat]: ((ord_less_eq_nat @ (power_power_nat @ (numeral_numeral_nat @ I) @ N) @ (semiri1382578993at_nat @ X)) = (ord_less_eq_nat @ (power_power_nat @ (numeral_numeral_nat @ I) @ N) @ X))))). % numeral_power_le_of_nat_cancel_iff
thf(fact_201_numeral__power__le__of__nat__cancel__iff, axiom,
    ((![I : num, N : nat, X : nat]: ((ord_less_eq_real @ (power_power_real @ (numeral_numeral_real @ I) @ N) @ (semiri2110766477t_real @ X)) = (ord_less_eq_nat @ (power_power_nat @ (numeral_numeral_nat @ I) @ N) @ X))))). % numeral_power_le_of_nat_cancel_iff
thf(fact_202_numeral__power__le__of__nat__cancel__iff, axiom,
    ((![I : num, N : nat, X : nat]: ((ord_less_eq_int @ (power_power_int @ (numeral_numeral_int @ I) @ N) @ (semiri2019852685at_int @ X)) = (ord_less_eq_nat @ (power_power_nat @ (numeral_numeral_nat @ I) @ N) @ X))))). % numeral_power_le_of_nat_cancel_iff
thf(fact_203_of__nat__le__numeral__power__cancel__iff, axiom,
    ((![X : nat, I : num, N : nat]: ((ord_less_eq_nat @ (semiri1382578993at_nat @ X) @ (power_power_nat @ (numeral_numeral_nat @ I) @ N)) = (ord_less_eq_nat @ X @ (power_power_nat @ (numeral_numeral_nat @ I) @ N)))))). % of_nat_le_numeral_power_cancel_iff
thf(fact_204_of__nat__le__numeral__power__cancel__iff, axiom,
    ((![X : nat, I : num, N : nat]: ((ord_less_eq_real @ (semiri2110766477t_real @ X) @ (power_power_real @ (numeral_numeral_real @ I) @ N)) = (ord_less_eq_nat @ X @ (power_power_nat @ (numeral_numeral_nat @ I) @ N)))))). % of_nat_le_numeral_power_cancel_iff
thf(fact_205_of__nat__le__numeral__power__cancel__iff, axiom,
    ((![X : nat, I : num, N : nat]: ((ord_less_eq_int @ (semiri2019852685at_int @ X) @ (power_power_int @ (numeral_numeral_int @ I) @ N)) = (ord_less_eq_nat @ X @ (power_power_nat @ (numeral_numeral_nat @ I) @ N)))))). % of_nat_le_numeral_power_cancel_iff
thf(fact_206_zero__le, axiom,
    ((![X : nat]: (ord_less_eq_nat @ zero_zero_nat @ X)))). % zero_le
thf(fact_207_bot__nat__0_Oextremum__uniqueI, axiom,
    ((![A : nat]: ((ord_less_eq_nat @ A @ zero_zero_nat) => (A = zero_zero_nat))))). % bot_nat_0.extremum_uniqueI
thf(fact_208_bot__nat__0_Oextremum__unique, axiom,
    ((![A : nat]: ((ord_less_eq_nat @ A @ zero_zero_nat) = (A = zero_zero_nat))))). % bot_nat_0.extremum_unique
thf(fact_209_le__0__eq, axiom,
    ((![N : nat]: ((ord_less_eq_nat @ N @ zero_zero_nat) = (N = zero_zero_nat))))). % le_0_eq
thf(fact_210_less__eq__nat_Osimps_I1_J, axiom,
    ((![N : nat]: (ord_less_eq_nat @ zero_zero_nat @ N)))). % less_eq_nat.simps(1)
thf(fact_211_less__eq__int__code_I1_J, axiom,
    ((ord_less_eq_int @ zero_zero_int @ zero_zero_int))). % less_eq_int_code(1)
thf(fact_212_le__numeral__extra_I3_J, axiom,
    ((ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat))). % le_numeral_extra(3)
thf(fact_213_le__numeral__extra_I3_J, axiom,
    ((ord_less_eq_real @ zero_zero_real @ zero_zero_real))). % le_numeral_extra(3)
thf(fact_214_le__numeral__extra_I3_J, axiom,
    ((ord_less_eq_int @ zero_zero_int @ zero_zero_int))). % le_numeral_extra(3)
thf(fact_215_verit__comp__simplify1_I3_J, axiom,
    ((![B3 : num, A3 : num]: ((~ ((ord_less_eq_num @ B3 @ A3))) = (ord_less_num @ A3 @ B3))))). % verit_comp_simplify1(3)
thf(fact_216_verit__comp__simplify1_I3_J, axiom,
    ((![B3 : nat, A3 : nat]: ((~ ((ord_less_eq_nat @ B3 @ A3))) = (ord_less_nat @ A3 @ B3))))). % verit_comp_simplify1(3)
thf(fact_217_verit__comp__simplify1_I3_J, axiom,
    ((![B3 : real, A3 : real]: ((~ ((ord_less_eq_real @ B3 @ A3))) = (ord_less_real @ A3 @ B3))))). % verit_comp_simplify1(3)
thf(fact_218_verit__comp__simplify1_I3_J, axiom,
    ((![B3 : int, A3 : int]: ((~ ((ord_less_eq_int @ B3 @ A3))) = (ord_less_int @ A3 @ B3))))). % verit_comp_simplify1(3)
thf(fact_219_le__num__One__iff, axiom,
    ((![X : num]: ((ord_less_eq_num @ X @ one) = (X = one))))). % le_num_One_iff
thf(fact_220_real__arch__simple, axiom,
    ((![X : real]: (?[N2 : nat]: (ord_less_eq_real @ X @ (semiri2110766477t_real @ N2)))))). % real_arch_simple
thf(fact_221_less__eq__real__def, axiom,
    ((ord_less_eq_real = (^[X4 : real]: (^[Y5 : real]: (((ord_less_real @ X4 @ Y5)) | ((X4 = Y5)))))))). % less_eq_real_def
thf(fact_222_less__mono__imp__le__mono, axiom,
    ((![F : nat > nat, I : nat, J : nat]: ((![I2 : nat, J2 : nat]: ((ord_less_nat @ I2 @ J2) => (ord_less_nat @ (F @ I2) @ (F @ J2)))) => ((ord_less_eq_nat @ I @ J) => (ord_less_eq_nat @ (F @ I) @ (F @ J))))))). % less_mono_imp_le_mono
thf(fact_223_le__neq__implies__less, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ M @ N) => ((~ ((M = N))) => (ord_less_nat @ M @ N)))))). % le_neq_implies_less
thf(fact_224_less__or__eq__imp__le, axiom,
    ((![M : nat, N : nat]: (((ord_less_nat @ M @ N) | (M = N)) => (ord_less_eq_nat @ M @ N))))). % less_or_eq_imp_le
thf(fact_225_le__eq__less__or__eq, axiom,
    ((ord_less_eq_nat = (^[M3 : nat]: (^[N3 : nat]: (((ord_less_nat @ M3 @ N3)) | ((M3 = N3)))))))). % le_eq_less_or_eq
thf(fact_226_less__imp__le__nat, axiom,
    ((![M : nat, N : nat]: ((ord_less_nat @ M @ N) => (ord_less_eq_nat @ M @ N))))). % less_imp_le_nat
thf(fact_227_nat__less__le, axiom,
    ((ord_less_nat = (^[M3 : nat]: (^[N3 : nat]: (((ord_less_eq_nat @ M3 @ N3)) & ((~ ((M3 = N3)))))))))). % nat_less_le
thf(fact_228_le__log2__of__power, axiom,
    ((![N : nat, M : nat]: ((ord_less_eq_nat @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N) @ M) => (ord_less_eq_real @ (semiri2110766477t_real @ N) @ (log @ (numeral_numeral_real @ (bit0 @ one)) @ (semiri2110766477t_real @ M))))))). % le_log2_of_power
thf(fact_229_verit__la__disequality, axiom,
    ((![A : num, B : num]: ((A = B) | ((~ ((ord_less_eq_num @ A @ B))) | (~ ((ord_less_eq_num @ B @ A)))))))). % verit_la_disequality
thf(fact_230_verit__la__disequality, axiom,
    ((![A : nat, B : nat]: ((A = B) | ((~ ((ord_less_eq_nat @ A @ B))) | (~ ((ord_less_eq_nat @ B @ A)))))))). % verit_la_disequality
thf(fact_231_verit__la__disequality, axiom,
    ((![A : real, B : real]: ((A = B) | ((~ ((ord_less_eq_real @ A @ B))) | (~ ((ord_less_eq_real @ B @ A)))))))). % verit_la_disequality
thf(fact_232_verit__la__disequality, axiom,
    ((![A : int, B : int]: ((A = B) | ((~ ((ord_less_eq_int @ A @ B))) | (~ ((ord_less_eq_int @ B @ A)))))))). % verit_la_disequality
thf(fact_233_of__nat__mono, axiom,
    ((![I : nat, J : nat]: ((ord_less_eq_nat @ I @ J) => (ord_less_eq_nat @ (semiri1382578993at_nat @ I) @ (semiri1382578993at_nat @ J)))))). % of_nat_mono
thf(fact_234_of__nat__mono, axiom,
    ((![I : nat, J : nat]: ((ord_less_eq_nat @ I @ J) => (ord_less_eq_real @ (semiri2110766477t_real @ I) @ (semiri2110766477t_real @ J)))))). % of_nat_mono
thf(fact_235_of__nat__mono, axiom,
    ((![I : nat, J : nat]: ((ord_less_eq_nat @ I @ J) => (ord_less_eq_int @ (semiri2019852685at_int @ I) @ (semiri2019852685at_int @ J)))))). % of_nat_mono
thf(fact_236_zle__int, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_int @ (semiri2019852685at_int @ M) @ (semiri2019852685at_int @ N)) = (ord_less_eq_nat @ M @ N))))). % zle_int
thf(fact_237_nat__int__comparison_I3_J, axiom,
    ((ord_less_eq_nat = (^[A2 : nat]: (^[B2 : nat]: (ord_less_eq_int @ (semiri2019852685at_int @ A2) @ (semiri2019852685at_int @ B2))))))). % nat_int_comparison(3)
thf(fact_238_ex__least__nat__le, axiom,
    ((![P : nat > $o, N : nat]: ((P @ N) => ((~ ((P @ zero_zero_nat))) => (?[K2 : nat]: ((ord_less_eq_nat @ K2 @ N) & ((![I3 : nat]: ((ord_less_nat @ I3 @ K2) => (~ ((P @ I3))))) & (P @ K2))))))))). % ex_least_nat_le
thf(fact_239_log2__of__power__le, axiom,
    ((![M : nat, N : nat]: ((ord_less_eq_nat @ M @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ N)) => ((ord_less_nat @ zero_zero_nat @ M) => (ord_less_eq_real @ (log @ (numeral_numeral_real @ (bit0 @ one)) @ (semiri2110766477t_real @ M)) @ (semiri2110766477t_real @ N))))))). % log2_of_power_le
thf(fact_240_nonneg__int__cases, axiom,
    ((![K : int]: ((ord_less_eq_int @ zero_zero_int @ K) => (~ ((![N2 : nat]: (~ ((K = (semiri2019852685at_int @ N2))))))))))). % nonneg_int_cases

% Helper facts (3)
thf(help_If_3_1_If_001t__Nat__Onat_T, axiom,
    ((![P : $o]: ((P = $true) | (P = $false))))).
thf(help_If_2_1_If_001t__Nat__Onat_T, axiom,
    ((![X : nat, Y : nat]: ((if_nat @ $false @ X @ Y) = Y)))).
thf(help_If_1_1_If_001t__Nat__Onat_T, axiom,
    ((![X : nat, Y : nat]: ((if_nat @ $true @ X @ Y) = X)))).

% Conjectures (2)
thf(conj_0, hypothesis,
    ((ord_less_nat @ i @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ k)))).
thf(conj_1, conjecture,
    (((fFT_Mirabelle_IDFT @ (power_power_nat @ (numeral_numeral_nat @ (bit0 @ one)) @ k) @ a @ i) = (fFT_Mirabelle_IFFT @ k @ a @ i)))).
