ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- RingOfIntegers.norm_algebraMap
theorem norm_algebraMap [Algebra.IsSeparable K L] (x : 𝓞 K) :
    norm K (algebraMap (𝓞 K) (𝓞 L) x) = x ^ finrank K L 
Arguments: {L : Type u_1} (K : Type u_2) [Field K] [Field L] [Algebra K L] [FiniteDimensional K L] [Algebra.IsSeparable K L] (x : 𝓞 K)
Goal: (RingOfIntegers.norm K) ((algebraMap (𝓞 K) (𝓞 L)) x) = x ^ FiniteDimensional.finrank K L
-- RingOfIntegers.isUnit_norm_of_isGalois
theorem isUnit_norm_of_isGalois [IsGalois K L] {x : 𝓞 L} : IsUnit (norm K x) ↔ IsUnit x 
Arguments: {L : Type u_1} (K : Type u_2) [Field K] [Field L] [Algebra K L] [FiniteDimensional K L] [IsGalois K L] {x : 𝓞 L}
Goal: IsUnit ((RingOfIntegers.norm K) x) ↔ IsUnit x
-- RingOfIntegers.dvd_norm
theorem dvd_norm [IsGalois K L] (x : 𝓞 L) : x ∣ algebraMap (𝓞 K) (𝓞 L) (norm K x) 
Arguments: {L : Type u_1} (K : Type u_2) [Field K] [Field L] [Algebra K L] [FiniteDimensional K L] [IsGalois K L] (x : 𝓞 L)
Goal: x ∣ (algebraMap (𝓞 K) (𝓞 L)) ((RingOfIntegers.norm K) x)
-- RingOfIntegers.norm_norm
theorem norm_norm [Algebra.IsSeparable K L] [Algebra F L] [Algebra.IsSeparable F L]
    [FiniteDimensional F L] [IsScalarTower K F L] (x : 𝓞 L) : norm K (norm F x) = norm K x 
Arguments: {L : Type u_1} (K : Type u_2) [Field K] [Field L] [Algebra K L] [FiniteDimensional K L] (F : Type u_3) [Field F] [Algebra K F] [Algebra.IsSeparable K F] [FiniteDimensional K F] [Algebra.IsSeparable K L] [Algebra F L] [Algebra.IsSeparable F L] [FiniteDimensional F L] [IsScalarTower K F L] (x : 𝓞 L)
Goal: (RingOfIntegers.norm K) ((RingOfIntegers.norm F) x) = (RingOfIntegers.norm K) x
-- RingOfIntegers.isUnit_norm
theorem isUnit_norm [CharZero K] {x : 𝓞 F} : IsUnit (norm K x) ↔ IsUnit x 
Arguments: (K : Type u_2) [Field K] {F : Type u_3} [Field F] [Algebra K F] [Algebra.IsSeparable K F] [FiniteDimensional K F] [CharZero K] {x : 𝓞 F}
Goal: IsUnit ((RingOfIntegers.norm K) x) ↔ IsUnit x
-- Fermat42.comm
theorem comm {a b c : ℤ} : Fermat42 a b c ↔ Fermat42 b a c 
Arguments: {a b c : ℤ}
Goal: Fermat42 a b c ↔ Fermat42 b a c
-- Fermat42.mul
theorem mul {a b c k : ℤ} (hk0 : k ≠ 0) :
    Fermat42 a b c ↔ Fermat42 (k * a) (k * b) (k ^ 2 * c) 
Arguments: {a b c k : ℤ} (hk0 : k ≠ 0)
Goal: Fermat42 a b c ↔ Fermat42 (k * a) (k * b) (k ^ 2 * c)
-- Fermat42.ne_zero
theorem ne_zero {a b c : ℤ} (h : Fermat42 a b c) : c ≠ 0 
Arguments: {a b c : ℤ} (h : Fermat42 a b c)
Goal: c ≠ 0
-- Fermat42.exists_minimal
theorem exists_minimal {a b c : ℤ} (h : Fermat42 a b c) : ∃ a0 b0 c0, Minimal a0 b0 c0 
Arguments: {a b c : ℤ} (h : Fermat42 a b c)
Goal: ∃ a0 b0 c0, Fermat42.Minimal a0 b0 c0
-- Fermat42.coprime_of_minimal
theorem coprime_of_minimal {a b c : ℤ} (h : Minimal a b c) : IsCoprime a b 
Arguments: {a b c : ℤ} (h : Fermat42.Minimal a b c)
Goal: IsCoprime a b
-- Fermat42.minimal_comm
theorem minimal_comm {a b c : ℤ} : Minimal a b c → Minimal b a c 
Arguments: {a b c : ℤ}
Goal: Fermat42.Minimal a b c → Fermat42.Minimal b a c
-- Fermat42.neg_of_minimal
theorem neg_of_minimal {a b c : ℤ} : Minimal a b c → Minimal a b (-c) 
Arguments: {a b c : ℤ}
Goal: Fermat42.Minimal a b c → Fermat42.Minimal a b (-c)
-- Fermat42.exists_odd_minimal
theorem exists_odd_minimal {a b c : ℤ} (h : Fermat42 a b c) :
    ∃ a0 b0 c0, Minimal a0 b0 c0 ∧ a0 % 2 = 1 
Arguments: {a b c : ℤ} (h : Fermat42 a b c)
Goal: ∃ a0 b0 c0, Fermat42.Minimal a0 b0 c0 ∧ a0 % 2 = 1
-- Fermat42.exists_pos_odd_minimal
theorem exists_pos_odd_minimal {a b c : ℤ} (h : Fermat42 a b c) :
    ∃ a0 b0 c0, Minimal a0 b0 c0 ∧ a0 % 2 = 1 ∧ 0 < c0 
Arguments: {a b c : ℤ} (h : Fermat42 a b c)
Goal: ∃ a0 b0 c0, Fermat42.Minimal a0 b0 c0 ∧ a0 % 2 = 1 ∧ 0 < c0
-- Int.coprime_of_sq_sum
theorem Int.coprime_of_sq_sum {r s : ℤ} (h2 : IsCoprime s r) : IsCoprime (r ^ 2 + s ^ 2) r 
Arguments: {r s : ℤ} (h2 : IsCoprime s r)
Goal: IsCoprime (r ^ 2 + s ^ 2) r
-- Int.coprime_of_sq_sum'
theorem Int.coprime_of_sq_sum' {r s : ℤ} (h : IsCoprime r s) :
    IsCoprime (r ^ 2 + s ^ 2) (r * s) 
Arguments: {r s : ℤ} (h : IsCoprime r s)
Goal: IsCoprime (r ^ 2 + s ^ 2) (r * s)
-- Fermat42.not_minimal
theorem not_minimal {a b c : ℤ} (h : Minimal a b c) (ha2 : a % 2 = 1) (hc : 0 < c) : False 
Arguments: {a b c : ℤ} (h : Fermat42.Minimal a b c) (ha2 : a % 2 = 1) (hc : 0 < c)
Goal: False
-- not_fermat_42
theorem not_fermat_42 {a b c : ℤ} (ha : a ≠ 0) (hb : b ≠ 0) : a ^ 4 + b ^ 4 ≠ c ^ 2 
Arguments: {a b c : ℤ} (ha : a ≠ 0) (hb : b ≠ 0)
Goal: a ^ 4 + b ^ 4 ≠ c ^ 2
-- fermatLastTheoremFour
theorem fermatLastTheoremFour : FermatLastTheoremFor 4 
Goal: FermatLastTheoremFor 4
-- FermatLastTheorem.of_odd_primes
theorem FermatLastTheorem.of_odd_primes
    (hprimes : ∀ p : ℕ, Nat.Prime p → Odd p → FermatLastTheoremFor p) : FermatLastTheorem 
Arguments: (hprimes : ∀ (p : ℕ), Nat.Prime p → Odd p → FermatLastTheoremFor p)
Goal: FermatLastTheorem
-- isEllSequence_id
lemma isEllSequence_id : IsEllSequence id 
Goal: IsEllSequence id
-- isDivSequence_id
lemma isDivSequence_id : IsDivSequence id 
Goal: IsDivSequence id
-- isEllDivSequence_id
theorem isEllDivSequence_id : IsEllDivSequence id 
Goal: IsEllDivSequence id
-- IsEllSequence.smul
lemma IsEllSequence.smul (h : IsEllSequence W) (x : R) : IsEllSequence (x • W) 
Arguments: {R : Type u} [CommRing R] {W : ℤ → R} (h : IsEllSequence W) (x : R)
Goal: IsEllSequence (x • W)
-- IsDivSequence.smul
lemma IsDivSequence.smul (h : IsDivSequence W) (x : R) : IsDivSequence (x • W) 
Arguments: {R : Type u} [CommRing R] {W : ℤ → R} (h : IsDivSequence W) (x : R)
Goal: IsDivSequence (x • W)
-- IsEllDivSequence.smul
lemma IsEllDivSequence.smul (h : IsEllDivSequence W) (x : R) : IsEllDivSequence (x • W) 
Arguments: {R : Type u} [CommRing R] {W : ℤ → R} (h : IsEllDivSequence W) (x : R)
Goal: IsEllDivSequence (x • W)
-- preNormEDS'_zero
@[simp]
lemma preNormEDS'_zero : preNormEDS' b c d 0 = 0 
Arguments: {R : Type u} [CommRing R] (b c d : R)
Goal: preNormEDS' b c d 0 = 0
-- preNormEDS'_one
@[simp]
lemma preNormEDS'_one : preNormEDS' b c d 1 = 1 
Arguments: {R : Type u} [CommRing R] (b c d : R)
Goal: preNormEDS' b c d 1 = 1
-- preNormEDS'_two
@[simp]
lemma preNormEDS'_two : preNormEDS' b c d 2 = 1 
Arguments: {R : Type u} [CommRing R] (b c d : R)
Goal: preNormEDS' b c d 2 = 1
-- preNormEDS'_three
@[simp]
lemma preNormEDS'_three : preNormEDS' b c d 3 = c 
Arguments: {R : Type u} [CommRing R] (b c d : R)
Goal: preNormEDS' b c d 3 = c
-- preNormEDS'_four
@[simp]
lemma preNormEDS'_four : preNormEDS' b c d 4 = d 
Arguments: {R : Type u} [CommRing R] (b c d : R)
Goal: preNormEDS' b c d 4 = d
-- preNormEDS'_odd
lemma preNormEDS'_odd (m : ℕ) : preNormEDS' b c d (2 * (m + 2) + 1) =
    preNormEDS' b c d (m + 4) * preNormEDS' b c d (m + 2) ^ 3 * (if Even m then b else 1) -
      preNormEDS' b c d (m + 1) * preNormEDS' b c d (m + 3) ^ 3 * (if Even m then 1 else b) 
Arguments: {R : Type u} [CommRing R] (b c d : R) (m : ℕ)
Goal: preNormEDS' b c d (2 * (m + 2) + 1) =
  (preNormEDS' b c d (m + 4) * preNormEDS' b c d (m + 2) ^ 3 * if Even m then b else 1) -
    preNormEDS' b c d (m + 1) * preNormEDS' b c d (m + 3) ^ 3 * if Even m then 1 else b
-- preNormEDS'_even
lemma preNormEDS'_even (m : ℕ) : preNormEDS' b c d (2 * (m + 3)) =
    preNormEDS' b c d (m + 2) ^ 2 * preNormEDS' b c d (m + 3) * preNormEDS' b c d (m + 5) -
      preNormEDS' b c d (m + 1) * preNormEDS' b c d (m + 3) * preNormEDS' b c d (m + 4) ^ 2 
Arguments: {R : Type u} [CommRing R] (b c d : R) (m : ℕ)
Goal: preNormEDS' b c d (2 * (m + 3)) =
  preNormEDS' b c d (m + 2) ^ 2 * preNormEDS' b c d (m + 3) * preNormEDS' b c d (m + 5) -
    preNormEDS' b c d (m + 1) * preNormEDS' b c d (m + 3) * preNormEDS' b c d (m + 4) ^ 2
-- preNormEDS_ofNat
@[simp]
lemma preNormEDS_ofNat (n : ℕ) : preNormEDS b c d n = preNormEDS' b c d n 
Arguments: {R : Type u} [CommRing R] (b c d : R) (n : ℕ)
Goal: preNormEDS b c d ↑n = preNormEDS' b c d n
-- preNormEDS_zero
@[simp]
lemma preNormEDS_zero : preNormEDS b c d 0 = 0 
Arguments: {R : Type u} [CommRing R] (b c d : R)
Goal: preNormEDS b c d 0 = 0
-- preNormEDS_one
@[simp]
lemma preNormEDS_one : preNormEDS b c d 1 = 1 
Arguments: {R : Type u} [CommRing R] (b c d : R)
Goal: preNormEDS b c d 1 = 1
-- preNormEDS_two
@[simp]
lemma preNormEDS_two : preNormEDS b c d 2 = 1 
Arguments: {R : Type u} [CommRing R] (b c d : R)
Goal: preNormEDS b c d 2 = 1
-- preNormEDS_three
@[simp]
lemma preNormEDS_three : preNormEDS b c d 3 = c 
Arguments: {R : Type u} [CommRing R] (b c d : R)
Goal: preNormEDS b c d 3 = c
-- preNormEDS_four
@[simp]
lemma preNormEDS_four : preNormEDS b c d 4 = d 
Arguments: {R : Type u} [CommRing R] (b c d : R)
Goal: preNormEDS b c d 4 = d
-- preNormEDS_odd
lemma preNormEDS_odd (m : ℕ) : preNormEDS b c d (2 * (m + 2) + 1) =
    preNormEDS b c d (m + 4) * preNormEDS b c d (m + 2) ^ 3 * (if Even m then b else 1) -
      preNormEDS b c d (m + 1) * preNormEDS b c d (m + 3) ^ 3 * (if Even m then 1 else b) 
Arguments: {R : Type u} [CommRing R] (b c d : R) (m : ℕ)
Goal: preNormEDS b c d (2 * (↑m + 2) + 1) =
  (preNormEDS b c d (↑m + 4) * preNormEDS b c d (↑m + 2) ^ 3 * if Even m then b else 1) -
    preNormEDS b c d (↑m + 1) * preNormEDS b c d (↑m + 3) ^ 3 * if Even m then 1 else b
-- preNormEDS_even
lemma preNormEDS_even (m : ℕ) : preNormEDS b c d (2 * (m + 3)) =
    preNormEDS b c d (m + 2) ^ 2 * preNormEDS b c d (m + 3) * preNormEDS b c d (m + 5) -
      preNormEDS b c d (m + 1) * preNormEDS b c d (m + 3) * preNormEDS b c d (m + 4) ^ 2 
Arguments: {R : Type u} [CommRing R] (b c d : R) (m : ℕ)
Goal: preNormEDS b c d (2 * (↑m + 3)) =
  preNormEDS b c d (↑m + 2) ^ 2 * preNormEDS b c d (↑m + 3) * preNormEDS b c d (↑m + 5) -
    preNormEDS b c d (↑m + 1) * preNormEDS b c d (↑m + 3) * preNormEDS b c d (↑m + 4) ^ 2
-- preNormEDS_neg
@[simp]
lemma preNormEDS_neg (n : ℤ) : preNormEDS b c d (-n) = -preNormEDS b c d n 
Arguments: {R : Type u} [CommRing R] (b c d : R) (n : ℤ)
Goal: preNormEDS b c d (-n) = -preNormEDS b c d n
-- normEDS_ofNat
@[simp]
lemma normEDS_ofNat (n : ℕ) :
    normEDS b c d n = preNormEDS' (b ^ 4) c d n * if Even n then b else 1 
Arguments: {R : Type u} [CommRing R] (b c d : R) (n : ℕ)
Goal: normEDS b c d ↑n = preNormEDS' (b ^ 4) c d n * if Even n then b else 1
-- normEDS_zero
@[simp]
lemma normEDS_zero : normEDS b c d 0 = 0 
Arguments: {R : Type u} [CommRing R] (b c d : R)
Goal: normEDS b c d 0 = 0
-- normEDS_one
@[simp]
lemma normEDS_one : normEDS b c d 1 = 1 
Arguments: {R : Type u} [CommRing R] (b c d : R)
Goal: normEDS b c d 1 = 1
-- normEDS_two
@[simp]
lemma normEDS_two : normEDS b c d 2 = b 
Arguments: {R : Type u} [CommRing R] (b c d : R)
Goal: normEDS b c d 2 = b
-- normEDS_three
@[simp]
lemma normEDS_three : normEDS b c d 3 = c 
Arguments: {R : Type u} [CommRing R] (b c d : R)
Goal: normEDS b c d 3 = c
-- normEDS_four
@[simp]
lemma normEDS_four : normEDS b c d 4 = d * b 
Arguments: {R : Type u} [CommRing R] (b c d : R)
Goal: normEDS b c d 4 = d * b
-- normEDS_odd
lemma normEDS_odd (m : ℕ) : normEDS b c d (2 * (m + 2) + 1) =
    normEDS b c d (m + 4) * normEDS b c d (m + 2) ^ 3 -
      normEDS b c d (m + 1) * normEDS b c d (m + 3) ^ 3 
Arguments: {R : Type u} [CommRing R] (b c d : R) (m : ℕ)
Goal: normEDS b c d (2 * (↑m + 2) + 1) =
  normEDS b c d (↑m + 4) * normEDS b c d (↑m + 2) ^ 3 - normEDS b c d (↑m + 1) * normEDS b c d (↑m + 3) ^ 3
-- normEDS_even
lemma normEDS_even (m : ℕ) : normEDS b c d (2 * (m + 3)) * b =
    normEDS b c d (m + 2) ^ 2 * normEDS b c d (m + 3) * normEDS b c d (m + 5) -
      normEDS b c d (m + 1) * normEDS b c d (m + 3) * normEDS b c d (m + 4) ^ 2 
Arguments: {R : Type u} [CommRing R] (b c d : R) (m : ℕ)
Goal: normEDS b c d (2 * (↑m + 3)) * b =
  normEDS b c d (↑m + 2) ^ 2 * normEDS b c d (↑m + 3) * normEDS b c d (↑m + 5) -
    normEDS b c d (↑m + 1) * normEDS b c d (↑m + 3) * normEDS b c d (↑m + 4) ^ 2
-- normEDS_neg
@[simp]
lemma normEDS_neg (n : ℤ) : normEDS b c d (-n) = -normEDS b c d n 
Arguments: {R : Type u} [CommRing R] (b c d : R) (n : ℤ)
Goal: normEDS b c d (-n) = -normEDS b c d n
-- map_preNormEDS'
lemma map_preNormEDS' (n : ℕ) : f (preNormEDS' b c d n) = preNormEDS' (f b) (f c) (f d) n 
Arguments: {R : Type u} {S : Type v} [CommRing R] [CommRing S] (f : R →+* S) (b c d : R) (n : ℕ)
Goal: f (preNormEDS' b c d n) = preNormEDS' (f b) (f c) (f d) n
-- map_preNormEDS
lemma map_preNormEDS (n : ℤ) : f (preNormEDS b c d n) = preNormEDS (f b) (f c) (f d) n 
Arguments: {R : Type u} {S : Type v} [CommRing R] [CommRing S] (f : R →+* S) (b c d : R) (n : ℤ)
Goal: f (preNormEDS b c d n) = preNormEDS (f b) (f c) (f d) n
-- map_normEDS
lemma map_normEDS (n : ℤ) : f (normEDS b c d n) = normEDS (f b) (f c) (f d) n 
Arguments: {R : Type u} {S : Type v} [CommRing R] [CommRing S] (f : R →+* S) (b c d : R) (n : ℤ)
Goal: f (normEDS b c d n) = normEDS (f b) (f c) (f d) n
-- LSeries.hasDerivAt_term
lemma LSeries.hasDerivAt_term (f : ℕ → ℂ) (n : ℕ) (s : ℂ) :
    HasDerivAt (fun z ↦ term f z n) (-(term (logMul f) s n)) s 
Arguments: (f : ℕ → ℂ) (n : ℕ) (s : ℂ)
Goal: HasDerivAt (fun z => LSeries.term f z n) (-LSeries.term (LSeries.logMul f) s n) s
-- _private.Mathlib.NumberTheory.LSeries.Deriv.0.LSeries.LSeriesSummable_logMul_and_hasDerivAt
private lemma LSeries.LSeriesSummable_logMul_and_hasDerivAt {f : ℕ → ℂ} {s : ℂ}
    (h : abscissaOfAbsConv f < s.re) :
    LSeriesSummable (logMul f) s ∧ HasDerivAt (LSeries f) (-LSeries (logMul f) s) s 
Arguments: {f : ℕ → ℂ} {s : ℂ} (h : LSeries.abscissaOfAbsConv f < ↑s.re)
Goal: LSeriesSummable (LSeries.logMul f) s ∧ HasDerivAt (LSeries f) (-LSeries (LSeries.logMul f) s) s
-- LSeries_hasDerivAt
lemma LSeries_hasDerivAt {f : ℕ → ℂ} {s : ℂ} (h : abscissaOfAbsConv f < s.re) :
    HasDerivAt (LSeries f) (- LSeries (logMul f) s) s 
Arguments: {f : ℕ → ℂ} {s : ℂ} (h : LSeries.abscissaOfAbsConv f < ↑s.re)
Goal: HasDerivAt (LSeries f) (-LSeries (LSeries.logMul f) s) s
-- LSeries_deriv
lemma LSeries_deriv {f : ℕ → ℂ} {s : ℂ} (h : abscissaOfAbsConv f < s.re) :
    deriv (LSeries f) s = - LSeries (logMul f) s 
Arguments: {f : ℕ → ℂ} {s : ℂ} (h : LSeries.abscissaOfAbsConv f < ↑s.re)
Goal: deriv (LSeries f) s = -LSeries (LSeries.logMul f) s
-- LSeries_deriv_eqOn
lemma LSeries_deriv_eqOn {f : ℕ → ℂ} :
    {s | abscissaOfAbsConv f < s.re}.EqOn (deriv (LSeries f)) (- LSeries (logMul f)) 
Arguments: {f : ℕ → ℂ}
Goal: Set.EqOn (deriv (LSeries f)) (-LSeries (LSeries.logMul f)) {s | LSeries.abscissaOfAbsConv f < ↑s.re}
-- LSeriesSummable_logMul_of_lt_re
lemma LSeriesSummable_logMul_of_lt_re {f : ℕ → ℂ} {s : ℂ} (h : abscissaOfAbsConv f < s.re) :
    LSeriesSummable (logMul f) s 
Arguments: {f : ℕ → ℂ} {s : ℂ} (h : LSeries.abscissaOfAbsConv f < ↑s.re)
Goal: LSeriesSummable (LSeries.logMul f) s
-- LSeries.abscissaOfAbsConv_logMul
@[simp]
lemma LSeries.abscissaOfAbsConv_logMul {f : ℕ → ℂ} :
    abscissaOfAbsConv (logMul f) = abscissaOfAbsConv f 
Arguments: {f : ℕ → ℂ}
Goal: LSeries.abscissaOfAbsConv (LSeries.logMul f) = LSeries.abscissaOfAbsConv f
-- LSeries.absicssaOfAbsConv_logPowMul
@[simp]
lemma LSeries.absicssaOfAbsConv_logPowMul {f : ℕ → ℂ} {m : ℕ} :
    abscissaOfAbsConv (logMul^[m] f) = abscissaOfAbsConv f 
Arguments: {f : ℕ → ℂ} {m : ℕ}
Goal: LSeries.abscissaOfAbsConv (LSeries.logMul^[m] f) = LSeries.abscissaOfAbsConv f
-- LSeries_iteratedDeriv
lemma LSeries_iteratedDeriv {f : ℕ → ℂ} (m : ℕ) {s : ℂ} (h : abscissaOfAbsConv f < s.re) :
    iteratedDeriv m (LSeries f) s = (-1) ^ m * LSeries (logMul^[m] f) s 
Arguments: {f : ℕ → ℂ} (m : ℕ) {s : ℂ} (h : LSeries.abscissaOfAbsConv f < ↑s.re)
Goal: iteratedDeriv m (LSeries f) s = (-1) ^ m * LSeries (LSeries.logMul^[m] f) s
-- LSeries_differentiableOn
lemma LSeries_differentiableOn (f : ℕ → ℂ) :
    DifferentiableOn ℂ (LSeries f) {s | abscissaOfAbsConv f < s.re} 
Arguments: (f : ℕ → ℂ)
Goal: DifferentiableOn ℂ (LSeries f) {s | LSeries.abscissaOfAbsConv f < ↑s.re}
-- LSeries_analyticOn
lemma LSeries_analyticOn (f : ℕ → ℂ) :
    AnalyticOn ℂ (LSeries f) {s | abscissaOfAbsConv f < s.re} 
Arguments: (f : ℕ → ℂ)
Goal: AnalyticOn ℂ (LSeries f) {s | LSeries.abscissaOfAbsConv f < ↑s.re}
-- Nat.Prime.sq_add_sq
theorem Nat.Prime.sq_add_sq {p : ℕ} [Fact p.Prime] (hp : p % 4 ≠ 3) :
    ∃ a b : ℕ, a ^ 2 + b ^ 2 = p 
Arguments: {p : ℕ} [Fact (Nat.Prime p)] (hp : p % 4 ≠ 3)
Goal: ∃ a b, a ^ 2 + b ^ 2 = p
-- sq_add_sq_mul
theorem sq_add_sq_mul {R} [CommRing R] {a b x y u v : R} (ha : a = x ^ 2 + y ^ 2)
    (hb : b = u ^ 2 + v ^ 2) : ∃ r s : R, a * b = r ^ 2 + s ^ 2 
Arguments: {R : Type u_1} [CommRing R] {a b x y u v : R} (ha : a = x ^ 2 + y ^ 2) (hb : b = u ^ 2 + v ^ 2)
Goal: ∃ r s, a * b = r ^ 2 + s ^ 2
-- Nat.sq_add_sq_mul
theorem Nat.sq_add_sq_mul {a b x y u v : ℕ} (ha : a = x ^ 2 + y ^ 2) (hb : b = u ^ 2 + v ^ 2) :
    ∃ r s : ℕ, a * b = r ^ 2 + s ^ 2 
Arguments: {a b x y u v : ℕ} (ha : a = x ^ 2 + y ^ 2) (hb : b = u ^ 2 + v ^ 2)
Goal: ∃ r s, a * b = r ^ 2 + s ^ 2
-- ZMod.isSquare_neg_one_of_dvd
theorem ZMod.isSquare_neg_one_of_dvd {m n : ℕ} (hd : m ∣ n) (hs : IsSquare (-1 : ZMod n)) :
    IsSquare (-1 : ZMod m) 
Arguments: {m n : ℕ} (hd : m ∣ n) (hs : IsSquare (-1))
Goal: IsSquare (-1)
-- ZMod.isSquare_neg_one_mul
theorem ZMod.isSquare_neg_one_mul {m n : ℕ} (hc : m.Coprime n) (hm : IsSquare (-1 : ZMod m))
    (hn : IsSquare (-1 : ZMod n)) : IsSquare (-1 : ZMod (m * n)) 
Arguments: {m n : ℕ} (hc : m.Coprime n) (hm : IsSquare (-1)) (hn : IsSquare (-1))
Goal: IsSquare (-1)
-- Nat.Prime.mod_four_ne_three_of_dvd_isSquare_neg_one
theorem Nat.Prime.mod_four_ne_three_of_dvd_isSquare_neg_one {p n : ℕ} (hpp : p.Prime) (hp : p ∣ n)
    (hs : IsSquare (-1 : ZMod n)) : p % 4 ≠ 3 
Arguments: {p n : ℕ} (hpp : Nat.Prime p) (hp : p ∣ n) (hs : IsSquare (-1))
Goal: p % 4 ≠ 3
-- ZMod.isSquare_neg_one_iff
theorem ZMod.isSquare_neg_one_iff {n : ℕ} (hn : Squarefree n) :
    IsSquare (-1 : ZMod n) ↔ ∀ {q : ℕ}, q.Prime → q ∣ n → q % 4 ≠ 3 
Arguments: {n : ℕ} (hn : Squarefree n)
Goal: IsSquare (-1) ↔ ∀ {q : ℕ}, Nat.Prime q → q ∣ n → q % 4 ≠ 3
-- ZMod.isSquare_neg_one_iff'
theorem ZMod.isSquare_neg_one_iff' {n : ℕ} (hn : Squarefree n) :
    IsSquare (-1 : ZMod n) ↔ ∀ {q : ℕ}, q ∣ n → q % 4 ≠ 3 
Arguments: {n : ℕ} (hn : Squarefree n)
Goal: IsSquare (-1) ↔ ∀ {q : ℕ}, q ∣ n → q % 4 ≠ 3
-- Nat.eq_sq_add_sq_of_isSquare_mod_neg_one
theorem Nat.eq_sq_add_sq_of_isSquare_mod_neg_one {n : ℕ} (h : IsSquare (-1 : ZMod n)) :
    ∃ x y : ℕ, n = x ^ 2 + y ^ 2 
Arguments: {n : ℕ} (h : IsSquare (-1))
Goal: ∃ x y, n = x ^ 2 + y ^ 2
-- ZMod.isSquare_neg_one_of_eq_sq_add_sq_of_isCoprime
theorem ZMod.isSquare_neg_one_of_eq_sq_add_sq_of_isCoprime {n x y : ℤ} (h : n = x ^ 2 + y ^ 2)
    (hc : IsCoprime x y) : IsSquare (-1 : ZMod n.natAbs) 
Arguments: {n x y : ℤ} (h : n = x ^ 2 + y ^ 2) (hc : IsCoprime x y)
Goal: IsSquare (-1)
-- ZMod.isSquare_neg_one_of_eq_sq_add_sq_of_coprime
theorem ZMod.isSquare_neg_one_of_eq_sq_add_sq_of_coprime {n x y : ℕ} (h : n = x ^ 2 + y ^ 2)
    (hc : x.Coprime y) : IsSquare (-1 : ZMod n) 
Arguments: {n x y : ℕ} (h : n = x ^ 2 + y ^ 2) (hc : x.Coprime y)
Goal: IsSquare (-1)
-- Nat.eq_sq_add_sq_iff_eq_sq_mul
theorem Nat.eq_sq_add_sq_iff_eq_sq_mul {n : ℕ} :
    (∃ x y : ℕ, n = x ^ 2 + y ^ 2) ↔ ∃ a b : ℕ, n = a ^ 2 * b ∧ IsSquare (-1 : ZMod b) 
Arguments: {n : ℕ}
Goal: (∃ x y, n = x ^ 2 + y ^ 2) ↔ ∃ a b, n = a ^ 2 * b ∧ IsSquare (-1)
-- Nat.eq_sq_add_sq_iff
theorem Nat.eq_sq_add_sq_iff {n : ℕ} :
    (∃ x y : ℕ, n = x ^ 2 + y ^ 2) ↔ ∀ {q : ℕ}, q.Prime → q % 4 = 3 → Even (padicValNat q n) 
Arguments: {n : ℕ}
Goal: (∃ x y, n = x ^ 2 + y ^ 2) ↔ ∀ {q : ℕ}, Nat.Prime q → q % 4 = 3 → Even (padicValNat q n)
-- toArithmeticFunction_congr
lemma toArithmeticFunction_congr {R : Type*} [Zero R] {f f' : ℕ → R}
    (h : ∀ {n}, n ≠ 0 → f n = f' n) :
    toArithmeticFunction f = toArithmeticFunction f' 
Arguments: {R : Type u_1} [Zero R] {f f' : ℕ → R} (h : ∀ {n : ℕ}, n ≠ 0 → f n = f' n)
Goal: toArithmeticFunction f = toArithmeticFunction f'
-- ArithmeticFunction.toArithmeticFunction_eq_self
@[simp]
lemma ArithmeticFunction.toArithmeticFunction_eq_self {R : Type*} [Zero R]
    (f : ArithmeticFunction R) :
    toArithmeticFunction f = f 
Arguments: {R : Type u_1} [Zero R] (f : ArithmeticFunction R)
Goal: toArithmeticFunction ⇑f = f
-- LSeries.convolution_congr
lemma LSeries.convolution_congr {R : Type*} [Semiring R] {f f' g g' : ℕ → R}
    (hf : ∀ {n}, n ≠ 0 → f n = f' n) (hg : ∀ {n}, n ≠ 0 → g n = g' n) :
    f ⍟ g = f' ⍟ g' 
Arguments: {R : Type u_1} [Semiring R] {f f' g g' : ℕ → R} (hf : ∀ {n : ℕ}, n ≠ 0 → f n = f' n) (hg : ∀ {n : ℕ}, n ≠ 0 → g n = g' n)
Goal: f ⍟ g = f' ⍟ g'
-- ArithmeticFunction.coe_mul
lemma ArithmeticFunction.coe_mul {R : Type*} [Semiring R] (f g : ArithmeticFunction R) :
    f ⍟ g = ⇑(f * g) 
Arguments: {R : Type u_1} [Semiring R] (f g : ArithmeticFunction R)
Goal: ⇑f ⍟ ⇑g = ⇑(f * g)
-- LSeries.convolution_def
lemma convolution_def {R : Type*} [Semiring R] (f g : ℕ → R) :
    f ⍟ g = fun n ↦ ∑ p ∈ n.divisorsAntidiagonal, f p.1 * g p.2 
Arguments: {R : Type u_1} [Semiring R] (f g : ℕ → R)
Goal: f ⍟ g = fun n => ∑ p ∈ n.divisorsAntidiagonal, f p.1 * g p.2
-- LSeries.convolution_map_zero
@[simp]
lemma convolution_map_zero {R : Type*} [Semiring R] (f g : ℕ → R) : (f ⍟ g) 0 = 0 
Arguments: {R : Type u_1} [Semiring R] (f g : ℕ → R)
Goal: (f ⍟ g) 0 = 0
-- LSeries.term_convolution
lemma term_convolution (f g : ℕ → ℂ) (s : ℂ) (n : ℕ) :
    term (f ⍟ g) s n = ∑ p ∈ n.divisorsAntidiagonal, term f s p.1 * term g s p.2 
Arguments: (f g : ℕ → ℂ) (s : ℂ) (n : ℕ)
Goal: LSeries.term (f ⍟ g) s n = ∑ p ∈ n.divisorsAntidiagonal, LSeries.term f s p.1 * LSeries.term g s p.2
-- LSeries.term_convolution'
lemma term_convolution' (f g : ℕ → ℂ) (s : ℂ) :
    term (f ⍟ g) s = fun n ↦
      ∑' (b : (fun p : ℕ × ℕ ↦ p.1 * p.2) ⁻¹' {n}), term f s b.val.1 * term g s b.val.2 
Arguments: (f g : ℕ → ℂ) (s : ℂ)
Goal: LSeries.term (f ⍟ g) s = fun n =>
  ∑' (b : ↑((fun p => p.1 * p.2) ⁻¹' {n})), LSeries.term f s (↑b).1 * LSeries.term g s (↑b).2
-- LSeriesHasSum.convolution
lemma LSeriesHasSum.convolution {f g : ℕ → ℂ} {s a b : ℂ} (hf : LSeriesHasSum f s a)
    (hg : LSeriesHasSum g s b) :
    LSeriesHasSum (f ⍟ g) s (a * b) 
Arguments: {f g : ℕ → ℂ} {s a b : ℂ} (hf : LSeriesHasSum f s a) (hg : LSeriesHasSum g s b)
Goal: LSeriesHasSum (f ⍟ g) s (a * b)
-- LSeries_convolution'
lemma LSeries_convolution' {f g : ℕ → ℂ} {s : ℂ} (hf : LSeriesSummable f s)
    (hg : LSeriesSummable g s) :
    LSeries (f ⍟ g) s = LSeries f s * LSeries g s 
Arguments: {f g : ℕ → ℂ} {s : ℂ} (hf : LSeriesSummable f s) (hg : LSeriesSummable g s)
Goal: L (f ⍟ g) s = L f s * L g s
-- LSeries_convolution
lemma LSeries_convolution {f g : ℕ → ℂ} {s : ℂ}
    (hf : abscissaOfAbsConv f < s.re) (hg : abscissaOfAbsConv g < s.re) :
    LSeries (f ⍟ g) s = LSeries f s * LSeries g s 
Arguments: {f g : ℕ → ℂ} {s : ℂ} (hf : LSeries.abscissaOfAbsConv f < ↑s.re) (hg : LSeries.abscissaOfAbsConv g < ↑s.re)
Goal: L (f ⍟ g) s = L f s * L g s
-- LSeriesSummable.convolution
lemma LSeriesSummable.convolution {f g : ℕ → ℂ} {s : ℂ} (hf : LSeriesSummable f s)
    (hg : LSeriesSummable g s) :
    LSeriesSummable (f ⍟ g) s 
Arguments: {f g : ℕ → ℂ} {s : ℂ} (hf : LSeriesSummable f s) (hg : LSeriesSummable g s)
Goal: LSeriesSummable (f ⍟ g) s
-- ArithmeticFunction.LSeriesHasSum_mul
lemma LSeriesHasSum_mul {f g : ArithmeticFunction ℂ} {s a b : ℂ} (hf : LSeriesHasSum ↗f s a)
    (hg : LSeriesHasSum ↗g s b) :
    LSeriesHasSum ↗(f * g) s (a * b) 
Arguments: {f g : ArithmeticFunction ℂ} {s a b : ℂ} (hf : LSeriesHasSum (fun n => f n) s a) (hg : LSeriesHasSum (fun n => g n) s b)
Goal: LSeriesHasSum (fun n => (f * g) n) s (a * b)
-- ArithmeticFunction.LSeries_mul'
lemma LSeries_mul' {f g : ArithmeticFunction ℂ} {s : ℂ} (hf : LSeriesSummable ↗f s)
    (hg : LSeriesSummable ↗g s) :
    LSeries ↗(f * g) s = LSeries ↗f s * LSeries ↗g s 
Arguments: {f g : ArithmeticFunction ℂ} {s : ℂ} (hf : LSeriesSummable (fun n => f n) s) (hg : LSeriesSummable (fun n => g n) s)
Goal: L (fun n => (f * g) n) s = L (fun n => f n) s * L (fun n => g n) s
-- ArithmeticFunction.LSeries_mul
lemma LSeries_mul {f g : ArithmeticFunction ℂ} {s : ℂ}
    (hf : abscissaOfAbsConv ↗f < s.re) (hg : abscissaOfAbsConv ↗g < s.re) :
    LSeries ↗(f * g) s = LSeries ↗f s * LSeries ↗g s 
Arguments: {f g : ArithmeticFunction ℂ} {s : ℂ} (hf : (LSeries.abscissaOfAbsConv fun n => f n) < ↑s.re) (hg : (LSeries.abscissaOfAbsConv fun n => g n) < ↑s.re)
Goal: L (fun n => (f * g) n) s = L (fun n => f n) s * L (fun n => g n) s