ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Ideal.finprod_not_dvd
theorem finprod_not_dvd (I : Ideal R) (hI : I ≠ 0) :
    ¬v.asIdeal ^ ((Associates.mk v.asIdeal).count (Associates.mk I).factors + 1) ∣
        ∏ᶠ v : HeightOneSpectrum R, v.maxPowDividing I 
Arguments: {R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : IsDedekindDomain.HeightOneSpectrum R) (I : Ideal R) (hI : I ≠ 0)
Goal: ¬v.asIdeal ^ ((Associates.mk v.asIdeal).count (Associates.mk I).factors + 1) ∣
    ∏ᶠ (v : IsDedekindDomain.HeightOneSpectrum R), v.maxPowDividing I
-- Associates.finprod_ne_zero
theorem Associates.finprod_ne_zero (I : Ideal R) :
    Associates.mk (∏ᶠ v : HeightOneSpectrum R, v.maxPowDividing I) ≠ 0 
Arguments: {R : Type u_1} [CommRing R] [IsDedekindDomain R] (I : Ideal R)
Goal: Associates.mk (∏ᶠ (v : IsDedekindDomain.HeightOneSpectrum R), v.maxPowDividing I) ≠ 0
-- Ideal.finprod_count
theorem finprod_count (I : Ideal R) (hI : I ≠ 0) : (Associates.mk v.asIdeal).count
    (Associates.mk (∏ᶠ v : HeightOneSpectrum R, v.maxPowDividing I)).factors =
    (Associates.mk v.asIdeal).count (Associates.mk I).factors 
Arguments: {R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : IsDedekindDomain.HeightOneSpectrum R) (I : Ideal R) (hI : I ≠ 0)
Goal: (Associates.mk v.asIdeal).count
    (Associates.mk (∏ᶠ (v : IsDedekindDomain.HeightOneSpectrum R), v.maxPowDividing I)).factors =
  (Associates.mk v.asIdeal).count (Associates.mk I).factors
-- Ideal.finprod_heightOneSpectrum_factorization
theorem finprod_heightOneSpectrum_factorization {I : Ideal R} (hI : I ≠ 0) :
    ∏ᶠ v : HeightOneSpectrum R, v.maxPowDividing I = I 
Arguments: {R : Type u_1} [CommRing R] [IsDedekindDomain R] {I : Ideal R} (hI : I ≠ 0)
Goal: ∏ᶠ (v : IsDedekindDomain.HeightOneSpectrum R), v.maxPowDividing I = I
-- Ideal.finprod_heightOneSpectrum_factorization_coe
theorem finprod_heightOneSpectrum_factorization_coe {I : Ideal R} (hI : I ≠ 0) :
    (∏ᶠ v : HeightOneSpectrum R, (v.asIdeal : FractionalIdeal R⁰ K) ^
      ((Associates.mk v.asIdeal).count (Associates.mk I).factors : ℤ)) = I 
Arguments: {R : Type u_1} [CommRing R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] [IsDedekindDomain R] {I : Ideal R} (hI : I ≠ 0)
Goal: ∏ᶠ (v : IsDedekindDomain.HeightOneSpectrum R),
    ↑v.asIdeal ^ ↑((Associates.mk v.asIdeal).count (Associates.mk I).factors) =
  ↑I
-- FractionalIdeal.finprod_heightOneSpectrum_factorization
theorem finprod_heightOneSpectrum_factorization {I : FractionalIdeal R⁰ K} (hI : I ≠ 0) {a : R}
    {J : Ideal R} (haJ : I = spanSingleton R⁰ ((algebraMap R K) a)⁻¹ * ↑J) :
    ∏ᶠ v : HeightOneSpectrum R, (v.asIdeal : FractionalIdeal R⁰ K) ^
      ((Associates.mk v.asIdeal).count (Associates.mk J).factors -
        (Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors : ℤ) = I 
Arguments: {R : Type u_1} [CommRing R] {K : Type u_2} [Field K] [Algebra R K] [IsFractionRing R K] [IsDedekindDomain R] {I : FractionalIdeal R⁰ K} (hI : I ≠ 0) {a : R} {J : Ideal R} (haJ : I = FractionalIdeal.spanSingleton R⁰ ((algebraMap R K) a)⁻¹ * ↑J)
Goal: ∏ᶠ (v : IsDedekindDomain.HeightOneSpectrum R),
    ↑v.asIdeal ^
      (↑((Associates.mk v.asIdeal).count (Associates.mk J).factors) -
        ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors)) =
  I
-- FractionalIdeal.finprod_heightOneSpectrum_factorization_principal_fraction
theorem finprod_heightOneSpectrum_factorization_principal_fraction {n : R} (hn : n ≠ 0) (d : ↥R⁰) :
    ∏ᶠ v : HeightOneSpectrum R, (v.asIdeal : FractionalIdeal R⁰ K) ^
      ((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {n} : Ideal R)).factors -
        (Associates.mk v.asIdeal).count (Associates.mk ((Ideal.span {(↑d : R)}) :
        Ideal R)).factors : ℤ) = spanSingleton R⁰ (mk' K n d) 
Arguments: {R : Type u_1} [CommRing R] {K : Type u_2} [Field K] [Algebra R K] [IsFractionRing R K] [IsDedekindDomain R] {n : R} (hn : n ≠ 0) (d : ↥R⁰)
Goal: ∏ᶠ (v : IsDedekindDomain.HeightOneSpectrum R),
    ↑v.asIdeal ^
      (↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {n})).factors) -
        ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {↑d})).factors)) =
  FractionalIdeal.spanSingleton R⁰ (IsLocalization.mk' K n d)
-- FractionalIdeal.finprod_heightOneSpectrum_factorization_principal
theorem finprod_heightOneSpectrum_factorization_principal {I : FractionalIdeal R⁰ K} (hI : I ≠ 0)
    (k : K) (hk : I = spanSingleton R⁰ k) :
    ∏ᶠ v : HeightOneSpectrum R, (v.asIdeal : FractionalIdeal R⁰ K) ^
      ((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {choose
          (mk'_surjective R⁰ k)} : Ideal R)).factors -
        (Associates.mk v.asIdeal).count (Associates.mk ((Ideal.span {(↑(choose
          (choose_spec (mk'_surjective R⁰ k)) : ↥R⁰) : R)}) : Ideal R)).factors : ℤ) = I 
Arguments: {R : Type u_1} [CommRing R] {K : Type u_2} [Field K] [Algebra R K] [IsFractionRing R K] [IsDedekindDomain R] {I : FractionalIdeal R⁰ K} (hI : I ≠ 0) (k : K) (hk : I = FractionalIdeal.spanSingleton R⁰ k)
Goal: ∏ᶠ (v : IsDedekindDomain.HeightOneSpectrum R),
    ↑v.asIdeal ^
      (↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {Classical.choose ⋯})).factors) -
        ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {↑(Classical.choose ⋯)})).factors)) =
  I
-- FractionalIdeal.count_zero
lemma count_zero : count K v (0 : FractionalIdeal R⁰ K) = 0 
Arguments: {R : Type u_1} [CommRing R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] [IsDedekindDomain R] (v : IsDedekindDomain.HeightOneSpectrum R)
Goal: FractionalIdeal.count K v 0 = 0
-- FractionalIdeal.count_ne_zero
lemma count_ne_zero {I : FractionalIdeal R⁰ K} (hI : I ≠ 0) :
    count K v I = ((Associates.mk v.asIdeal).count (Associates.mk
      (choose (choose_spec (exists_eq_spanSingleton_mul I)))).factors -
      (Associates.mk v.asIdeal).count
        (Associates.mk (Ideal.span {choose (exists_eq_spanSingleton_mul I)})).factors : ℤ) 
Arguments: {R : Type u_1} [CommRing R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] [IsDedekindDomain R] (v : IsDedekindDomain.HeightOneSpectrum R) {I : FractionalIdeal R⁰ K} (hI : I ≠ 0)
Goal: FractionalIdeal.count K v I =
  ↑((Associates.mk v.asIdeal).count (Associates.mk (Classical.choose ⋯)).factors) -
    ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {Classical.choose ⋯})).factors)
-- FractionalIdeal.count_well_defined
theorem count_well_defined {I : FractionalIdeal R⁰ K} (hI : I ≠ 0) {a : R}
    {J : Ideal R} (h_aJ : I = spanSingleton R⁰ ((algebraMap R K) a)⁻¹ * ↑J) :
    count K v I = ((Associates.mk v.asIdeal).count (Associates.mk J).factors -
      (Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors : ℤ) 
Arguments: {R : Type u_1} [CommRing R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] [IsDedekindDomain R] (v : IsDedekindDomain.HeightOneSpectrum R) {I : FractionalIdeal R⁰ K} (hI : I ≠ 0) {a : R} {J : Ideal R} (h_aJ : I = FractionalIdeal.spanSingleton R⁰ ((algebraMap R K) a)⁻¹ * ↑J)
Goal: FractionalIdeal.count K v I =
  ↑((Associates.mk v.asIdeal).count (Associates.mk J).factors) -
    ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors)
-- FractionalIdeal.count_mul
theorem count_mul {I I' : FractionalIdeal R⁰ K} (hI : I ≠ 0) (hI' : I' ≠ 0) :
    count K v (I * I') = count K v I + count K v I' 
Arguments: {R : Type u_1} [CommRing R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] [IsDedekindDomain R] (v : IsDedekindDomain.HeightOneSpectrum R) {I I' : FractionalIdeal R⁰ K} (hI : I ≠ 0) (hI' : I' ≠ 0)
Goal: FractionalIdeal.count K v (I * I') = FractionalIdeal.count K v I + FractionalIdeal.count K v I'
-- FractionalIdeal.count_mul'
theorem count_mul' (I I' : FractionalIdeal R⁰ K) :
    count K v (I * I') = if I ≠ 0 ∧ I' ≠ 0 then count K v I + count K v I' else 0 
Arguments: {R : Type u_1} [CommRing R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] [IsDedekindDomain R] (v : IsDedekindDomain.HeightOneSpectrum R) (I I' : FractionalIdeal R⁰ K)
Goal: FractionalIdeal.count K v (I * I') =
  if I ≠ 0 ∧ I' ≠ 0 then FractionalIdeal.count K v I + FractionalIdeal.count K v I' else 0
-- FractionalIdeal.count_one
theorem count_one : count K v (1 : FractionalIdeal R⁰ K) = 0 
Arguments: {R : Type u_1} [CommRing R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] [IsDedekindDomain R] (v : IsDedekindDomain.HeightOneSpectrum R)
Goal: FractionalIdeal.count K v 1 = 0
-- FractionalIdeal.count_prod
theorem count_prod {ι} (s : Finset ι) (I : ι → FractionalIdeal R⁰ K) (hS : ∀ i ∈ s, I i ≠ 0) :
    count K v (∏ i ∈ s, I i) = ∑ i ∈ s, count K v (I i) 
Arguments: {R : Type u_1} [CommRing R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] [IsDedekindDomain R] (v : IsDedekindDomain.HeightOneSpectrum R) {ι : Type u_3} (s : Finset ι) (I : ι → FractionalIdeal R⁰ K) (hS : ∀ i ∈ s, I i ≠ 0)
Goal: FractionalIdeal.count K v (∏ i ∈ s, I i) = ∑ i ∈ s, FractionalIdeal.count K v (I i)
-- FractionalIdeal.count_pow
theorem count_pow (n : ℕ) (I : FractionalIdeal R⁰ K) :
    count K v (I ^ n) = n * count K v I 
Arguments: {R : Type u_1} [CommRing R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] [IsDedekindDomain R] (v : IsDedekindDomain.HeightOneSpectrum R) (n : ℕ) (I : FractionalIdeal R⁰ K)
Goal: FractionalIdeal.count K v (I ^ n) = ↑n * FractionalIdeal.count K v I
-- FractionalIdeal.count_self
theorem count_self : count K v (v.asIdeal : FractionalIdeal R⁰ K) = 1 
Arguments: {R : Type u_1} [CommRing R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] [IsDedekindDomain R] (v : IsDedekindDomain.HeightOneSpectrum R)
Goal: FractionalIdeal.count K v ↑v.asIdeal = 1
-- FractionalIdeal.count_pow_self
theorem count_pow_self (n : ℕ) :
    count K v ((v.asIdeal : FractionalIdeal R⁰ K) ^ n) = n 
Arguments: {R : Type u_1} [CommRing R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] [IsDedekindDomain R] (v : IsDedekindDomain.HeightOneSpectrum R) (n : ℕ)
Goal: FractionalIdeal.count K v (↑v.asIdeal ^ n) = ↑n
-- FractionalIdeal.count_neg_zpow
theorem count_neg_zpow (n : ℤ) (I : FractionalIdeal R⁰ K) :
    count K v (I ^ (-n)) = - count K v (I ^ n) 
Arguments: {R : Type u_1} [CommRing R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] [IsDedekindDomain R] (v : IsDedekindDomain.HeightOneSpectrum R) (n : ℤ) (I : FractionalIdeal R⁰ K)
Goal: FractionalIdeal.count K v (I ^ (-n)) = -FractionalIdeal.count K v (I ^ n)
-- FractionalIdeal.count_inv
theorem count_inv  (I : FractionalIdeal R⁰ K) :
    count K v (I⁻¹) = - count K v I 
Arguments: {R : Type u_1} [CommRing R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] [IsDedekindDomain R] (v : IsDedekindDomain.HeightOneSpectrum R) (I : FractionalIdeal R⁰ K)
Goal: FractionalIdeal.count K v I⁻¹ = -FractionalIdeal.count K v I
-- FractionalIdeal.count_zpow
theorem count_zpow (n : ℤ) (I : FractionalIdeal R⁰ K) :
    count K v (I ^ n) = n * count K v I 
Arguments: {R : Type u_1} [CommRing R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] [IsDedekindDomain R] (v : IsDedekindDomain.HeightOneSpectrum R) (n : ℤ) (I : FractionalIdeal R⁰ K)
Goal: FractionalIdeal.count K v (I ^ n) = n * FractionalIdeal.count K v I
-- FractionalIdeal.count_zpow_self
theorem count_zpow_self (n : ℤ) :
    count K v ((v.asIdeal : FractionalIdeal R⁰ K) ^ n) = n 
Arguments: {R : Type u_1} [CommRing R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] [IsDedekindDomain R] (v : IsDedekindDomain.HeightOneSpectrum R) (n : ℤ)
Goal: FractionalIdeal.count K v (↑v.asIdeal ^ n) = n
-- FractionalIdeal.count_maximal_coprime
theorem count_maximal_coprime {w : HeightOneSpectrum R} (hw : w ≠ v) :
    count K v (w.asIdeal : FractionalIdeal R⁰ K) = 0 
Arguments: {R : Type u_1} [CommRing R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] [IsDedekindDomain R] (v : IsDedekindDomain.HeightOneSpectrum R) {w : IsDedekindDomain.HeightOneSpectrum R} (hw : w ≠ v)
Goal: FractionalIdeal.count K v ↑w.asIdeal = 0
-- FractionalIdeal.count_maximal
theorem count_maximal (w : HeightOneSpectrum R) :
    count K v (w.asIdeal : FractionalIdeal R⁰ K) = if w = v then 1 else 0 
Arguments: {R : Type u_1} [CommRing R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] [IsDedekindDomain R] (v w : IsDedekindDomain.HeightOneSpectrum R)
Goal: FractionalIdeal.count K v ↑w.asIdeal = if w = v then 1 else 0
-- FractionalIdeal.count_finprod_coprime
theorem count_finprod_coprime (exps : HeightOneSpectrum R → ℤ) :
    count K v (∏ᶠ (w : HeightOneSpectrum R) (_ : w ≠ v),
      (w.asIdeal : (FractionalIdeal R⁰ K)) ^ exps w) = 0 
Arguments: {R : Type u_1} [CommRing R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] [IsDedekindDomain R] (v : IsDedekindDomain.HeightOneSpectrum R) (exps : IsDedekindDomain.HeightOneSpectrum R → ℤ)
Goal: FractionalIdeal.count K v (∏ᶠ (w : IsDedekindDomain.HeightOneSpectrum R) (_ : w ≠ v), ↑w.asIdeal ^ exps w) = 0
-- FractionalIdeal.count_finsupp_prod
theorem count_finsupp_prod (exps : HeightOneSpectrum R →₀ ℤ) :
    count K v (exps.prod (HeightOneSpectrum.asIdeal · ^ ·)) = exps v 
Arguments: {R : Type u_1} [CommRing R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] [IsDedekindDomain R] (v : IsDedekindDomain.HeightOneSpectrum R) (exps : IsDedekindDomain.HeightOneSpectrum R →₀ ℤ)
Goal: FractionalIdeal.count K v (exps.prod fun x x_1 => ↑x.asIdeal ^ x_1) = exps v
-- FractionalIdeal.count_finprod
theorem count_finprod (exps : HeightOneSpectrum R → ℤ)
    (h_exps : ∀ᶠ v : HeightOneSpectrum R in Filter.cofinite, exps v = 0) :
    count K v (∏ᶠ v : HeightOneSpectrum R,
      (v.asIdeal : FractionalIdeal R⁰ K) ^ exps v) = exps v 
Arguments: {R : Type u_1} [CommRing R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] [IsDedekindDomain R] (v : IsDedekindDomain.HeightOneSpectrum R) (exps : IsDedekindDomain.HeightOneSpectrum R → ℤ) (h_exps : ∀ᶠ (v : IsDedekindDomain.HeightOneSpectrum R) in Filter.cofinite, exps v = 0)
Goal: FractionalIdeal.count K v (∏ᶠ (v : IsDedekindDomain.HeightOneSpectrum R), ↑v.asIdeal ^ exps v) = exps v
-- FractionalIdeal.count_coe
theorem count_coe {J : Ideal R} (hJ : J ≠ 0) :
    count K v J = (Associates.mk v.asIdeal).count (Associates.mk J).factors 
Arguments: {R : Type u_1} [CommRing R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] [IsDedekindDomain R] (v : IsDedekindDomain.HeightOneSpectrum R) {J : Ideal R} (hJ : J ≠ 0)
Goal: FractionalIdeal.count K v ↑J = ↑((Associates.mk v.asIdeal).count (Associates.mk J).factors)
-- FractionalIdeal.count_coe_nonneg
theorem count_coe_nonneg (J : Ideal R) : 0 ≤ count K v J 
Arguments: {R : Type u_1} [CommRing R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] [IsDedekindDomain R] (v : IsDedekindDomain.HeightOneSpectrum R) (J : Ideal R)
Goal: 0 ≤ FractionalIdeal.count K v ↑J
-- FractionalIdeal.count_mono
theorem count_mono {I J} (hI : I ≠ 0) (h : I ≤ J) : count K v J ≤ count K v I 
Arguments: {R : Type u_1} [CommRing R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] [IsDedekindDomain R] (v : IsDedekindDomain.HeightOneSpectrum R) {I J : FractionalIdeal R⁰ K} (hI : I ≠ 0) (h : I ≤ J)
Goal: FractionalIdeal.count K v J ≤ FractionalIdeal.count K v I
-- FractionalIdeal.finprod_heightOneSpectrum_factorization'
theorem finprod_heightOneSpectrum_factorization' {I : FractionalIdeal R⁰ K} (hI : I ≠ 0) :
    ∏ᶠ v : HeightOneSpectrum R, (v.asIdeal : FractionalIdeal R⁰ K) ^ (count K v I) = I 
Arguments: {R : Type u_1} [CommRing R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] [IsDedekindDomain R] {I : FractionalIdeal R⁰ K} (hI : I ≠ 0)
Goal: ∏ᶠ (v : IsDedekindDomain.HeightOneSpectrum R), ↑v.asIdeal ^ FractionalIdeal.count K v I = I
-- FractionalIdeal.finite_factors'
theorem finite_factors' {I : FractionalIdeal R⁰ K} (hI : I ≠ 0) {a : R}
    {J : Ideal R} (haJ : I = spanSingleton R⁰ ((algebraMap R K) a)⁻¹ * ↑J) :
    ∀ᶠ v : HeightOneSpectrum R in Filter.cofinite,
      ((Associates.mk v.asIdeal).count (Associates.mk J).factors : ℤ) -
        (Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors = 0 
Arguments: {R : Type u_1} [CommRing R] {K : Type u_2} [Field K] [Algebra R K] [IsFractionRing R K] [IsDedekindDomain R] {I : FractionalIdeal R⁰ K} (hI : I ≠ 0) {a : R} {J : Ideal R} (haJ : I = FractionalIdeal.spanSingleton R⁰ ((algebraMap R K) a)⁻¹ * ↑J)
Goal: ∀ᶠ (v : IsDedekindDomain.HeightOneSpectrum R) in Filter.cofinite,
  ↑((Associates.mk v.asIdeal).count (Associates.mk J).factors) -
      ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {a})).factors) =
    0
-- FractionalIdeal.finite_factors
theorem finite_factors (I : FractionalIdeal R⁰ K) :
    ∀ᶠ v : HeightOneSpectrum R in Filter.cofinite, count K v I = 0 
Arguments: {R : Type u_1} [CommRing R] {K : Type u_2} [Field K] [Algebra R K] [IsFractionRing R K] [IsDedekindDomain R] (I : FractionalIdeal R⁰ K)
Goal: ∀ᶠ (v : IsDedekindDomain.HeightOneSpectrum R) in Filter.cofinite, FractionalIdeal.count K v I = 0
-- IsNilpotent.mk
theorem IsNilpotent.mk [Zero R] [Pow R ℕ] (x : R) (n : ℕ) (e : x ^ n = 0) : IsNilpotent x 
Arguments: {R : Type u_1} [Zero R] [Pow R ℕ] (x : R) (n : ℕ) (e : x ^ n = 0)
Goal: IsNilpotent x
-- isNilpotent_of_subsingleton
@[simp] lemma isNilpotent_of_subsingleton [Zero R] [Pow R ℕ] [Subsingleton R] : IsNilpotent x 
Arguments: {R : Type u_1} {x : R} [Zero R] [Pow R ℕ] [Subsingleton R]
Goal: IsNilpotent x
-- IsNilpotent.zero
@[simp] theorem IsNilpotent.zero [MonoidWithZero R] : IsNilpotent (0 : R) 
Arguments: {R : Type u_1} [MonoidWithZero R]
Goal: IsNilpotent 0
-- not_isNilpotent_one
theorem not_isNilpotent_one [MonoidWithZero R] [Nontrivial R] :
    ¬ IsNilpotent (1 : R) 
Arguments: {R : Type u_1} [MonoidWithZero R] [Nontrivial R]
Goal: ¬IsNilpotent 1
-- IsNilpotent.pow_succ
lemma IsNilpotent.pow_succ (n : ℕ) {S : Type*} [MonoidWithZero S] {x : S}
    (hx : IsNilpotent x) : IsNilpotent (x ^ n.succ) 
Arguments: (n : ℕ) {S : Type u_3} [MonoidWithZero S] {x : S} (hx : IsNilpotent x)
Goal: IsNilpotent (x ^ n.succ)
-- IsNilpotent.of_pow
theorem  IsNilpotent.of_pow [MonoidWithZero R] {x : R} {m : ℕ}
    (h : IsNilpotent (x ^ m)) : IsNilpotent x 
Arguments: {R : Type u_1} [MonoidWithZero R] {x : R} {m : ℕ} (h : IsNilpotent (x ^ m))
Goal: IsNilpotent x
-- IsNilpotent.pow_of_pos
lemma IsNilpotent.pow_of_pos {n} {S : Type*} [MonoidWithZero S] {x : S}
    (hx : IsNilpotent x) (hn : n ≠ 0) : IsNilpotent (x ^ n) 
Arguments: {n : ℕ} {S : Type u_3} [MonoidWithZero S] {x : S} (hx : IsNilpotent x) (hn : n ≠ 0)
Goal: IsNilpotent (x ^ n)
-- IsNilpotent.pow_iff_pos
@[simp]
lemma IsNilpotent.pow_iff_pos {n} {S : Type*} [MonoidWithZero S] {x : S}
    (hn : n ≠ 0) : IsNilpotent (x ^ n) ↔ IsNilpotent x 
Arguments: {n : ℕ} {S : Type u_3} [MonoidWithZero S] {x : S} (hn : n ≠ 0)
Goal: IsNilpotent (x ^ n) ↔ IsNilpotent x
-- IsNilpotent.map
theorem IsNilpotent.map [MonoidWithZero R] [MonoidWithZero S] {r : R} {F : Type*}
    [FunLike F R S] [MonoidWithZeroHomClass F R S] (hr : IsNilpotent r) (f : F) :
    IsNilpotent (f r) 
Arguments: {R : Type u_1} {S : Type u_2} [MonoidWithZero R] [MonoidWithZero S] {r : R} {F : Type u_3} [FunLike F R S] [MonoidWithZeroHomClass F R S] (hr : IsNilpotent r) (f : F)
Goal: IsNilpotent (f r)
-- IsNilpotent.map_iff
lemma IsNilpotent.map_iff [MonoidWithZero R] [MonoidWithZero S] {r : R} {F : Type*}
    [FunLike F R S] [MonoidWithZeroHomClass F R S] {f : F} (hf : Function.Injective f) :
    IsNilpotent (f r) ↔ IsNilpotent r 
Arguments: {R : Type u_1} {S : Type u_2} [MonoidWithZero R] [MonoidWithZero S] {r : R} {F : Type u_3} [FunLike F R S] [MonoidWithZeroHomClass F R S] {f : F} (hf : Function.Injective ⇑f)
Goal: IsNilpotent (f r) ↔ IsNilpotent r
-- IsUnit.isNilpotent_mul_unit_of_commute_iff
theorem IsUnit.isNilpotent_mul_unit_of_commute_iff [MonoidWithZero R] {r u : R}
    (hu : IsUnit u) (h_comm : Commute r u) :
    IsNilpotent (r * u) ↔ IsNilpotent r 
Arguments: {R : Type u_1} [MonoidWithZero R] {r u : R} (hu : IsUnit u) (h_comm : Commute r u)
Goal: IsNilpotent (r * u) ↔ IsNilpotent r
-- IsUnit.isNilpotent_unit_mul_of_commute_iff
theorem IsUnit.isNilpotent_unit_mul_of_commute_iff [MonoidWithZero R] {r u : R}
    (hu : IsUnit u) (h_comm : Commute r u) :
    IsNilpotent (u * r) ↔ IsNilpotent r 
Arguments: {R : Type u_1} [MonoidWithZero R] {r u : R} (hu : IsUnit u) (h_comm : Commute r u)
Goal: IsNilpotent (u * r) ↔ IsNilpotent r
-- nilpotencyClass_eq_zero_of_subsingleton
@[simp] lemma nilpotencyClass_eq_zero_of_subsingleton [Subsingleton R] :
    nilpotencyClass x = 0 
Arguments: {R : Type u_1} {x : R} [Zero R] [Pow R ℕ] [Subsingleton R]
Goal: nilpotencyClass x = 0
-- isNilpotent_of_pos_nilpotencyClass
lemma isNilpotent_of_pos_nilpotencyClass (hx : 0 < nilpotencyClass x) :
    IsNilpotent x 
Arguments: {R : Type u_1} {x : R} [Zero R] [Pow R ℕ] (hx : 0 < nilpotencyClass x)
Goal: IsNilpotent x
-- pow_nilpotencyClass
lemma pow_nilpotencyClass (hx : IsNilpotent x) : x ^ (nilpotencyClass x) = 0 
Arguments: {R : Type u_1} {x : R} [Zero R] [Pow R ℕ] (hx : IsNilpotent x)
Goal: x ^ nilpotencyClass x = 0
-- nilpotencyClass_eq_succ_iff
lemma nilpotencyClass_eq_succ_iff {k : ℕ} :
    nilpotencyClass x = k + 1 ↔ x ^ (k + 1) = 0 ∧ x ^ k ≠ 0 
Arguments: {R : Type u_1} {x : R} [MonoidWithZero R] {k : ℕ}
Goal: nilpotencyClass x = k + 1 ↔ x ^ (k + 1) = 0 ∧ x ^ k ≠ 0
-- nilpotencyClass_zero
@[simp] lemma nilpotencyClass_zero [Nontrivial R] :
    nilpotencyClass (0 : R) = 1 
Arguments: {R : Type u_1} [MonoidWithZero R] [Nontrivial R]
Goal: nilpotencyClass 0 = 1
-- pos_nilpotencyClass_iff
@[simp] lemma pos_nilpotencyClass_iff [Nontrivial R] :
    0 < nilpotencyClass x ↔ IsNilpotent x 
Arguments: {R : Type u_1} {x : R} [MonoidWithZero R] [Nontrivial R]
Goal: 0 < nilpotencyClass x ↔ IsNilpotent x
-- pow_pred_nilpotencyClass
lemma pow_pred_nilpotencyClass [Nontrivial R] (hx : IsNilpotent x) :
    x ^ (nilpotencyClass x - 1) ≠ 0 
Arguments: {R : Type u_1} {x : R} [MonoidWithZero R] [Nontrivial R] (hx : IsNilpotent x)
Goal: x ^ (nilpotencyClass x - 1) ≠ 0
-- eq_zero_of_nilpotencyClass_eq_one
lemma eq_zero_of_nilpotencyClass_eq_one (hx : nilpotencyClass x = 1) :
    x = 0 
Arguments: {R : Type u_1} {x : R} [MonoidWithZero R] (hx : nilpotencyClass x = 1)
Goal: x = 0
-- nilpotencyClass_eq_one
@[simp] lemma nilpotencyClass_eq_one [Nontrivial R] :
    nilpotencyClass x = 1 ↔ x = 0 
Arguments: {R : Type u_1} {x : R} [MonoidWithZero R] [Nontrivial R]
Goal: nilpotencyClass x = 1 ↔ x = 0
-- IsReduced.eq_zero
@[mk_iff]
class IsReduced (R : Type*) [Zero R] [Pow R ℕ] : Prop where
    eq_zero : ∀ x : R, IsNilpotent x → x = 0
Arguments: {R : Type u_3} [Zero R] [Pow R ℕ] [self : IsReduced R] (x : R)
Goal: IsNilpotent x → x = 0
-- isReduced_iff
@[mk_iff]
class IsReduced (R : Type*) [Zero R] [Pow R ℕ] : Prop where
    eq_zero : ∀ x : R, IsNilpotent x → x = 0
Arguments: (R : Type u_3) [Zero R] [Pow R ℕ]
Goal: IsReduced R ↔ ∀ (x : R), IsNilpotent x → x = 0
-- IsNilpotent.eq_zero
theorem IsNilpotent.eq_zero [Zero R] [Pow R ℕ] [IsReduced R] (h : IsNilpotent x) : x = 0 
Arguments: {R : Type u_1} {x : R} [Zero R] [Pow R ℕ] [IsReduced R] (h : IsNilpotent x)
Goal: x = 0
-- isNilpotent_iff_eq_zero
@[simp]
theorem isNilpotent_iff_eq_zero [MonoidWithZero R] [IsReduced R] : IsNilpotent x ↔ x = 0 
Arguments: {R : Type u_1} {x : R} [MonoidWithZero R] [IsReduced R]
Goal: IsNilpotent x ↔ x = 0
-- isReduced_of_injective
theorem isReduced_of_injective [MonoidWithZero R] [MonoidWithZero S] {F : Type*}
    [FunLike F R S] [MonoidWithZeroHomClass F R S]
    (f : F) (hf : Function.Injective f) [IsReduced S] :
    IsReduced R 
Arguments: {R : Type u_1} {S : Type u_2} [MonoidWithZero R] [MonoidWithZero S] {F : Type u_3} [FunLike F R S] [MonoidWithZeroHomClass F R S] (f : F) (hf : Function.Injective ⇑f) [IsReduced S]
Goal: IsReduced R
-- isRadical_iff_pow_one_lt
theorem isRadical_iff_pow_one_lt [MonoidWithZero R] (k : ℕ) (hk : 1 < k) :
    IsRadical y ↔ ∀ x, y ∣ x ^ k → y ∣ x 
Arguments: {R : Type u_1} {y : R} [MonoidWithZero R] (k : ℕ) (hk : 1 < k)
Goal: IsRadical y ↔ ∀ (x : R), y ∣ x ^ k → y ∣ x
-- Commute.isNilpotent_mul_left
theorem isNilpotent_mul_left (h : IsNilpotent x) : IsNilpotent (x * y) 
Arguments: {R : Type u_1} {x y : R} [Semiring R] (h_comm : Commute x y) (h : IsNilpotent x)
Goal: IsNilpotent (x * y)