ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Polynomial.coe_injective
theorem coe_injective : Function.Injective (Coe.coe : R[X] → PowerSeries R) 
Arguments: (R : Type u_2) [CommSemiring R]
Goal: Function.Injective Coe.coe
-- Polynomial.coe_inj
@[simp, norm_cast]
theorem coe_inj : (φ : PowerSeries R) = ψ ↔ φ = ψ 
Arguments: {R : Type u_2} [CommSemiring R] {φ ψ : Polynomial R}
Goal: ↑φ = ↑ψ ↔ φ = ψ
-- Polynomial.coe_eq_zero_iff
@[simp]
theorem coe_eq_zero_iff : (φ : PowerSeries R) = 0 ↔ φ = 0 
Arguments: {R : Type u_2} [CommSemiring R] {φ : Polynomial R}
Goal: ↑φ = 0 ↔ φ = 0
-- Polynomial.coe_eq_one_iff
@[simp]
theorem coe_eq_one_iff : (φ : PowerSeries R) = 1 ↔ φ = 1 
Arguments: {R : Type u_2} [CommSemiring R] {φ : Polynomial R}
Goal: ↑φ = 1 ↔ φ = 1
-- Polynomial.coeToPowerSeries.ringHom_apply
@[simp]
theorem coeToPowerSeries.ringHom_apply : coeToPowerSeries.ringHom φ = φ 
Arguments: {R : Type u_2} [CommSemiring R] (φ : Polynomial R)
Goal: Polynomial.coeToPowerSeries.ringHom φ = ↑φ
-- Polynomial.coe_pow
@[simp, norm_cast]
theorem coe_pow (n : ℕ) : ((φ ^ n : R[X]) : PowerSeries R) = (φ : PowerSeries R) ^ n 
Arguments: {R : Type u_2} [CommSemiring R] (φ : Polynomial R) (n : ℕ)
Goal: ↑(φ ^ n) = ↑φ ^ n
-- Polynomial.eval₂_C_X_eq_coe
theorem eval₂_C_X_eq_coe : φ.eval₂ (PowerSeries.C R) PowerSeries.X = ↑φ 
Arguments: {R : Type u_2} [CommSemiring R] (φ : Polynomial R)
Goal: Polynomial.eval₂ (PowerSeries.C R) PowerSeries.X φ = ↑φ
-- Polynomial.coeToPowerSeries.algHom_apply
@[simp]
theorem coeToPowerSeries.algHom_apply :
    coeToPowerSeries.algHom A φ = PowerSeries.map (algebraMap R A) ↑φ 
Arguments: {R : Type u_2} [CommSemiring R] (φ : Polynomial R) (A : Type u_3) [Semiring A] [Algebra R A]
Goal: (Polynomial.coeToPowerSeries.algHom A) φ = (PowerSeries.map (algebraMap R A)) ↑φ
-- PowerSeries.algebraMap_apply'
theorem algebraMap_apply' (p : R[X]) : algebraMap R[X] A⟦X⟧ p = map (algebraMap R A) p 
Arguments: {R : Type u_1} (A : Type u_2) [CommSemiring R] [CommSemiring A] [Algebra R A] (p : Polynomial R)
Goal: (algebraMap (Polynomial R) (PowerSeries A)) p = (PowerSeries.map (algebraMap R A)) ↑p
-- PowerSeries.algebraMap_apply''
theorem algebraMap_apply'' :
    algebraMap R⟦X⟧ A⟦X⟧ f = map (algebraMap R A) f 
Arguments: {R : Type u_1} (A : Type u_2) [CommSemiring R] [CommSemiring A] [Algebra R A] (f : PowerSeries R)
Goal: (algebraMap (PowerSeries R) (PowerSeries A)) f = (PowerSeries.map (algebraMap R A)) f
-- HahnSeries.addVal_apply
theorem addVal_apply {x : HahnSeries Γ R} :
    addVal Γ R x = x.orderTop 
Arguments: {Γ : Type u_1} {R : Type u_2} [LinearOrderedCancelAddCommMonoid Γ] [Ring R] [IsDomain R] {x : HahnSeries Γ R}
Goal: (HahnSeries.addVal Γ R) x = x.orderTop
-- HahnSeries.addVal_apply_of_ne
@[simp]
theorem addVal_apply_of_ne {x : HahnSeries Γ R} (hx : x ≠ 0) : addVal Γ R x = x.order 
Arguments: {Γ : Type u_1} {R : Type u_2} [LinearOrderedCancelAddCommMonoid Γ] [Ring R] [IsDomain R] {x : HahnSeries Γ R} (hx : x ≠ 0)
Goal: (HahnSeries.addVal Γ R) x = ↑x.order
-- HahnSeries.addVal_le_of_coeff_ne_zero
theorem addVal_le_of_coeff_ne_zero {x : HahnSeries Γ R} {g : Γ} (h : x.coeff g ≠ 0) :
    addVal Γ R x ≤ g 
Arguments: {Γ : Type u_1} {R : Type u_2} [LinearOrderedCancelAddCommMonoid Γ] [Ring R] [IsDomain R] {x : HahnSeries Γ R} {g : Γ} (h : x.coeff g ≠ 0)
Goal: (HahnSeries.addVal Γ R) x ≤ ↑g
-- IsDedekindDomain.HeightOneSpectrum.intValuationDef_if_pos
theorem intValuationDef_if_pos {r : R} (hr : r = 0) : v.intValuationDef r = 0 
Arguments: {R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : IsDedekindDomain.HeightOneSpectrum R) {r : R} (hr : r = 0)
Goal: v.intValuationDef r = 0
-- IsDedekindDomain.HeightOneSpectrum.intValuationDef_zero
@[simp]
theorem intValuationDef_zero : v.intValuationDef 0 = 0 
Arguments: {R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : IsDedekindDomain.HeightOneSpectrum R)
Goal: v.intValuationDef 0 = 0
-- IsDedekindDomain.HeightOneSpectrum.intValuationDef_if_neg
theorem intValuationDef_if_neg {r : R} (hr : r ≠ 0) :
    v.intValuationDef r =
      Multiplicative.ofAdd
        (-(Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {r} : Ideal R)).factors : ℤ) 
Arguments: {R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : IsDedekindDomain.HeightOneSpectrum R) {r : R} (hr : r ≠ 0)
Goal: v.intValuationDef r =
  ↑(Multiplicative.ofAdd (-↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {r})).factors)))
-- IsDedekindDomain.HeightOneSpectrum.intValuation_ne_zero
theorem intValuation_ne_zero (x : R) (hx : x ≠ 0) : v.intValuationDef x ≠ 0 
Arguments: {R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : IsDedekindDomain.HeightOneSpectrum R) (x : R) (hx : x ≠ 0)
Goal: v.intValuationDef x ≠ 0
-- IsDedekindDomain.HeightOneSpectrum.intValuation_ne_zero'
theorem intValuation_ne_zero' (x : nonZeroDivisors R) : v.intValuationDef x ≠ 0 
Arguments: {R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : IsDedekindDomain.HeightOneSpectrum R) (x : ↥(nonZeroDivisors R))
Goal: v.intValuationDef ↑x ≠ 0
-- IsDedekindDomain.HeightOneSpectrum.intValuation_zero_le
theorem intValuation_zero_le (x : nonZeroDivisors R) : 0 < v.intValuationDef x 
Arguments: {R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : IsDedekindDomain.HeightOneSpectrum R) (x : ↥(nonZeroDivisors R))
Goal: 0 < v.intValuationDef ↑x
-- IsDedekindDomain.HeightOneSpectrum.intValuation_le_one
theorem intValuation_le_one (x : R) : v.intValuationDef x ≤ 1 
Arguments: {R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : IsDedekindDomain.HeightOneSpectrum R) (x : R)
Goal: v.intValuationDef x ≤ 1
-- IsDedekindDomain.HeightOneSpectrum.intValuation_lt_one_iff_dvd
theorem intValuation_lt_one_iff_dvd (r : R) :
    v.intValuationDef r < 1 ↔ v.asIdeal ∣ Ideal.span {r} 
Arguments: {R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : IsDedekindDomain.HeightOneSpectrum R) (r : R)
Goal: v.intValuationDef r < 1 ↔ v.asIdeal ∣ Ideal.span {r}
-- IsDedekindDomain.HeightOneSpectrum.intValuation_le_pow_iff_dvd
theorem intValuation_le_pow_iff_dvd (r : R) (n : ℕ) :
    v.intValuationDef r ≤ Multiplicative.ofAdd (-(n : ℤ)) ↔ v.asIdeal ^ n ∣ Ideal.span {r} 
Arguments: {R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : IsDedekindDomain.HeightOneSpectrum R) (r : R) (n : ℕ)
Goal: v.intValuationDef r ≤ ↑(Multiplicative.ofAdd (-↑n)) ↔ v.asIdeal ^ n ∣ Ideal.span {r}
-- IsDedekindDomain.HeightOneSpectrum.intValuation.map_zero'
theorem intValuation.map_zero' : v.intValuationDef 0 = 0 
Arguments: {R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : IsDedekindDomain.HeightOneSpectrum R)
Goal: v.intValuationDef 0 = 0
-- IsDedekindDomain.HeightOneSpectrum.intValuation.map_one'
theorem intValuation.map_one' : v.intValuationDef 1 = 1 
Arguments: {R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : IsDedekindDomain.HeightOneSpectrum R)
Goal: v.intValuationDef 1 = 1
-- IsDedekindDomain.HeightOneSpectrum.intValuation.map_mul'
theorem intValuation.map_mul' (x y : R) :
    v.intValuationDef (x * y) = v.intValuationDef x * v.intValuationDef y 
Arguments: {R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : IsDedekindDomain.HeightOneSpectrum R) (x y : R)
Goal: v.intValuationDef (x * y) = v.intValuationDef x * v.intValuationDef y
-- IsDedekindDomain.HeightOneSpectrum.intValuation.le_max_iff_min_le
theorem intValuation.le_max_iff_min_le {a b c : ℕ} :
    Multiplicative.ofAdd (-c : ℤ) ≤
      max (Multiplicative.ofAdd (-a : ℤ)) (Multiplicative.ofAdd (-b : ℤ)) ↔
      min a b ≤ c 
Arguments: {a b c : ℕ}
Goal: Multiplicative.ofAdd (-↑c) ≤ max (Multiplicative.ofAdd (-↑a)) (Multiplicative.ofAdd (-↑b)) ↔ min a b ≤ c
-- IsDedekindDomain.HeightOneSpectrum.intValuation.map_add_le_max'
theorem intValuation.map_add_le_max' (x y : R) :
    v.intValuationDef (x + y) ≤ max (v.intValuationDef x) (v.intValuationDef y) 
Arguments: {R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : IsDedekindDomain.HeightOneSpectrum R) (x y : R)
Goal: v.intValuationDef (x + y) ≤ max (v.intValuationDef x) (v.intValuationDef y)
-- IsDedekindDomain.HeightOneSpectrum.intValuation_toFun
@[simps]
def intValuation : Valuation R ℤₘ₀ 
Arguments: {R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : IsDedekindDomain.HeightOneSpectrum R) (r : R)
Goal: v.intValuation r = v.intValuationDef r
-- IsDedekindDomain.HeightOneSpectrum.intValuation_apply
theorem intValuation_apply {r : R} (v : IsDedekindDomain.HeightOneSpectrum R) :
    intValuation v r = intValuationDef v r 
Arguments: {R : Type u_1} [CommRing R] [IsDedekindDomain R] {r : R} (v : IsDedekindDomain.HeightOneSpectrum R)
Goal: v.intValuation r = v.intValuationDef r
-- IsDedekindDomain.HeightOneSpectrum.intValuation_exists_uniformizer
theorem intValuation_exists_uniformizer :
    ∃ π : R, v.intValuationDef π = Multiplicative.ofAdd (-1 : ℤ) 
Arguments: {R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : IsDedekindDomain.HeightOneSpectrum R)
Goal: ∃ π, v.intValuationDef π = ↑(Multiplicative.ofAdd (-1))
-- IsDedekindDomain.HeightOneSpectrum.intValuation_singleton
theorem intValuation_singleton {r : R} (hr : r ≠ 0) (hv : v.asIdeal = Ideal.span {r}) :
    v.intValuation r = Multiplicative.ofAdd (-1 : ℤ) 
Arguments: {R : Type u_1} [CommRing R] [IsDedekindDomain R] (v : IsDedekindDomain.HeightOneSpectrum R) {r : R} (hr : r ≠ 0) (hv : v.asIdeal = Ideal.span {r})
Goal: v.intValuation r = ↑(Multiplicative.ofAdd (-1))
-- IsDedekindDomain.HeightOneSpectrum.valuation_def
theorem valuation_def (x : K) :
    v.valuation x =
      v.intValuation.extendToLocalization
        (fun r hr => Set.mem_compl (v.intValuation_ne_zero' ⟨r, hr⟩)) K x 
Arguments: {R : Type u_1} [CommRing R] [IsDedekindDomain R] {K : Type u_2} [Field K] [Algebra R K] [IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R) (x : K)
Goal: v.valuation x = (v.intValuation.extendToLocalization ⋯ K) x
-- IsDedekindDomain.HeightOneSpectrum.valuation_of_mk'
theorem valuation_of_mk' {r : R} {s : nonZeroDivisors R} :
    v.valuation (IsLocalization.mk' K r s) = v.intValuation r / v.intValuation s 
Arguments: {R : Type u_1} [CommRing R] [IsDedekindDomain R] {K : Type u_2} [Field K] [Algebra R K] [IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R) {r : R} {s : ↥(nonZeroDivisors R)}
Goal: v.valuation (IsLocalization.mk' K r s) = v.intValuation r / v.intValuation ↑s
-- IsDedekindDomain.HeightOneSpectrum.valuation_of_algebraMap
theorem valuation_of_algebraMap (r : R) : v.valuation (algebraMap R K r) = v.intValuation r 
Arguments: {R : Type u_1} [CommRing R] [IsDedekindDomain R] {K : Type u_2} [Field K] [Algebra R K] [IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R) (r : R)
Goal: v.valuation ((algebraMap R K) r) = v.intValuation r
-- IsDedekindDomain.HeightOneSpectrum.valuation_eq_intValuationDef
lemma valuation_eq_intValuationDef (r : R) : v.valuation (r : K) = v.intValuationDef r 
Arguments: {R : Type u_1} [CommRing R] [IsDedekindDomain R] {K : Type u_2} [Field K] [Algebra R K] [IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R) (r : R)
Goal: v.valuation ↑r = v.intValuationDef r
-- IsDedekindDomain.HeightOneSpectrum.valuation_le_one
theorem valuation_le_one (r : R) : v.valuation (algebraMap R K r) ≤ 1 
Arguments: {R : Type u_1} [CommRing R] [IsDedekindDomain R] {K : Type u_2} [Field K] [Algebra R K] [IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R) (r : R)
Goal: v.valuation ((algebraMap R K) r) ≤ 1
-- IsDedekindDomain.HeightOneSpectrum.valuation_lt_one_iff_dvd
theorem valuation_lt_one_iff_dvd (r : R) :
    v.valuation (algebraMap R K r) < 1 ↔ v.asIdeal ∣ Ideal.span {r} 
Arguments: {R : Type u_1} [CommRing R] [IsDedekindDomain R] {K : Type u_2} [Field K] [Algebra R K] [IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R) (r : R)
Goal: v.valuation ((algebraMap R K) r) < 1 ↔ v.asIdeal ∣ Ideal.span {r}
-- IsDedekindDomain.HeightOneSpectrum.valuation_exists_uniformizer
theorem valuation_exists_uniformizer : ∃ π : K, v.valuation π = Multiplicative.ofAdd (-1 : ℤ) 
Arguments: {R : Type u_1} [CommRing R] [IsDedekindDomain R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R)
Goal: ∃ π, v.valuation π = ↑(Multiplicative.ofAdd (-1))
-- IsDedekindDomain.HeightOneSpectrum.valuation_uniformizer_ne_zero
theorem valuation_uniformizer_ne_zero : Classical.choose (v.valuation_exists_uniformizer K) ≠ 0 
Arguments: {R : Type u_1} [CommRing R] [IsDedekindDomain R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R)
Goal: Classical.choose ⋯ ≠ 0
-- IsDedekindDomain.HeightOneSpectrum.adicValued_apply
theorem adicValued_apply {x : K} : (v.adicValued.v : _) x = v.valuation x 
Arguments: {R : Type u_1} [CommRing R] [IsDedekindDomain R] {K : Type u_2} [Field K] [Algebra R K] [IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R) {x : K}
Goal: Valued.v x = v.valuation x
-- IsDedekindDomain.HeightOneSpectrum.valuedAdicCompletion_def
theorem valuedAdicCompletion_def {x : v.adicCompletion K} :
    Valued.v x = @Valued.extension K _ _ _ (adicValued v) x 
Arguments: {R : Type u_1} [CommRing R] [IsDedekindDomain R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R) {x : IsDedekindDomain.HeightOneSpectrum.adicCompletion K v}
Goal: Valued.v x = Valued.extension x
-- IsDedekindDomain.HeightOneSpectrum.mem_adicCompletionIntegers
theorem mem_adicCompletionIntegers {x : v.adicCompletion K} :
    x ∈ v.adicCompletionIntegers K ↔ (Valued.v x : ℤₘ₀) ≤ 1 
Arguments: (R : Type u_1) [CommRing R] [IsDedekindDomain R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R) {x : IsDedekindDomain.HeightOneSpectrum.adicCompletion K v}
Goal: x ∈ IsDedekindDomain.HeightOneSpectrum.adicCompletionIntegers K v ↔ Valued.v x ≤ 1
-- IsDedekindDomain.HeightOneSpectrum.not_mem_adicCompletionIntegers
theorem not_mem_adicCompletionIntegers {x : v.adicCompletion K} :
    x ∉ v.adicCompletionIntegers K ↔ 1 < (Valued.v x : ℤₘ₀) 
Arguments: (R : Type u_1) [CommRing R] [IsDedekindDomain R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R) {x : IsDedekindDomain.HeightOneSpectrum.adicCompletion K v}
Goal: x ∉ IsDedekindDomain.HeightOneSpectrum.adicCompletionIntegers K v ↔ 1 < Valued.v x
-- IsDedekindDomain.HeightOneSpectrum.coe_smul_adicCompletion
theorem coe_smul_adicCompletion (r : R) (x : K) :
    (↑(r • x) : v.adicCompletion K) = r • (↑x : v.adicCompletion K) 
Arguments: (R : Type u_1) [CommRing R] [IsDedekindDomain R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R) (r : R) (x : K)
Goal: ↑K (r • x) = r • ↑K x
-- IsDedekindDomain.HeightOneSpectrum.algebraMap_adicCompletion'
theorem algebraMap_adicCompletion' :
    ⇑(algebraMap R <| v.adicCompletion K) = (↑) ∘ algebraMap R K 
Arguments: (R : Type u_1) [CommRing R] [IsDedekindDomain R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R)
Goal: ⇑(algebraMap R (IsDedekindDomain.HeightOneSpectrum.adicCompletion K v)) = ↑K ∘ ⇑(algebraMap R K)
-- IsDedekindDomain.HeightOneSpectrum.algebraMap_adicCompletion
theorem algebraMap_adicCompletion :
    ⇑(algebraMap K <| v.adicCompletion K) = ((↑) : K → adicCompletion K v) 
Arguments: (R : Type u_1) [CommRing R] [IsDedekindDomain R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R)
Goal: ⇑(algebraMap K (IsDedekindDomain.HeightOneSpectrum.adicCompletion K v)) = ↑K
-- IsDedekindDomain.HeightOneSpectrum.valuedAdicCompletion_eq_valuation
theorem valuedAdicCompletion_eq_valuation (r : R) :
    Valued.v (r : v.adicCompletion K) = v.valuation (r : K) 
Arguments: {R : Type u_1} [CommRing R] [IsDedekindDomain R] {K : Type u_2} [Field K] [Algebra R K] [IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R) (r : R)
Goal: Valued.v ↑r = v.valuation ↑r
-- IsDedekindDomain.HeightOneSpectrum.valuedAdicCompletion_eq_valuation'
theorem valuedAdicCompletion_eq_valuation' (k : K) :
    Valued.v (k : v.adicCompletion K) = v.valuation k 
Arguments: {R : Type u_1} [CommRing R] [IsDedekindDomain R] {K : Type u_2} [Field K] [Algebra R K] [IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R) (k : K)
Goal: Valued.v (↑K k) = v.valuation k
-- IsDedekindDomain.HeightOneSpectrum.coe_mem_adicCompletionIntegers
lemma coe_mem_adicCompletionIntegers (r : R) :
    (r : adicCompletion K v) ∈ adicCompletionIntegers K v 
Arguments: {R : Type u_1} [CommRing R] [IsDedekindDomain R] {K : Type u_2} [Field K] [Algebra R K] [IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R) (r : R)
Goal: ↑r ∈ IsDedekindDomain.HeightOneSpectrum.adicCompletionIntegers K v
-- IsDedekindDomain.HeightOneSpectrum.coe_smul_adicCompletionIntegers
@[simp]
theorem coe_smul_adicCompletionIntegers (r : R) (x : v.adicCompletionIntegers K) :
    (↑(r • x) : v.adicCompletion K) = r • (x : v.adicCompletion K) 
Arguments: (R : Type u_1) [CommRing R] [IsDedekindDomain R] (K : Type u_2) [Field K] [Algebra R K] [IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R) (r : R) (x : ↥(IsDedekindDomain.HeightOneSpectrum.adicCompletionIntegers K v))
Goal: ↑(r • x) = r • ↑x
-- IsDedekindDomain.HeightOneSpectrum.adicCompletion.mul_nonZeroDivisor_mem_adicCompletionIntegers
lemma adicCompletion.mul_nonZeroDivisor_mem_adicCompletionIntegers (v : HeightOneSpectrum R)
    (a : v.adicCompletion K) : ∃ b ∈ R⁰, a * b ∈ v.adicCompletionIntegers K 
Arguments: {R : Type u_1} [CommRing R] [IsDedekindDomain R] {K : Type u_2} [Field K] [Algebra R K] [IsFractionRing R K] (v : IsDedekindDomain.HeightOneSpectrum R) (a : IsDedekindDomain.HeightOneSpectrum.adicCompletion K v)
Goal: ∃ b ∈ nonZeroDivisors R, a * ↑b ∈ IsDedekindDomain.HeightOneSpectrum.adicCompletionIntegers K v
-- mul_right_bijective_of_finite₀
theorem mul_right_bijective_of_finite₀ {a : M} (ha : a ≠ 0) : Bijective fun b => a * b 
Arguments: {M : Type u_1} [CancelMonoidWithZero M] [Finite M] {a : M} (ha : a ≠ 0)
Goal: Function.Bijective fun b => a * b
-- mul_left_bijective_of_finite₀
theorem mul_left_bijective_of_finite₀ {a : M} (ha : a ≠ 0) : Bijective fun b => b * a 
Arguments: {M : Type u_1} [CancelMonoidWithZero M] [Finite M] {a : M} (ha : a ≠ 0)
Goal: Function.Bijective fun b => b * a
-- exists_eq_pow_of_mul_eq_pow_of_coprime
theorem exists_eq_pow_of_mul_eq_pow_of_coprime {R : Type*} [CommSemiring R] [IsDomain R]
    [GCDMonoid R] [Unique Rˣ] {a b c : R} {n : ℕ} (cp : IsCoprime a b) (h : a * b = c ^ n) :
    ∃ d : R, a = d ^ n 
Arguments: {R : Type u_2} [CommSemiring R] [IsDomain R] [GCDMonoid R] [Unique Rˣ] {a b c : R} {n : ℕ} (cp : IsCoprime a b) (h : a * b = c ^ n)
Goal: ∃ d, a = d ^ n
-- Finset.exists_eq_pow_of_mul_eq_pow_of_coprime
nonrec
theorem Finset.exists_eq_pow_of_mul_eq_pow_of_coprime {ι R : Type*} [CommSemiring R] [IsDomain R]
    [GCDMonoid R] [Unique Rˣ] {n : ℕ} {c : R} {s : Finset ι} {f : ι → R}
    (h : ∀ i ∈ s, ∀ j ∈ s, i ≠ j → IsCoprime (f i) (f j))
    (hprod : ∏ i ∈ s, f i = c ^ n) : ∀ i ∈ s, ∃ d : R, f i = d ^ n 
Arguments: {ι : Type u_2} {R : Type u_3} [CommSemiring R] [IsDomain R] [GCDMonoid R] [Unique Rˣ] {n : ℕ} {c : R} {s : Finset ι} {f : ι → R} (h : ∀ i ∈ s, ∀ j ∈ s, i ≠ j → IsCoprime (f i) (f j)) (hprod : ∏ i ∈ s, f i = c ^ n) (i : ι)
Goal: i ∈ s → ∃ d, f i = d ^ n
-- Finite.isField_of_domain
theorem Finite.isField_of_domain (R) [CommRing R] [IsDomain R] [Finite R] : IsField R 
Arguments: (R : Type u_3) [CommRing R] [IsDomain R] [Finite R]
Goal: IsField R
-- card_nthRoots_subgroup_units
theorem card_nthRoots_subgroup_units [Fintype G] [DecidableEq G] (f : G →* R) (hf : Injective f)
    {n : ℕ} (hn : 0 < n) (g₀ : G) :
    Finset.card (Finset.univ.filter (fun g ↦ g^n = g₀)) ≤ Multiset.card (nthRoots n (f g₀)) 
Arguments: {R : Type u_1} {G : Type u_2} [CommRing R] [IsDomain R] [Group G] [Fintype G] [DecidableEq G] (f : G →* R) (hf : Function.Injective ⇑f) {n : ℕ} (hn : 0 < n) (g₀ : G)
Goal: (Finset.filter (fun g => g ^ n = g₀) Finset.univ).card ≤ Multiset.card (Polynomial.nthRoots n (f g₀))
-- isCyclic_of_subgroup_isDomain
theorem isCyclic_of_subgroup_isDomain [Finite G] (f : G →* R) (hf : Injective f) : IsCyclic G 
Arguments: {R : Type u_1} {G : Type u_2} [CommRing R] [IsDomain R] [Group G] [Finite G] (f : G →* R) (hf : Function.Injective ⇑f)
Goal: IsCyclic G
-- Polynomial.div_eq_quo_add_rem_div
theorem div_eq_quo_add_rem_div (f : R[X]) {g : R[X]} (hg : g.Monic) :
    ∃ q r : R[X], r.degree < g.degree ∧
      (algebraMap R[X] K f) / (algebraMap R[X] K g) =
        algebraMap R[X] K q + (algebraMap R[X] K r) / (algebraMap R[X] K g) 
Arguments: {R : Type u_1} [CommRing R] [IsDomain R] (K : Type) [Field K] [Algebra (Polynomial R) K] [IsFractionRing (Polynomial R) K] (f : Polynomial R) {g : Polynomial R} (hg : g.Monic)
Goal: ∃ q r,
  r.degree < g.degree ∧
    (algebraMap (Polynomial R) K) f / (algebraMap (Polynomial R) K) g =
      (algebraMap (Polynomial R) K) q + (algebraMap (Polynomial R) K) r / (algebraMap (Polynomial R) K) g
-- sum_hom_units_eq_zero
theorem sum_hom_units_eq_zero (f : G →* R) (hf : f ≠ 1) : ∑ g : G, f g = 0 
Arguments: {R : Type u_1} {G : Type u_2} [CommRing R] [IsDomain R] [Group G] [Fintype G] (f : G →* R) (hf : f ≠ 1)
Goal: ∑ g : G, f g = 0
-- sum_hom_units
theorem sum_hom_units (f : G →* R) [Decidable (f = 1)] :
    ∑ g : G, f g = if f = 1 then Fintype.card G else 0 
Arguments: {R : Type u_1} {G : Type u_2} [CommRing R] [IsDomain R] [Group G] [Fintype G] (f : G →* R) [Decidable (f = 1)]
Goal: ∑ g : G, f g = ↑(if f = 1 then Fintype.card G else 0)
-- MatrixEquivTensor.toFunBilinear_apply
@[simp]
theorem toFunBilinear_apply (a : A) (m : Matrix n n R) :
    toFunBilinear R A n a m = a • m.map (algebraMap R A) 
Arguments: (R : Type u) [CommSemiring R] (A : Type v) [Semiring A] [Algebra R A] (n : Type w) (a : A) (m : Matrix n n R)
Goal: ((MatrixEquivTensor.toFunBilinear R A n) a) m = a • m.map ⇑(algebraMap R A)
-- MatrixEquivTensor.toFunAlgHom_apply
@[simp]
theorem toFunAlgHom_apply (a : A) (m : Matrix n n R) :
    toFunAlgHom R A n (a ⊗ₜ m) = a • m.map (algebraMap R A) 
Arguments: (R : Type u) [CommSemiring R] (A : Type v) [Semiring A] [Algebra R A] (n : Type w) [DecidableEq n] [Fintype n] (a : A) (m : Matrix n n R)
Goal: (MatrixEquivTensor.toFunAlgHom R A n) (a ⊗ₜ[R] m) = a • m.map ⇑(algebraMap R A)
-- MatrixEquivTensor.invFun_zero
@[simp]
theorem invFun_zero : invFun R A n 0 = 0 
Arguments: (R : Type u) [CommSemiring R] (A : Type v) [Semiring A] [Algebra R A] (n : Type w) [DecidableEq n] [Fintype n]
Goal: MatrixEquivTensor.invFun R A n 0 = 0
-- MatrixEquivTensor.invFun_add
@[simp]
theorem invFun_add (M N : Matrix n n A) :
    invFun R A n (M + N) = invFun R A n M + invFun R A n N 
Arguments: (R : Type u) [CommSemiring R] (A : Type v) [Semiring A] [Algebra R A] (n : Type w) [DecidableEq n] [Fintype n] (M N : Matrix n n A)
Goal: MatrixEquivTensor.invFun R A n (M + N) = MatrixEquivTensor.invFun R A n M + MatrixEquivTensor.invFun R A n N
-- MatrixEquivTensor.invFun_smul
@[simp]
theorem invFun_smul (a : A) (M : Matrix n n A) :
    invFun R A n (a • M) = a ⊗ₜ 1 * invFun R A n M 
Arguments: (R : Type u) [CommSemiring R] (A : Type v) [Semiring A] [Algebra R A] (n : Type w) [DecidableEq n] [Fintype n] (a : A) (M : Matrix n n A)
Goal: MatrixEquivTensor.invFun R A n (a • M) = a ⊗ₜ[R] 1 * MatrixEquivTensor.invFun R A n M
-- MatrixEquivTensor.invFun_algebraMap
@[simp]
theorem invFun_algebraMap (M : Matrix n n R) : invFun R A n (M.map (algebraMap R A)) = 1 ⊗ₜ M 
Arguments: (R : Type u) [CommSemiring R] (A : Type v) [Semiring A] [Algebra R A] (n : Type w) [DecidableEq n] [Fintype n] (M : Matrix n n R)
Goal: MatrixEquivTensor.invFun R A n (M.map ⇑(algebraMap R A)) = 1 ⊗ₜ[R] M
-- MatrixEquivTensor.right_inv
theorem right_inv (M : Matrix n n A) : (toFunAlgHom R A n) (invFun R A n M) = M 
Arguments: (R : Type u) [CommSemiring R] (A : Type v) [Semiring A] [Algebra R A] (n : Type w) [DecidableEq n] [Fintype n] (M : Matrix n n A)
Goal: (MatrixEquivTensor.toFunAlgHom R A n) (MatrixEquivTensor.invFun R A n M) = M
-- MatrixEquivTensor.left_inv
theorem left_inv (M : A ⊗[R] Matrix n n R) : invFun R A n (toFunAlgHom R A n M) = M 
Arguments: (R : Type u) [CommSemiring R] (A : Type v) [Semiring A] [Algebra R A] (n : Type w) [DecidableEq n] [Fintype n] (M : A ⊗[R] Matrix n n R)
Goal: MatrixEquivTensor.invFun R A n ((MatrixEquivTensor.toFunAlgHom R A n) M) = M