ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- MeasureTheory.stoppedValue_sub_eq_sum
theorem stoppedValue_sub_eq_sum [AddCommGroup β] (hle : τ ≤ π) :
    stoppedValue u π - stoppedValue u τ = fun ω =>
      (∑ i ∈ Finset.Ico (τ ω) (π ω), (u (i + 1) - u i)) ω 
Arguments: {Ω : Type u_1} {β : Type u_2} {u : ℕ → Ω → β} {τ π : Ω → ℕ} [AddCommGroup β] (hle : τ ≤ π)
Goal: MeasureTheory.stoppedValue u π - MeasureTheory.stoppedValue u τ = fun ω =>
  (∑ i ∈ Finset.Ico (τ ω) (π ω), (u (i + 1) - u i)) ω
-- MeasureTheory.stoppedValue_sub_eq_sum'
theorem stoppedValue_sub_eq_sum' [AddCommGroup β] (hle : τ ≤ π) {N : ℕ} (hbdd : ∀ ω, π ω ≤ N) :
    stoppedValue u π - stoppedValue u τ = fun ω =>
      (∑ i ∈ Finset.range (N + 1), Set.indicator {ω | τ ω ≤ i ∧ i < π ω} (u (i + 1) - u i)) ω 
Arguments: {Ω : Type u_1} {β : Type u_2} {u : ℕ → Ω → β} {τ π : Ω → ℕ} [AddCommGroup β] (hle : τ ≤ π) {N : ℕ} (hbdd : ∀ (ω : Ω), π ω ≤ N)
Goal: MeasureTheory.stoppedValue u π - MeasureTheory.stoppedValue u τ = fun ω =>
  (∑ i ∈ Finset.range (N + 1), {ω | τ ω ≤ i ∧ i < π ω}.indicator (u (i + 1) - u i)) ω
-- MeasureTheory.stoppedValue_eq
theorem stoppedValue_eq {N : ℕ} (hbdd : ∀ ω, τ ω ≤ N) : stoppedValue u τ = fun x =>
    (∑ i ∈ Finset.range (N + 1), Set.indicator {ω | τ ω = i} (u i)) x 
Arguments: {Ω : Type u_1} {β : Type u_2} {u : ℕ → Ω → β} {τ : Ω → ℕ} [AddCommMonoid β] {N : ℕ} (hbdd : ∀ (ω : Ω), τ ω ≤ N)
Goal: MeasureTheory.stoppedValue u τ = fun x => (∑ i ∈ Finset.range (N + 1), {ω | τ ω = i}.indicator (u i)) x
-- MeasureTheory.stoppedProcess_eq
theorem stoppedProcess_eq (n : ℕ) : stoppedProcess u τ n = Set.indicator {a | n ≤ τ a} (u n) +
    ∑ i ∈ Finset.range n, Set.indicator {ω | τ ω = i} (u i) 
Arguments: {Ω : Type u_1} {β : Type u_2} {u : ℕ → Ω → β} {τ : Ω → ℕ} [AddCommMonoid β] (n : ℕ)
Goal: MeasureTheory.stoppedProcess u τ n = {a | n ≤ τ a}.indicator (u n) + ∑ i ∈ Finset.range n, {ω | τ ω = i}.indicator (u i)
-- MeasureTheory.stoppedProcess_eq'
theorem stoppedProcess_eq' (n : ℕ) : stoppedProcess u τ n = Set.indicator {a | n + 1 ≤ τ a} (u n) +
    ∑ i ∈ Finset.range (n + 1), Set.indicator {a | τ a = i} (u i) 
Arguments: {Ω : Type u_1} {β : Type u_2} {u : ℕ → Ω → β} {τ : Ω → ℕ} [AddCommMonoid β] (n : ℕ)
Goal: MeasureTheory.stoppedProcess u τ n =
  {a | n + 1 ≤ τ a}.indicator (u n) + ∑ i ∈ Finset.range (n + 1), {a | τ a = i}.indicator (u i)
-- MeasureTheory.IsStoppingTime.piecewise_of_le
theorem IsStoppingTime.piecewise_of_le (hτ_st : IsStoppingTime 𝒢 τ) (hη_st : IsStoppingTime 𝒢 η)
    (hτ : ∀ ω, i ≤ τ ω) (hη : ∀ ω, i ≤ η ω) (hs : MeasurableSet[𝒢 i] s) :
    IsStoppingTime 𝒢 (s.piecewise τ η) 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [Preorder ι] {𝒢 : MeasureTheory.Filtration ι m} {τ η : Ω → ι} {i : ι} {s : Set Ω} [DecidablePred fun x => x ∈ s] (hτ_st : MeasureTheory.IsStoppingTime 𝒢 τ) (hη_st : MeasureTheory.IsStoppingTime 𝒢 η) (hτ : ∀ (ω : Ω), i ≤ τ ω) (hη : ∀ (ω : Ω), i ≤ η ω) (hs : MeasurableSet s)
Goal: MeasureTheory.IsStoppingTime 𝒢 (s.piecewise τ η)
-- MeasureTheory.isStoppingTime_piecewise_const
theorem isStoppingTime_piecewise_const (hij : i ≤ j) (hs : MeasurableSet[𝒢 i] s) :
    IsStoppingTime 𝒢 (s.piecewise (fun _ => i) fun _ => j) 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [Preorder ι] {𝒢 : MeasureTheory.Filtration ι m} {i j : ι} {s : Set Ω} [DecidablePred fun x => x ∈ s] (hij : i ≤ j) (hs : MeasurableSet s)
Goal: MeasureTheory.IsStoppingTime 𝒢 (s.piecewise (fun x => i) fun x => j)
-- MeasureTheory.stoppedValue_piecewise_const
theorem stoppedValue_piecewise_const {ι' : Type*} {i j : ι'} {f : ι' → Ω → ℝ} :
    stoppedValue f (s.piecewise (fun _ => i) fun _ => j) = s.piecewise (f i) (f j) 
Arguments: {Ω : Type u_1} {s : Set Ω} [DecidablePred fun x => x ∈ s] {ι' : Type u_4} {i j : ι'} {f : ι' → Ω → ℝ}
Goal: MeasureTheory.stoppedValue f (s.piecewise (fun x => i) fun x => j) = s.piecewise (f i) (f j)
-- MeasureTheory.stoppedValue_piecewise_const'
theorem stoppedValue_piecewise_const' {ι' : Type*} {i j : ι'} {f : ι' → Ω → ℝ} :
    stoppedValue f (s.piecewise (fun _ => i) fun _ => j) =
    s.indicator (f i) + sᶜ.indicator (f j) 
Arguments: {Ω : Type u_1} {s : Set Ω} [DecidablePred fun x => x ∈ s] {ι' : Type u_4} {i j : ι'} {f : ι' → Ω → ℝ}
Goal: MeasureTheory.stoppedValue f (s.piecewise (fun x => i) fun x => j) = s.indicator (f i) + sᶜ.indicator (f j)
-- MeasureTheory.condexp_stopping_time_ae_eq_restrict_eq_of_countable_range
theorem condexp_stopping_time_ae_eq_restrict_eq_of_countable_range [SigmaFiniteFiltration μ ℱ]
    (hτ : IsStoppingTime ℱ τ) (h_countable : (Set.range τ).Countable)
    [SigmaFinite (μ.trim (hτ.measurableSpace_le_of_countable_range h_countable))] (i : ι) :
    μ[f|hτ.measurableSpace] =ᵐ[μ.restrict {x | τ x = i}] μ[f|ℱ i] 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [LinearOrder ι] {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ι m} {τ : Ω → ι} {E : Type u_4} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : Ω → E} [MeasureTheory.SigmaFiniteFiltration μ ℱ] (hτ : MeasureTheory.IsStoppingTime ℱ τ) (h_countable : (Set.range τ).Countable) [MeasureTheory.SigmaFinite (μ.trim ⋯)] (i : ι)
Goal: μ[f|hτ.measurableSpace] =ᵐ[μ.restrict {x | τ x = i}] μ[f|↑ℱ i]
-- MeasureTheory.condexp_stopping_time_ae_eq_restrict_eq_of_countable
theorem condexp_stopping_time_ae_eq_restrict_eq_of_countable [Countable ι]
    [SigmaFiniteFiltration μ ℱ] (hτ : IsStoppingTime ℱ τ)
    [SigmaFinite (μ.trim hτ.measurableSpace_le_of_countable)] (i : ι) :
    μ[f|hτ.measurableSpace] =ᵐ[μ.restrict {x | τ x = i}] μ[f|ℱ i] 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [LinearOrder ι] {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ι m} {τ : Ω → ι} {E : Type u_4} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : Ω → E} [Countable ι] [MeasureTheory.SigmaFiniteFiltration μ ℱ] (hτ : MeasureTheory.IsStoppingTime ℱ τ) [MeasureTheory.SigmaFinite (μ.trim ⋯)] (i : ι)
Goal: μ[f|hτ.measurableSpace] =ᵐ[μ.restrict {x | τ x = i}] μ[f|↑ℱ i]
-- MeasureTheory.condexp_min_stopping_time_ae_eq_restrict_le_const
theorem condexp_min_stopping_time_ae_eq_restrict_le_const (hτ : IsStoppingTime ℱ τ) (i : ι)
    [SigmaFinite (μ.trim (hτ.min_const i).measurableSpace_le)] :
    μ[f|(hτ.min_const i).measurableSpace] =ᵐ[μ.restrict {x | τ x ≤ i}] μ[f|hτ.measurableSpace] 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [LinearOrder ι] {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ι m} {τ : Ω → ι} {E : Type u_4} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : Ω → E} [Filter.atTop.IsCountablyGenerated] (hτ : MeasureTheory.IsStoppingTime ℱ τ) (i : ι) [MeasureTheory.SigmaFinite (μ.trim ⋯)]
Goal: μ[f|⋯.measurableSpace] =ᵐ[μ.restrict {x | τ x ≤ i}] μ[f|hτ.measurableSpace]
-- MeasureTheory.condexp_stopping_time_ae_eq_restrict_eq
theorem condexp_stopping_time_ae_eq_restrict_eq [FirstCountableTopology ι]
    [SigmaFiniteFiltration μ ℱ] (hτ : IsStoppingTime ℱ τ)
    [SigmaFinite (μ.trim hτ.measurableSpace_le)] (i : ι) :
    μ[f|hτ.measurableSpace] =ᵐ[μ.restrict {x | τ x = i}] μ[f|ℱ i] 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [LinearOrder ι] {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ι m} {τ : Ω → ι} {E : Type u_4} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : Ω → E} [Filter.atTop.IsCountablyGenerated] [TopologicalSpace ι] [OrderTopology ι] [FirstCountableTopology ι] [MeasureTheory.SigmaFiniteFiltration μ ℱ] (hτ : MeasureTheory.IsStoppingTime ℱ τ) [MeasureTheory.SigmaFinite (μ.trim ⋯)] (i : ι)
Goal: μ[f|hτ.measurableSpace] =ᵐ[μ.restrict {x | τ x = i}] μ[f|↑ℱ i]
-- MeasureTheory.condexp_min_stopping_time_ae_eq_restrict_le
theorem condexp_min_stopping_time_ae_eq_restrict_le [MeasurableSpace ι] [SecondCountableTopology ι]
    [BorelSpace ι] (hτ : IsStoppingTime ℱ τ) (hσ : IsStoppingTime ℱ σ)
    [SigmaFinite (μ.trim (hτ.min hσ).measurableSpace_le)] :
    μ[f|(hτ.min hσ).measurableSpace] =ᵐ[μ.restrict {x | τ x ≤ σ x}] μ[f|hτ.measurableSpace] 
Arguments: {Ω : Type u_1} {ι : Type u_3} {m : MeasurableSpace Ω} [LinearOrder ι] {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ι m} {τ σ : Ω → ι} {E : Type u_4} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {f : Ω → E} [Filter.atTop.IsCountablyGenerated] [TopologicalSpace ι] [OrderTopology ι] [MeasurableSpace ι] [SecondCountableTopology ι] [BorelSpace ι] (hτ : MeasureTheory.IsStoppingTime ℱ τ) (hσ : MeasureTheory.IsStoppingTime ℱ σ) [MeasureTheory.SigmaFinite (μ.trim ⋯)]
Goal: μ[f|⋯.measurableSpace] =ᵐ[μ.restrict {x | τ x ≤ σ x}] μ[f|hτ.measurableSpace]
-- RingHom.finiteType_stableUnderComposition
theorem finiteType_stableUnderComposition : StableUnderComposition @FiniteType 
Goal: RingHom.StableUnderComposition @RingHom.FiniteType
-- RingHom.finiteType_holdsForLocalizationAway
theorem finiteType_holdsForLocalizationAway : HoldsForLocalizationAway @FiniteType 
Goal: RingHom.HoldsForLocalizationAway @RingHom.FiniteType
-- RingHom.finiteType_ofLocalizationSpanTarget
theorem finiteType_ofLocalizationSpanTarget : OfLocalizationSpanTarget @FiniteType 
Goal: RingHom.OfLocalizationSpanTarget @RingHom.FiniteType
-- RingHom.finiteType_is_local
theorem finiteType_is_local : PropertyIsLocal @FiniteType 
Goal: RingHom.PropertyIsLocal @RingHom.FiniteType
-- RingHom.finiteType_respectsIso
theorem finiteType_respectsIso : RingHom.RespectsIso @RingHom.FiniteType 
Goal: RingHom.RespectsIso @RingHom.FiniteType
-- RingHom.finiteType_stableUnderBaseChange
theorem finiteType_stableUnderBaseChange : StableUnderBaseChange @FiniteType 
Goal: RingHom.StableUnderBaseChange @RingHom.FiniteType
-- Valuation.mem_integer_iff
lemma mem_integer_iff (r : R) : r ∈ v.integer ↔ v r ≤ 1 
Arguments: {R : Type u} {Γ₀ : Type v} [Ring R] [LinearOrderedCommGroupWithZero Γ₀] (v : Valuation R Γ₀) (r : R)
Goal: r ∈ v.integer ↔ v r ≤ 1
-- Valuation.Integers.exists_of_le_one
structure Integers : Prop where
  hom_inj : Function.Injective (algebraMap O R)
  map_le_one : ∀ x, v (algebraMap O R x) ≤ 1
  exists_of_le_one : ∀ ⦃r⦄, v r ≤ 1 → ∃ x, algebraMap O R x = r
Arguments: {R : Type u} {Γ₀ : Type v} [CommRing R] [LinearOrderedCommGroupWithZero Γ₀] {v : Valuation R Γ₀} {O : Type w} [CommRing O] [Algebra O R] (self : v.Integers O) ⦃r : R⦄
Goal: v r ≤ 1 → ∃ x, (algebraMap O R) x = r
-- Valuation.Integers.map_le_one
structure Integers : Prop where
  hom_inj : Function.Injective (algebraMap O R)
  map_le_one : ∀ x, v (algebraMap O R x) ≤ 1
  exists_of_le_one : ∀ ⦃r⦄, v r ≤ 1 → ∃ x, algebraMap O R x = r
Arguments: {R : Type u} {Γ₀ : Type v} [CommRing R] [LinearOrderedCommGroupWithZero Γ₀] {v : Valuation R Γ₀} {O : Type w} [CommRing O] [Algebra O R] (self : v.Integers O) (x : O)
Goal: v ((algebraMap O R) x) ≤ 1
-- Valuation.Integers.hom_inj
structure Integers : Prop where
  hom_inj : Function.Injective (algebraMap O R)
  map_le_one : ∀ x, v (algebraMap O R x) ≤ 1
  exists_of_le_one : ∀ ⦃r⦄, v r ≤ 1 → ∃ x, algebraMap O R x = r
Arguments: {R : Type u} {Γ₀ : Type v} [CommRing R] [LinearOrderedCommGroupWithZero Γ₀] {v : Valuation R Γ₀} {O : Type w} [CommRing O] [Algebra O R] (self : v.Integers O)
Goal: Function.Injective ⇑(algebraMap O R)
-- Valuation.integer.integers
theorem integer.integers : v.Integers v.integer 
Arguments: {R : Type u} {Γ₀ : Type v} [CommRing R] [LinearOrderedCommGroupWithZero Γ₀] (v : Valuation R Γ₀)
Goal: v.Integers ↥v.integer
-- Valuation.Integers.one_of_isUnit'
theorem one_of_isUnit' {x : O} (hx : IsUnit x) (H : ∀ x, v (algebraMap O R x) ≤ 1) :
    v (algebraMap O R x) = 1 
Arguments: {R : Type u} {Γ₀ : Type v} [CommRing R] [LinearOrderedCommGroupWithZero Γ₀] {v : Valuation R Γ₀} {O : Type w} [CommRing O] [Algebra O R] {x : O} (hx : IsUnit x) (H : ∀ (x : O), v ((algebraMap O R) x) ≤ 1)
Goal: v ((algebraMap O R) x) = 1
-- Valuation.Integers.one_of_isUnit
theorem one_of_isUnit (hv : Integers v O) {x : O} (hx : IsUnit x) : v (algebraMap O R x) = 1 
Arguments: {R : Type u} {Γ₀ : Type v} [CommRing R] [LinearOrderedCommGroupWithZero Γ₀] {v : Valuation R Γ₀} {O : Type w} [CommRing O] [Algebra O R] (hv : v.Integers O) {x : O} (hx : IsUnit x)
Goal: v ((algebraMap O R) x) = 1
-- Valuation.Integers.isUnit_of_one
theorem isUnit_of_one (hv : Integers v O) {x : O} (hx : IsUnit (algebraMap O R x))
    (hvx : v (algebraMap O R x) = 1) : IsUnit x 
Arguments: {R : Type u} {Γ₀ : Type v} [CommRing R] [LinearOrderedCommGroupWithZero Γ₀] {v : Valuation R Γ₀} {O : Type w} [CommRing O] [Algebra O R] (hv : v.Integers O) {x : O} (hx : IsUnit ((algebraMap O R) x)) (hvx : v ((algebraMap O R) x) = 1)
Goal: IsUnit x
-- Valuation.Integers.le_of_dvd
theorem le_of_dvd (hv : Integers v O) {x y : O} (h : x ∣ y) :
    v (algebraMap O R y) ≤ v (algebraMap O R x) 
Arguments: {R : Type u} {Γ₀ : Type v} [CommRing R] [LinearOrderedCommGroupWithZero Γ₀] {v : Valuation R Γ₀} {O : Type w} [CommRing O] [Algebra O R] (hv : v.Integers O) {x y : O} (h : x ∣ y)
Goal: v ((algebraMap O R) y) ≤ v ((algebraMap O R) x)
-- Valuation.Integers.dvd_of_le
theorem dvd_of_le (hv : Integers v O) {x y : O}
    (h : v (algebraMap O F x) ≤ v (algebraMap O F y)) : y ∣ x 
Arguments: {F : Type u} {Γ₀ : Type v} [Field F] [LinearOrderedCommGroupWithZero Γ₀] {v : Valuation F Γ₀} {O : Type w} [CommRing O] [Algebra O F] (hv : v.Integers O) {x y : O} (h : v ((algebraMap O F) x) ≤ v ((algebraMap O F) y))
Goal: y ∣ x
-- Valuation.Integers.dvd_iff_le
theorem dvd_iff_le (hv : Integers v O) {x y : O} :
    x ∣ y ↔ v (algebraMap O F y) ≤ v (algebraMap O F x) 
Arguments: {F : Type u} {Γ₀ : Type v} [Field F] [LinearOrderedCommGroupWithZero Γ₀] {v : Valuation F Γ₀} {O : Type w} [CommRing O] [Algebra O F] (hv : v.Integers O) {x y : O}
Goal: x ∣ y ↔ v ((algebraMap O F) y) ≤ v ((algebraMap O F) x)
-- Valuation.Integers.le_iff_dvd
theorem le_iff_dvd (hv : Integers v O) {x y : O} :
    v (algebraMap O F x) ≤ v (algebraMap O F y) ↔ y ∣ x 
Arguments: {F : Type u} {Γ₀ : Type v} [Field F] [LinearOrderedCommGroupWithZero Γ₀] {v : Valuation F Γ₀} {O : Type w} [CommRing O] [Algebra O F] (hv : v.Integers O) {x y : O}
Goal: v ((algebraMap O F) x) ≤ v ((algebraMap O F) y) ↔ y ∣ x
-- Valuation.Integers.isUnit_of_one'
theorem isUnit_of_one' (hv : Integers v O) {x : O} (hvx : v (algebraMap O F x) = 1) : IsUnit x 
Arguments: {F : Type u} {Γ₀ : Type v} [Field F] [LinearOrderedCommGroupWithZero Γ₀] {v : Valuation F Γ₀} {O : Type w} [CommRing O] [Algebra O F] (hv : v.Integers O) {x : O} (hvx : v ((algebraMap O F) x) = 1)
Goal: IsUnit x
-- Valuation.Integers.eq_algebraMap_or_inv_eq_algebraMap
theorem eq_algebraMap_or_inv_eq_algebraMap (hv : Integers v O) (x : F) :
    ∃ a : O, x = algebraMap O F a ∨ x⁻¹ = algebraMap O F a 
Arguments: {F : Type u} {Γ₀ : Type v} [Field F] [LinearOrderedCommGroupWithZero Γ₀] {v : Valuation F Γ₀} {O : Type w} [CommRing O] [Algebra O F] (hv : v.Integers O) (x : F)
Goal: ∃ a, x = (algebraMap O F) a ∨ x⁻¹ = (algebraMap O F) a
-- Valuation.extendToLocalization_apply_map_apply
@[simp]
theorem Valuation.extendToLocalization_apply_map_apply (a : A) :
    v.extendToLocalization hS B (algebraMap A B a) = v a 
Arguments: {A : Type u_1} [CommRing A] {Γ : Type u_2} [LinearOrderedCommGroupWithZero Γ] (v : Valuation A Γ) {S : Submonoid A} (hS : S ≤ v.supp.primeCompl) (B : Type u_3) [CommRing B] [Algebra A B] [IsLocalization S B] (a : A)
Goal: (v.extendToLocalization hS B) ((algebraMap A B) a) = v a
-- bernsteinPolynomial.eq_zero_of_lt
theorem eq_zero_of_lt {n ν : ℕ} (h : n < ν) : bernsteinPolynomial R n ν = 0 
Arguments: (R : Type u_1) [CommRing R] {n ν : ℕ} (h : n < ν)
Goal: bernsteinPolynomial R n ν = 0
-- bernsteinPolynomial.map
@[simp]
theorem map (f : R →+* S) (n ν : ℕ) :
    (bernsteinPolynomial R n ν).map f = bernsteinPolynomial S n ν 
Arguments: {R : Type u_1} [CommRing R] {S : Type u_2} [CommRing S] (f : R →+* S) (n ν : ℕ)
Goal: Polynomial.map f (bernsteinPolynomial R n ν) = bernsteinPolynomial S n ν
-- bernsteinPolynomial.flip
theorem flip (n ν : ℕ) (h : ν ≤ n) :
    (bernsteinPolynomial R n ν).comp (1 - X) = bernsteinPolynomial R n (n - ν) 
Arguments: (R : Type u_1) [CommRing R] (n ν : ℕ) (h : ν ≤ n)
Goal: (bernsteinPolynomial R n ν).comp (1 - Polynomial.X) = bernsteinPolynomial R n (n - ν)
-- bernsteinPolynomial.flip'
theorem flip' (n ν : ℕ) (h : ν ≤ n) :
    bernsteinPolynomial R n ν = (bernsteinPolynomial R n (n - ν)).comp (1 - X) 
Arguments: (R : Type u_1) [CommRing R] (n ν : ℕ) (h : ν ≤ n)
Goal: bernsteinPolynomial R n ν = (bernsteinPolynomial R n (n - ν)).comp (1 - Polynomial.X)
-- bernsteinPolynomial.eval_at_0
theorem eval_at_0 (n ν : ℕ) : (bernsteinPolynomial R n ν).eval 0 = if ν = 0 then 1 else 0 
Arguments: (R : Type u_1) [CommRing R] (n ν : ℕ)
Goal: Polynomial.eval 0 (bernsteinPolynomial R n ν) = if ν = 0 then 1 else 0
-- bernsteinPolynomial.eval_at_1
theorem eval_at_1 (n ν : ℕ) : (bernsteinPolynomial R n ν).eval 1 = if ν = n then 1 else 0 
Arguments: (R : Type u_1) [CommRing R] (n ν : ℕ)
Goal: Polynomial.eval 1 (bernsteinPolynomial R n ν) = if ν = n then 1 else 0
-- bernsteinPolynomial.derivative_succ_aux
theorem derivative_succ_aux (n ν : ℕ) :
    Polynomial.derivative (bernsteinPolynomial R (n + 1) (ν + 1)) =
      (n + 1) * (bernsteinPolynomial R n ν - bernsteinPolynomial R n (ν + 1)) 
Arguments: (R : Type u_1) [CommRing R] (n ν : ℕ)
Goal: Polynomial.derivative (bernsteinPolynomial R (n + 1) (ν + 1)) =
  (↑n + 1) * (bernsteinPolynomial R n ν - bernsteinPolynomial R n (ν + 1))
-- bernsteinPolynomial.derivative_succ
theorem derivative_succ (n ν : ℕ) : Polynomial.derivative (bernsteinPolynomial R n (ν + 1)) =
    n * (bernsteinPolynomial R (n - 1) ν - bernsteinPolynomial R (n - 1) (ν + 1)) 
Arguments: (R : Type u_1) [CommRing R] (n ν : ℕ)
Goal: Polynomial.derivative (bernsteinPolynomial R n (ν + 1)) =
  ↑n * (bernsteinPolynomial R (n - 1) ν - bernsteinPolynomial R (n - 1) (ν + 1))
-- bernsteinPolynomial.derivative_zero
theorem derivative_zero (n : ℕ) :
    Polynomial.derivative (bernsteinPolynomial R n 0) = -n * bernsteinPolynomial R (n - 1) 0 
Arguments: (R : Type u_1) [CommRing R] (n : ℕ)
Goal: Polynomial.derivative (bernsteinPolynomial R n 0) = -↑n * bernsteinPolynomial R (n - 1) 0
-- bernsteinPolynomial.iterate_derivative_at_0_eq_zero_of_lt
theorem iterate_derivative_at_0_eq_zero_of_lt (n : ℕ) {ν k : ℕ} :
    k < ν → (Polynomial.derivative^[k] (bernsteinPolynomial R n ν)).eval 0 = 0 
Arguments: (R : Type u_1) [CommRing R] (n : ℕ) {ν k : ℕ}
Goal: k < ν → Polynomial.eval 0 ((⇑Polynomial.derivative)^[k] (bernsteinPolynomial R n ν)) = 0
-- bernsteinPolynomial.iterate_derivative_succ_at_0_eq_zero
@[simp]
theorem iterate_derivative_succ_at_0_eq_zero (n ν : ℕ) :
    (Polynomial.derivative^[ν] (bernsteinPolynomial R n (ν + 1))).eval 0 = 0 
Arguments: (R : Type u_1) [CommRing R] (n ν : ℕ)
Goal: Polynomial.eval 0 ((⇑Polynomial.derivative)^[ν] (bernsteinPolynomial R n (ν + 1))) = 0
-- bernsteinPolynomial.iterate_derivative_at_0
@[simp]
theorem iterate_derivative_at_0 (n ν : ℕ) :
    (Polynomial.derivative^[ν] (bernsteinPolynomial R n ν)).eval 0 =
      (ascPochhammer R ν).eval ((n - (ν - 1) : ℕ) : R) 
Arguments: (R : Type u_1) [CommRing R] (n ν : ℕ)
Goal: Polynomial.eval 0 ((⇑Polynomial.derivative)^[ν] (bernsteinPolynomial R n ν)) =
  Polynomial.eval (↑(n - (ν - 1))) (ascPochhammer R ν)
-- bernsteinPolynomial.iterate_derivative_at_0_ne_zero
theorem iterate_derivative_at_0_ne_zero [CharZero R] (n ν : ℕ) (h : ν ≤ n) :
    (Polynomial.derivative^[ν] (bernsteinPolynomial R n ν)).eval 0 ≠ 0 
Arguments: (R : Type u_1) [CommRing R] [CharZero R] (n ν : ℕ) (h : ν ≤ n)
Goal: Polynomial.eval 0 ((⇑Polynomial.derivative)^[ν] (bernsteinPolynomial R n ν)) ≠ 0
-- bernsteinPolynomial.iterate_derivative_at_1_eq_zero_of_lt
theorem iterate_derivative_at_1_eq_zero_of_lt (n : ℕ) {ν k : ℕ} :
    k < n - ν → (Polynomial.derivative^[k] (bernsteinPolynomial R n ν)).eval 1 = 0 
Arguments: (R : Type u_1) [CommRing R] (n : ℕ) {ν k : ℕ}
Goal: k < n - ν → Polynomial.eval 1 ((⇑Polynomial.derivative)^[k] (bernsteinPolynomial R n ν)) = 0
-- bernsteinPolynomial.iterate_derivative_at_1
@[simp]
theorem iterate_derivative_at_1 (n ν : ℕ) (h : ν ≤ n) :
    (Polynomial.derivative^[n - ν] (bernsteinPolynomial R n ν)).eval 1 =
      (-1) ^ (n - ν) * (ascPochhammer R (n - ν)).eval (ν + 1 : R) 
Arguments: (R : Type u_1) [CommRing R] (n ν : ℕ) (h : ν ≤ n)
Goal: Polynomial.eval 1 ((⇑Polynomial.derivative)^[n - ν] (bernsteinPolynomial R n ν)) =
  (-1) ^ (n - ν) * Polynomial.eval (↑ν + 1) (ascPochhammer R (n - ν))
-- bernsteinPolynomial.iterate_derivative_at_1_ne_zero
theorem iterate_derivative_at_1_ne_zero [CharZero R] (n ν : ℕ) (h : ν ≤ n) :
    (Polynomial.derivative^[n - ν] (bernsteinPolynomial R n ν)).eval 1 ≠ 0 
Arguments: (R : Type u_1) [CommRing R] [CharZero R] (n ν : ℕ) (h : ν ≤ n)
Goal: Polynomial.eval 1 ((⇑Polynomial.derivative)^[n - ν] (bernsteinPolynomial R n ν)) ≠ 0
-- bernsteinPolynomial.linearIndependent_aux
theorem linearIndependent_aux (n k : ℕ) (h : k ≤ n + 1) :
    LinearIndependent ℚ fun ν : Fin k => bernsteinPolynomial ℚ n ν 
Arguments: (n k : ℕ) (h : k ≤ n + 1)
Goal: LinearIndependent ℚ fun ν => bernsteinPolynomial ℚ n ↑ν
-- bernsteinPolynomial.linearIndependent
theorem linearIndependent (n : ℕ) :
    LinearIndependent ℚ fun ν : Fin (n + 1) => bernsteinPolynomial ℚ n ν 
Arguments: (n : ℕ)
Goal: LinearIndependent ℚ fun ν => bernsteinPolynomial ℚ n ↑ν
-- bernsteinPolynomial.sum
theorem sum (n : ℕ) : (∑ ν ∈ Finset.range (n + 1), bernsteinPolynomial R n ν) = 1 
Arguments: (R : Type u_1) [CommRing R] (n : ℕ)
Goal: ∑ ν ∈ Finset.range (n + 1), bernsteinPolynomial R n ν = 1
-- bernsteinPolynomial.sum_smul
theorem sum_smul (n : ℕ) :
    (∑ ν ∈ Finset.range (n + 1), ν • bernsteinPolynomial R n ν) = n • X 
Arguments: (R : Type u_1) [CommRing R] (n : ℕ)
Goal: ∑ ν ∈ Finset.range (n + 1), ν • bernsteinPolynomial R n ν = n • Polynomial.X
-- bernsteinPolynomial.sum_mul_smul
theorem sum_mul_smul (n : ℕ) :
    (∑ ν ∈ Finset.range (n + 1), (ν * (ν - 1)) • bernsteinPolynomial R n ν) =
      (n * (n - 1)) • X ^ 2 
Arguments: (R : Type u_1) [CommRing R] (n : ℕ)
Goal: ∑ ν ∈ Finset.range (n + 1), (ν * (ν - 1)) • bernsteinPolynomial R n ν = (n * (n - 1)) • Polynomial.X ^ 2
-- bernsteinPolynomial.variance
theorem variance (n : ℕ) :
    (∑ ν ∈ Finset.range (n + 1), (n • Polynomial.X - (ν : R[X])) ^ 2 * bernsteinPolynomial R n ν) =
      n • Polynomial.X * ((1 : R[X]) - Polynomial.X) 
Arguments: (R : Type u_1) [CommRing R] (n : ℕ)
Goal: ∑ ν ∈ Finset.range (n + 1), (n • Polynomial.X - ↑ν) ^ 2 * bernsteinPolynomial R n ν =
  n • Polynomial.X * (1 - Polynomial.X)
-- Valuation.RankOne.nontrivial'
class RankOne (v : Valuation R Γ₀) where
    hom : Γ₀ →*₀ ℝ≥0
  strictMono' : StrictMono hom
  nontrivial' : ∃ r : R, v r ≠ 0 ∧ v r ≠ 1
Arguments: {R : Type u_1} [Ring R] {Γ₀ : Type u_2} [LinearOrderedCommGroupWithZero Γ₀] {v : Valuation R Γ₀} [self : v.RankOne]
Goal: ∃ r, v r ≠ 0 ∧ v r ≠ 1
-- Valuation.RankOne.strictMono'
class RankOne (v : Valuation R Γ₀) where
    hom : Γ₀ →*₀ ℝ≥0
  strictMono' : StrictMono hom
  nontrivial' : ∃ r : R, v r ≠ 0 ∧ v r ≠ 1
Arguments: {R : Type u_1} [Ring R] {Γ₀ : Type u_2} [LinearOrderedCommGroupWithZero Γ₀] {v : Valuation R Γ₀} [self : v.RankOne]
Goal: StrictMono ⇑(Valuation.RankOne.hom v)
-- Valuation.RankOne.strictMono
lemma strictMono : StrictMono (hom v) 
Arguments: {R : Type u_1} [Ring R] {Γ₀ : Type u_2} [LinearOrderedCommGroupWithZero Γ₀] (v : Valuation R Γ₀) [v.RankOne]
Goal: StrictMono ⇑(Valuation.RankOne.hom v)
-- Valuation.RankOne.nontrivial
lemma nontrivial : ∃ r : R, v r ≠ 0 ∧ v r ≠ 1 
Arguments: {R : Type u_1} [Ring R] {Γ₀ : Type u_2} [LinearOrderedCommGroupWithZero Γ₀] (v : Valuation R Γ₀) [v.RankOne]
Goal: ∃ r, v r ≠ 0 ∧ v r ≠ 1
-- Valuation.RankOne.zero_of_hom_zero
theorem zero_of_hom_zero {x : Γ₀} (hx : hom v x = 0) : x = 0 
Arguments: {R : Type u_1} [Ring R] {Γ₀ : Type u_2} [LinearOrderedCommGroupWithZero Γ₀] (v : Valuation R Γ₀) [v.RankOne] {x : Γ₀} (hx : (Valuation.RankOne.hom v) x = 0)
Goal: x = 0
-- Valuation.RankOne.hom_eq_zero_iff
theorem hom_eq_zero_iff {x : Γ₀} : RankOne.hom v x = 0 ↔ x = 0 
Arguments: {R : Type u_1} [Ring R] {Γ₀ : Type u_2} [LinearOrderedCommGroupWithZero Γ₀] (v : Valuation R Γ₀) [v.RankOne] {x : Γ₀}
Goal: (Valuation.RankOne.hom v) x = 0 ↔ x = 0
-- Valuation.RankOne.unit_ne_one
theorem unit_ne_one : unit v ≠ 1 
Arguments: {R : Type u_1} [Ring R] {Γ₀ : Type u_2} [LinearOrderedCommGroupWithZero Γ₀] (v : Valuation R Γ₀) [v.RankOne]
Goal: Valuation.RankOne.unit v ≠ 1