ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- Polynomial.dickson_one_one_charP
theorem dickson_one_one_charP (p : ℕ) [Fact p.Prime] [CharP R p] : dickson 1 (1 : R) p = X ^ p 
Arguments: (R : Type u_1) [CommRing R] (p : ℕ) [Fact (Nat.Prime p)] [CharP R p]
Goal: Polynomial.dickson 1 1 p = Polynomial.X ^ p
-- MeasureTheory.Adapted.isStoppingTime_leastGE
theorem Adapted.isStoppingTime_leastGE (r : ℝ) (n : ℕ) (hf : Adapted ℱ f) :
    IsStoppingTime ℱ (leastGE f r n) 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {ℱ : MeasureTheory.Filtration ℕ m0} {f : ℕ → Ω → ℝ} (r : ℝ) (n : ℕ) (hf : MeasureTheory.Adapted ℱ f)
Goal: MeasureTheory.IsStoppingTime ℱ (MeasureTheory.leastGE f r n)
-- MeasureTheory.leastGE_le
theorem leastGE_le {i : ℕ} {r : ℝ} (ω : Ω) : leastGE f r i ω ≤ i 
Arguments: {Ω : Type u_1} {f : ℕ → Ω → ℝ} {i : ℕ} {r : ℝ} (ω : Ω)
Goal: MeasureTheory.leastGE f r i ω ≤ i
-- MeasureTheory.leastGE_mono
theorem leastGE_mono {n m : ℕ} (hnm : n ≤ m) (r : ℝ) (ω : Ω) : leastGE f r n ω ≤ leastGE f r m ω 
Arguments: {Ω : Type u_1} {f : ℕ → Ω → ℝ} {n m : ℕ} (hnm : n ≤ m) (r : ℝ) (ω : Ω)
Goal: MeasureTheory.leastGE f r n ω ≤ MeasureTheory.leastGE f r m ω
-- MeasureTheory.leastGE_eq_min
theorem leastGE_eq_min (π : Ω → ℕ) (r : ℝ) (ω : Ω) {n : ℕ} (hπn : ∀ ω, π ω ≤ n) :
    leastGE f r (π ω) ω = min (π ω) (leastGE f r n ω) 
Arguments: {Ω : Type u_1} {f : ℕ → Ω → ℝ} (π : Ω → ℕ) (r : ℝ) (ω : Ω) {n : ℕ} (hπn : ∀ (ω : Ω), π ω ≤ n)
Goal: MeasureTheory.leastGE f r (π ω) ω = min (π ω) (MeasureTheory.leastGE f r n ω)
-- MeasureTheory.stoppedValue_stoppedValue_leastGE
theorem stoppedValue_stoppedValue_leastGE (f : ℕ → Ω → ℝ) (π : Ω → ℕ) (r : ℝ) {n : ℕ}
    (hπn : ∀ ω, π ω ≤ n) : stoppedValue (fun i => stoppedValue f (leastGE f r i)) π =
      stoppedValue (stoppedProcess f (leastGE f r n)) π 
Arguments: {Ω : Type u_1} (f : ℕ → Ω → ℝ) (π : Ω → ℕ) (r : ℝ) {n : ℕ} (hπn : ∀ (ω : Ω), π ω ≤ n)
Goal: MeasureTheory.stoppedValue (fun i => MeasureTheory.stoppedValue f (MeasureTheory.leastGE f r i)) π =
  MeasureTheory.stoppedValue (MeasureTheory.stoppedProcess f (MeasureTheory.leastGE f r n)) π
-- MeasureTheory.Submartingale.stoppedValue_leastGE
theorem Submartingale.stoppedValue_leastGE [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ) (r : ℝ) :
    Submartingale (fun i => stoppedValue f (leastGE f r i)) ℱ μ 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ℕ m0} {f : ℕ → Ω → ℝ} [MeasureTheory.IsFiniteMeasure μ] (hf : MeasureTheory.Submartingale f ℱ μ) (r : ℝ)
Goal: MeasureTheory.Submartingale (fun i => MeasureTheory.stoppedValue f (MeasureTheory.leastGE f r i)) ℱ μ
-- MeasureTheory.norm_stoppedValue_leastGE_le
theorem norm_stoppedValue_leastGE_le (hr : 0 ≤ r) (hf0 : f 0 = 0)
    (hbdd : ∀ᵐ ω ∂μ, ∀ i, |f (i + 1) ω - f i ω| ≤ R) (i : ℕ) :
    ∀ᵐ ω ∂μ, stoppedValue f (leastGE f r i) ω ≤ r + R 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {f : ℕ → Ω → ℝ} {r : ℝ} {R : ℝ≥0} (hr : 0 ≤ r) (hf0 : f 0 = 0) (hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R) (i : ℕ)
Goal: ∀ᵐ (ω : Ω) ∂μ, MeasureTheory.stoppedValue f (MeasureTheory.leastGE f r i) ω ≤ r + ↑R
-- MeasureTheory.Submartingale.stoppedValue_leastGE_eLpNorm_le
theorem Submartingale.stoppedValue_leastGE_eLpNorm_le [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
    (hr : 0 ≤ r) (hf0 : f 0 = 0) (hbdd : ∀ᵐ ω ∂μ, ∀ i, |f (i + 1) ω - f i ω| ≤ R) (i : ℕ) :
    eLpNorm (stoppedValue f (leastGE f r i)) 1 μ ≤ 2 * μ Set.univ * ENNReal.ofReal (r + R) 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ℕ m0} {f : ℕ → Ω → ℝ} {r : ℝ} {R : ℝ≥0} [MeasureTheory.IsFiniteMeasure μ] (hf : MeasureTheory.Submartingale f ℱ μ) (hr : 0 ≤ r) (hf0 : f 0 = 0) (hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R) (i : ℕ)
Goal: MeasureTheory.eLpNorm (MeasureTheory.stoppedValue f (MeasureTheory.leastGE f r i)) 1 μ ≤
  2 * μ Set.univ * ENNReal.ofReal (r + ↑R)
-- MeasureTheory.Submartingale.stoppedValue_leastGE_eLpNorm_le'
theorem Submartingale.stoppedValue_leastGE_eLpNorm_le' [IsFiniteMeasure μ]
    (hf : Submartingale f ℱ μ) (hr : 0 ≤ r) (hf0 : f 0 = 0)
    (hbdd : ∀ᵐ ω ∂μ, ∀ i, |f (i + 1) ω - f i ω| ≤ R) (i : ℕ) :
    eLpNorm (stoppedValue f (leastGE f r i)) 1 μ ≤
      ENNReal.toNNReal (2 * μ Set.univ * ENNReal.ofReal (r + R)) 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ℕ m0} {f : ℕ → Ω → ℝ} {r : ℝ} {R : ℝ≥0} [MeasureTheory.IsFiniteMeasure μ] (hf : MeasureTheory.Submartingale f ℱ μ) (hr : 0 ≤ r) (hf0 : f 0 = 0) (hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R) (i : ℕ)
Goal: MeasureTheory.eLpNorm (MeasureTheory.stoppedValue f (MeasureTheory.leastGE f r i)) 1 μ ≤
  ↑(2 * μ Set.univ * ENNReal.ofReal (r + ↑R)).toNNReal
-- MeasureTheory.Submartingale.exists_tendsto_of_abs_bddAbove_aux
theorem Submartingale.exists_tendsto_of_abs_bddAbove_aux [IsFiniteMeasure μ]
    (hf : Submartingale f ℱ μ) (hf0 : f 0 = 0) (hbdd : ∀ᵐ ω ∂μ, ∀ i, |f (i + 1) ω - f i ω| ≤ R) :
    ∀ᵐ ω ∂μ, BddAbove (Set.range fun n => f n ω) → ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ℕ m0} {f : ℕ → Ω → ℝ} {R : ℝ≥0} [MeasureTheory.IsFiniteMeasure μ] (hf : MeasureTheory.Submartingale f ℱ μ) (hf0 : f 0 = 0) (hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R)
Goal: ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n => f n ω) → ∃ c, Filter.Tendsto (fun n => f n ω) Filter.atTop (𝓝 c)
-- MeasureTheory.Submartingale.bddAbove_iff_exists_tendsto_aux
theorem Submartingale.bddAbove_iff_exists_tendsto_aux [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
    (hf0 : f 0 = 0) (hbdd : ∀ᵐ ω ∂μ, ∀ i, |f (i + 1) ω - f i ω| ≤ R) :
    ∀ᵐ ω ∂μ, BddAbove (Set.range fun n => f n ω) ↔ ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ℕ m0} {f : ℕ → Ω → ℝ} {R : ℝ≥0} [MeasureTheory.IsFiniteMeasure μ] (hf : MeasureTheory.Submartingale f ℱ μ) (hf0 : f 0 = 0) (hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R)
Goal: ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n => f n ω) ↔ ∃ c, Filter.Tendsto (fun n => f n ω) Filter.atTop (𝓝 c)
-- MeasureTheory.Submartingale.bddAbove_iff_exists_tendsto
theorem Submartingale.bddAbove_iff_exists_tendsto [IsFiniteMeasure μ] (hf : Submartingale f ℱ μ)
    (hbdd : ∀ᵐ ω ∂μ, ∀ i, |f (i + 1) ω - f i ω| ≤ R) :
    ∀ᵐ ω ∂μ, BddAbove (Set.range fun n => f n ω) ↔ ∃ c, Tendsto (fun n => f n ω) atTop (𝓝 c) 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ℕ m0} {f : ℕ → Ω → ℝ} {R : ℝ≥0} [MeasureTheory.IsFiniteMeasure μ] (hf : MeasureTheory.Submartingale f ℱ μ) (hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R)
Goal: ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n => f n ω) ↔ ∃ c, Filter.Tendsto (fun n => f n ω) Filter.atTop (𝓝 c)
-- MeasureTheory.Martingale.bddAbove_range_iff_bddBelow_range
theorem Martingale.bddAbove_range_iff_bddBelow_range [IsFiniteMeasure μ] (hf : Martingale f ℱ μ)
    (hbdd : ∀ᵐ ω ∂μ, ∀ i, |f (i + 1) ω - f i ω| ≤ R) :
    ∀ᵐ ω ∂μ, BddAbove (Set.range fun n => f n ω) ↔ BddBelow (Set.range fun n => f n ω) 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ℕ m0} {f : ℕ → Ω → ℝ} {R : ℝ≥0} [MeasureTheory.IsFiniteMeasure μ] (hf : MeasureTheory.Martingale f ℱ μ) (hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R)
Goal: ∀ᵐ (ω : Ω) ∂μ, BddAbove (Set.range fun n => f n ω) ↔ BddBelow (Set.range fun n => f n ω)
-- MeasureTheory.Martingale.ae_not_tendsto_atTop_atTop
theorem Martingale.ae_not_tendsto_atTop_atTop [IsFiniteMeasure μ] (hf : Martingale f ℱ μ)
    (hbdd : ∀ᵐ ω ∂μ, ∀ i, |f (i + 1) ω - f i ω| ≤ R) :
    ∀ᵐ ω ∂μ, ¬Tendsto (fun n => f n ω) atTop atTop 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ℕ m0} {f : ℕ → Ω → ℝ} {R : ℝ≥0} [MeasureTheory.IsFiniteMeasure μ] (hf : MeasureTheory.Martingale f ℱ μ) (hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R)
Goal: ∀ᵐ (ω : Ω) ∂μ, ¬Filter.Tendsto (fun n => f n ω) Filter.atTop Filter.atTop
-- MeasureTheory.Martingale.ae_not_tendsto_atTop_atBot
theorem Martingale.ae_not_tendsto_atTop_atBot [IsFiniteMeasure μ] (hf : Martingale f ℱ μ)
    (hbdd : ∀ᵐ ω ∂μ, ∀ i, |f (i + 1) ω - f i ω| ≤ R) :
    ∀ᵐ ω ∂μ, ¬Tendsto (fun n => f n ω) atTop atBot 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ℕ m0} {f : ℕ → Ω → ℝ} {R : ℝ≥0} [MeasureTheory.IsFiniteMeasure μ] (hf : MeasureTheory.Martingale f ℱ μ) (hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (i : ℕ), |f (i + 1) ω - f i ω| ≤ ↑R)
Goal: ∀ᵐ (ω : Ω) ∂μ, ¬Filter.Tendsto (fun n => f n ω) Filter.atTop Filter.atBot
-- MeasureTheory.BorelCantelli.process_zero
theorem process_zero : process s 0 = 0 
Arguments: {Ω : Type u_1} {s : ℕ → Set Ω}
Goal: MeasureTheory.BorelCantelli.process s 0 = 0
-- MeasureTheory.BorelCantelli.adapted_process
theorem adapted_process (hs : ∀ n, MeasurableSet[ℱ n] (s n)) : Adapted ℱ (process s) 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {ℱ : MeasureTheory.Filtration ℕ m0} {s : ℕ → Set Ω} (hs : ∀ (n : ℕ), MeasurableSet (s n))
Goal: MeasureTheory.Adapted ℱ (MeasureTheory.BorelCantelli.process s)
-- MeasureTheory.BorelCantelli.martingalePart_process_ae_eq
theorem martingalePart_process_ae_eq (ℱ : Filtration ℕ m0) (μ : Measure Ω) (s : ℕ → Set Ω) (n : ℕ) :
    martingalePart (process s) ℱ μ n =
      ∑ k ∈ Finset.range n, ((s (k + 1)).indicator 1 - μ[(s (k + 1)).indicator 1|ℱ k]) 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} (ℱ : MeasureTheory.Filtration ℕ m0) (μ : MeasureTheory.Measure Ω) (s : ℕ → Set Ω) (n : ℕ)
Goal: MeasureTheory.martingalePart (MeasureTheory.BorelCantelli.process s) ℱ μ n =
  ∑ k ∈ Finset.range n, ((s (k + 1)).indicator 1 - μ[(s (k + 1)).indicator 1|↑ℱ k])
-- MeasureTheory.BorelCantelli.predictablePart_process_ae_eq
theorem predictablePart_process_ae_eq (ℱ : Filtration ℕ m0) (μ : Measure Ω) (s : ℕ → Set Ω)
    (n : ℕ) : predictablePart (process s) ℱ μ n =
    ∑ k ∈ Finset.range n, μ[(s (k + 1)).indicator (1 : Ω → ℝ)|ℱ k] 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} (ℱ : MeasureTheory.Filtration ℕ m0) (μ : MeasureTheory.Measure Ω) (s : ℕ → Set Ω) (n : ℕ)
Goal: MeasureTheory.predictablePart (MeasureTheory.BorelCantelli.process s) ℱ μ n =
  ∑ k ∈ Finset.range n, μ[(s (k + 1)).indicator 1|↑ℱ k]
-- MeasureTheory.BorelCantelli.process_difference_le
theorem process_difference_le (s : ℕ → Set Ω) (ω : Ω) (n : ℕ) :
    |process s (n + 1) ω - process s n ω| ≤ (1 : ℝ≥0) 
Arguments: {Ω : Type u_1} (s : ℕ → Set Ω) (ω : Ω) (n : ℕ)
Goal: |MeasureTheory.BorelCantelli.process s (n + 1) ω - MeasureTheory.BorelCantelli.process s n ω| ≤ ↑1
-- MeasureTheory.BorelCantelli.integrable_process
theorem integrable_process (μ : Measure Ω) [IsFiniteMeasure μ] (hs : ∀ n, MeasurableSet[ℱ n] (s n))
    (n : ℕ) : Integrable (process s n) μ 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {ℱ : MeasureTheory.Filtration ℕ m0} {s : ℕ → Set Ω} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] (hs : ∀ (n : ℕ), MeasurableSet (s n)) (n : ℕ)
Goal: MeasureTheory.Integrable (MeasureTheory.BorelCantelli.process s n) μ
-- MeasureTheory.tendsto_sum_indicator_atTop_iff
theorem tendsto_sum_indicator_atTop_iff [IsFiniteMeasure μ]
    (hfmono : ∀ᵐ ω ∂μ, ∀ n, f n ω ≤ f (n + 1) ω) (hf : Adapted ℱ f) (hint : ∀ n, Integrable (f n) μ)
    (hbdd : ∀ᵐ ω ∂μ, ∀ n, |f (n + 1) ω - f n ω| ≤ R) :
    ∀ᵐ ω ∂μ, Tendsto (fun n => f n ω) atTop atTop ↔
      Tendsto (fun n => predictablePart f ℱ μ n ω) atTop atTop 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ℕ m0} {f : ℕ → Ω → ℝ} {R : ℝ≥0} [MeasureTheory.IsFiniteMeasure μ] (hfmono : ∀ᵐ (ω : Ω) ∂μ, ∀ (n : ℕ), f n ω ≤ f (n + 1) ω) (hf : MeasureTheory.Adapted ℱ f) (hint : ∀ (n : ℕ), MeasureTheory.Integrable (f n) μ) (hbdd : ∀ᵐ (ω : Ω) ∂μ, ∀ (n : ℕ), |f (n + 1) ω - f n ω| ≤ ↑R)
Goal: ∀ᵐ (ω : Ω) ∂μ,
  Filter.Tendsto (fun n => f n ω) Filter.atTop Filter.atTop ↔
    Filter.Tendsto (fun n => MeasureTheory.predictablePart f ℱ μ n ω) Filter.atTop Filter.atTop
-- MeasureTheory.tendsto_sum_indicator_atTop_iff'
theorem tendsto_sum_indicator_atTop_iff' [IsFiniteMeasure μ] {s : ℕ → Set Ω}
    (hs : ∀ n, MeasurableSet[ℱ n] (s n)) : ∀ᵐ ω ∂μ,
    Tendsto (fun n => ∑ k ∈ Finset.range n,
      (s (k + 1)).indicator (1 : Ω → ℝ) ω) atTop atTop ↔
    Tendsto (fun n => ∑ k ∈ Finset.range n,
      (μ[(s (k + 1)).indicator (1 : Ω → ℝ)|ℱ k]) ω) atTop atTop 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {μ : MeasureTheory.Measure Ω} {ℱ : MeasureTheory.Filtration ℕ m0} [MeasureTheory.IsFiniteMeasure μ] {s : ℕ → Set Ω} (hs : ∀ (n : ℕ), MeasurableSet (s n))
Goal: ∀ᵐ (ω : Ω) ∂μ,
  Filter.Tendsto (fun n => ∑ k ∈ Finset.range n, (s (k + 1)).indicator 1 ω) Filter.atTop Filter.atTop ↔
    Filter.Tendsto (fun n => ∑ k ∈ Finset.range n, (μ[(s (k + 1)).indicator 1|↑ℱ k]) ω) Filter.atTop Filter.atTop
-- MeasureTheory.ae_mem_limsup_atTop_iff
theorem ae_mem_limsup_atTop_iff (μ : Measure Ω) [IsFiniteMeasure μ] {s : ℕ → Set Ω}
    (hs : ∀ n, MeasurableSet[ℱ n] (s n)) : ∀ᵐ ω ∂μ, ω ∈ limsup s atTop ↔
    Tendsto (fun n => ∑ k ∈ Finset.range n,
      (μ[(s (k + 1)).indicator (1 : Ω → ℝ)|ℱ k]) ω) atTop atTop 
Arguments: {Ω : Type u_1} {m0 : MeasurableSpace Ω} {ℱ : MeasureTheory.Filtration ℕ m0} (μ : MeasureTheory.Measure Ω) [MeasureTheory.IsFiniteMeasure μ] {s : ℕ → Set Ω} (hs : ∀ (n : ℕ), MeasurableSet (s n))
Goal: ∀ᵐ (ω : Ω) ∂μ,
  ω ∈ Filter.limsup s Filter.atTop ↔
    Filter.Tendsto (fun n => ∑ k ∈ Finset.range n, (μ[(s (k + 1)).indicator 1|↑ℱ k]) ω) Filter.atTop Filter.atTop
-- Polynomial.wronskianBilin_apply
@[simp]
theorem wronskianBilin_apply (a b : R[X]) : wronskianBilin R a b = wronskian a b 
Arguments: {R : Type u_1} [CommRing R] (a b : R[X])
Goal: ((Polynomial.wronskianBilin R) a) b = a.wronskian b
-- Polynomial.wronskian_zero_left
@[simp]
theorem wronskian_zero_left (a : R[X]) : wronskian 0 a = 0 
Arguments: {R : Type u_1} [CommRing R] (a : R[X])
Goal: Polynomial.wronskian 0 a = 0
-- Polynomial.wronskian_zero_right
@[simp]
theorem wronskian_zero_right (a : R[X]) : wronskian a 0 = 0 
Arguments: {R : Type u_1} [CommRing R] (a : R[X])
Goal: a.wronskian 0 = 0
-- Polynomial.wronskian_neg_left
theorem wronskian_neg_left (a b : R[X]) : wronskian (-a) b = -wronskian a b 
Arguments: {R : Type u_1} [CommRing R] (a b : R[X])
Goal: (-a).wronskian b = -a.wronskian b
-- Polynomial.wronskian_neg_right
theorem wronskian_neg_right (a b : R[X]) : wronskian a (-b) = -wronskian a b 
Arguments: {R : Type u_1} [CommRing R] (a b : R[X])
Goal: a.wronskian (-b) = -a.wronskian b
-- Polynomial.wronskian_add_right
theorem wronskian_add_right (a b c : R[X]) : wronskian a (b + c) = wronskian a b + wronskian a c 
Arguments: {R : Type u_1} [CommRing R] (a b c : R[X])
Goal: a.wronskian (b + c) = a.wronskian b + a.wronskian c
-- Polynomial.wronskian_add_left
theorem wronskian_add_left (a b c : R[X]) : wronskian (a + b) c = wronskian a c + wronskian b c 
Arguments: {R : Type u_1} [CommRing R] (a b c : R[X])
Goal: (a + b).wronskian c = a.wronskian c + b.wronskian c
-- Polynomial.wronskian_self_eq_zero
theorem wronskian_self_eq_zero (a : R[X]) : wronskian a a = 0 
Arguments: {R : Type u_1} [CommRing R] (a : R[X])
Goal: a.wronskian a = 0
-- Polynomial.isAlt_wronskianBilin
theorem isAlt_wronskianBilin : (wronskianBilin R).IsAlt 
Arguments: {R : Type u_1} [CommRing R]
Goal: (Polynomial.wronskianBilin R).IsAlt
-- Polynomial.wronskian_neg_eq
theorem wronskian_neg_eq (a b : R[X]) : -wronskian a b = wronskian b a 
Arguments: {R : Type u_1} [CommRing R] (a b : R[X])
Goal: -a.wronskian b = b.wronskian a
-- Polynomial.wronskian_eq_of_sum_zero
theorem wronskian_eq_of_sum_zero {a b c : R[X]} (hAdd : a + b + c = 0) :
    wronskian a b = wronskian b c 
Arguments: {R : Type u_1} [CommRing R] {a b c : R[X]} (hAdd : a + b + c = 0)
Goal: a.wronskian b = b.wronskian c
-- Polynomial.degree_wronskian_lt_add
theorem degree_wronskian_lt_add {a b : R[X]} (ha : a ≠ 0) (hb : b ≠ 0) :
    (wronskian a b).degree < a.degree + b.degree 
Arguments: {R : Type u_1} [CommRing R] {a b : R[X]} (ha : a ≠ 0) (hb : b ≠ 0)
Goal: (a.wronskian b).degree < a.degree + b.degree
-- Polynomial.natDegree_wronskian_lt_add
theorem natDegree_wronskian_lt_add {a b : R[X]} (hw : wronskian a b ≠ 0) :
    (wronskian a b).natDegree < a.natDegree + b.natDegree 
Arguments: {R : Type u_1} [CommRing R] {a b : R[X]} (hw : a.wronskian b ≠ 0)
Goal: (a.wronskian b).natDegree < a.natDegree + b.natDegree
-- Polynomial.integralNormalization_zero
@[simp]
theorem integralNormalization_zero : integralNormalization (0 : R[X]) = 0 
Arguments: {R : Type u} [Semiring R]
Goal: Polynomial.integralNormalization 0 = 0
-- Polynomial.integralNormalization_coeff
theorem integralNormalization_coeff {f : R[X]} {i : ℕ} :
    (integralNormalization f).coeff i =
      if f.degree = i then 1 else coeff f i * f.leadingCoeff ^ (f.natDegree - 1 - i) 
Arguments: {R : Type u} [Semiring R] {f : R[X]} {i : ℕ}
Goal: f.integralNormalization.coeff i = if f.degree = ↑i then 1 else f.coeff i * f.leadingCoeff ^ (f.natDegree - 1 - i)
-- Polynomial.integralNormalization_support
theorem integralNormalization_support {f : R[X]} :
    (integralNormalization f).support ⊆ f.support 
Arguments: {R : Type u} [Semiring R] {f : R[X]}
Goal: f.integralNormalization.support ⊆ f.support
-- Polynomial.integralNormalization_coeff_degree
theorem integralNormalization_coeff_degree {f : R[X]} {i : ℕ} (hi : f.degree = i) :
    (integralNormalization f).coeff i = 1 
Arguments: {R : Type u} [Semiring R] {f : R[X]} {i : ℕ} (hi : f.degree = ↑i)
Goal: f.integralNormalization.coeff i = 1
-- Polynomial.integralNormalization_coeff_natDegree
theorem integralNormalization_coeff_natDegree {f : R[X]} (hf : f ≠ 0) :
    (integralNormalization f).coeff (natDegree f) = 1 
Arguments: {R : Type u} [Semiring R] {f : R[X]} (hf : f ≠ 0)
Goal: f.integralNormalization.coeff f.natDegree = 1
-- Polynomial.integralNormalization_coeff_ne_degree
theorem integralNormalization_coeff_ne_degree {f : R[X]} {i : ℕ} (hi : f.degree ≠ i) :
    coeff (integralNormalization f) i = coeff f i * f.leadingCoeff ^ (f.natDegree - 1 - i) 
Arguments: {R : Type u} [Semiring R] {f : R[X]} {i : ℕ} (hi : f.degree ≠ ↑i)
Goal: f.integralNormalization.coeff i = f.coeff i * f.leadingCoeff ^ (f.natDegree - 1 - i)
-- Polynomial.integralNormalization_coeff_ne_natDegree
theorem integralNormalization_coeff_ne_natDegree {f : R[X]} {i : ℕ} (hi : i ≠ natDegree f) :
    coeff (integralNormalization f) i = coeff f i * f.leadingCoeff ^ (f.natDegree - 1 - i) 
Arguments: {R : Type u} [Semiring R] {f : R[X]} {i : ℕ} (hi : i ≠ f.natDegree)
Goal: f.integralNormalization.coeff i = f.coeff i * f.leadingCoeff ^ (f.natDegree - 1 - i)
-- Polynomial.monic_integralNormalization
theorem monic_integralNormalization {f : R[X]} (hf : f ≠ 0) : Monic (integralNormalization f) 
Arguments: {R : Type u} [Semiring R] {f : R[X]} (hf : f ≠ 0)
Goal: f.integralNormalization.Monic
-- Polynomial.support_integralNormalization
@[simp]
theorem support_integralNormalization {f : R[X]} :
    (integralNormalization f).support = f.support 
Arguments: {R : Type u} [Ring R] [IsDomain R] {f : R[X]}
Goal: f.integralNormalization.support = f.support
-- Polynomial.integralNormalization_eval₂_eq_zero
theorem integralNormalization_eval₂_eq_zero {p : R[X]} (f : R →+* S) {z : S} (hz : eval₂ f z p = 0)
    (inj : ∀ x : R, f x = 0 → x = 0) :
    eval₂ f (z * f p.leadingCoeff) (integralNormalization p) = 0 
Arguments: {R : Type u} {S : Type v} [CommRing R] [IsDomain R] [CommSemiring S] {p : R[X]} (f : R →+* S) {z : S} (hz : Polynomial.eval₂ f z p = 0) (inj : ∀ (x : R), f x = 0 → x = 0)
Goal: Polynomial.eval₂ f (z * f p.leadingCoeff) p.integralNormalization = 0
-- Polynomial.integralNormalization_aeval_eq_zero
theorem integralNormalization_aeval_eq_zero [Algebra R S] {f : R[X]} {z : S} (hz : aeval z f = 0)
    (inj : ∀ x : R, algebraMap R S x = 0 → x = 0) :
    aeval (z * algebraMap R S f.leadingCoeff) (integralNormalization f) = 0 
Arguments: {R : Type u} {S : Type v} [CommRing R] [IsDomain R] [CommSemiring S] [Algebra R S] {f : R[X]} {z : S} (hz : (Polynomial.aeval z) f = 0) (inj : ∀ (x : R), (algebraMap R S) x = 0 → x = 0)
Goal: (Polynomial.aeval (z * (algebraMap R S) f.leadingCoeff)) f.integralNormalization = 0
-- MeasureTheory.upperCrossingTime_zero
@[simp]
theorem upperCrossingTime_zero : upperCrossingTime a b f N 0 = ⊥ 
Arguments: {Ω : Type u_1} {ι : Type u_2} [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ} {N : ι}
Goal: MeasureTheory.upperCrossingTime a b f N 0 = ⊥
-- MeasureTheory.lowerCrossingTime_zero
@[simp]
theorem lowerCrossingTime_zero : lowerCrossingTime a b f N 0 = hitting f (Set.Iic a) ⊥ N 
Arguments: {Ω : Type u_1} {ι : Type u_2} [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ} {N : ι}
Goal: MeasureTheory.lowerCrossingTime a b f N 0 = MeasureTheory.hitting f (Set.Iic a) ⊥ N
-- MeasureTheory.upperCrossingTime_succ
theorem upperCrossingTime_succ : upperCrossingTime a b f N (n + 1) ω =
    hitting f (Set.Ici b) (lowerCrossingTimeAux a f (upperCrossingTime a b f N n ω) N ω) N ω 
Arguments: {Ω : Type u_1} {ι : Type u_2} [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n : ℕ} {ω : Ω}
Goal: MeasureTheory.upperCrossingTime a b f N (n + 1) ω =
  MeasureTheory.hitting f (Set.Ici b)
    (MeasureTheory.lowerCrossingTimeAux a f (MeasureTheory.upperCrossingTime a b f N n ω) N ω) N ω
-- MeasureTheory.upperCrossingTime_succ_eq
theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
    hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω 
Arguments: {Ω : Type u_1} {ι : Type u_2} [Preorder ι] [OrderBot ι] [InfSet ι] {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n : ℕ} (ω : Ω)
Goal: MeasureTheory.upperCrossingTime a b f N (n + 1) ω =
  MeasureTheory.hitting f (Set.Ici b) (MeasureTheory.lowerCrossingTime a b f N n ω) N ω
-- MeasureTheory.upperCrossingTime_le
theorem upperCrossingTime_le : upperCrossingTime a b f N n ω ≤ N 
Arguments: {Ω : Type u_1} {ι : Type u_2} [ConditionallyCompleteLinearOrderBot ι] {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n : ℕ} {ω : Ω}
Goal: MeasureTheory.upperCrossingTime a b f N n ω ≤ N
-- MeasureTheory.upperCrossingTime_zero'
@[simp]
theorem upperCrossingTime_zero' : upperCrossingTime a b f ⊥ n ω = ⊥ 
Arguments: {Ω : Type u_1} {ι : Type u_2} [ConditionallyCompleteLinearOrderBot ι] {a b : ℝ} {f : ι → Ω → ℝ} {n : ℕ} {ω : Ω}
Goal: MeasureTheory.upperCrossingTime a b f ⊥ n ω = ⊥
-- MeasureTheory.lowerCrossingTime_le
theorem lowerCrossingTime_le : lowerCrossingTime a b f N n ω ≤ N 
Arguments: {Ω : Type u_1} {ι : Type u_2} [ConditionallyCompleteLinearOrderBot ι] {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n : ℕ} {ω : Ω}
Goal: MeasureTheory.lowerCrossingTime a b f N n ω ≤ N
-- MeasureTheory.upperCrossingTime_le_lowerCrossingTime
theorem upperCrossingTime_le_lowerCrossingTime :
    upperCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N n ω 
Arguments: {Ω : Type u_1} {ι : Type u_2} [ConditionallyCompleteLinearOrderBot ι] {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n : ℕ} {ω : Ω}
Goal: MeasureTheory.upperCrossingTime a b f N n ω ≤ MeasureTheory.lowerCrossingTime a b f N n ω
-- MeasureTheory.lowerCrossingTime_le_upperCrossingTime_succ
theorem lowerCrossingTime_le_upperCrossingTime_succ :
    lowerCrossingTime a b f N n ω ≤ upperCrossingTime a b f N (n + 1) ω 
Arguments: {Ω : Type u_1} {ι : Type u_2} [ConditionallyCompleteLinearOrderBot ι] {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n : ℕ} {ω : Ω}
Goal: MeasureTheory.lowerCrossingTime a b f N n ω ≤ MeasureTheory.upperCrossingTime a b f N (n + 1) ω
-- MeasureTheory.lowerCrossingTime_mono
theorem lowerCrossingTime_mono (hnm : n ≤ m) :
    lowerCrossingTime a b f N n ω ≤ lowerCrossingTime a b f N m ω 
Arguments: {Ω : Type u_1} {ι : Type u_2} [ConditionallyCompleteLinearOrderBot ι] {a b : ℝ} {f : ι → Ω → ℝ} {N : ι} {n m : ℕ} {ω : Ω} (hnm : n ≤ m)
Goal: MeasureTheory.lowerCrossingTime a b f N n ω ≤ MeasureTheory.lowerCrossingTime a b f N m ω