ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- CPolynomialOn_congr'
theorem CPolynomialOn_congr' {s : Set E} (h : f =ᶠ[𝓝ˢ s] g) :
    CPolynomialOn 𝕜 f s ↔ CPolynomialOn 𝕜 g s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f g : E → F} {s : Set E} (h : f =ᶠ[𝓝ˢ s] g)
Goal: CPolynomialOn 𝕜 f s ↔ CPolynomialOn 𝕜 g s
-- CPolynomialOn.congr
theorem CPolynomialOn.congr {s : Set E} (hs : IsOpen s) (hf : CPolynomialOn 𝕜 f s)
    (hg : s.EqOn f g) : CPolynomialOn 𝕜 g s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f g : E → F} {s : Set E} (hs : IsOpen s) (hf : CPolynomialOn 𝕜 f s) (hg : Set.EqOn f g s)
Goal: CPolynomialOn 𝕜 g s
-- CPolynomialOn_congr
theorem CPolynomialOn_congr {s : Set E} (hs : IsOpen s) (h : s.EqOn f g) :
    CPolynomialOn 𝕜 f s ↔ CPolynomialOn 𝕜 g s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f g : E → F} {s : Set E} (hs : IsOpen s) (h : Set.EqOn f g s)
Goal: CPolynomialOn 𝕜 f s ↔ CPolynomialOn 𝕜 g s
-- CPolynomialOn.add
theorem CPolynomialOn.add {s : Set E} (hf : CPolynomialOn 𝕜 f s) (hg : CPolynomialOn 𝕜 g s) :
    CPolynomialOn 𝕜 (f + g) s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f g : E → F} {s : Set E} (hf : CPolynomialOn 𝕜 f s) (hg : CPolynomialOn 𝕜 g s)
Goal: CPolynomialOn 𝕜 (f + g) s
-- CPolynomialOn.sub
theorem CPolynomialOn.sub {s : Set E} (hf : CPolynomialOn 𝕜 f s) (hg : CPolynomialOn 𝕜 g s) :
    CPolynomialOn 𝕜 (f - g) s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f g : E → F} {s : Set E} (hf : CPolynomialOn 𝕜 f s) (hg : CPolynomialOn 𝕜 g s)
Goal: CPolynomialOn 𝕜 (f - g) s
-- ContinuousLinearMap.comp_hasFiniteFPowerSeriesOnBall
theorem ContinuousLinearMap.comp_hasFiniteFPowerSeriesOnBall (g : F →L[𝕜] G)
    (h : HasFiniteFPowerSeriesOnBall f p x n r) :
    HasFiniteFPowerSeriesOnBall (g ∘ f) (g.compFormalMultilinearSeries p) x n r 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x : E} {r : ℝ≥0∞} {n : ℕ} (g : F →L[𝕜] G) (h : HasFiniteFPowerSeriesOnBall f p x n r)
Goal: HasFiniteFPowerSeriesOnBall (⇑g ∘ f) (g.compFormalMultilinearSeries p) x n r
-- ContinuousLinearMap.comp_cPolynomialOn
theorem ContinuousLinearMap.comp_cPolynomialOn {s : Set E} (g : F →L[𝕜] G)
    (h : CPolynomialOn 𝕜 f s) : CPolynomialOn 𝕜 (g ∘ f) s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] [NormedAddCommGroup G] [NormedSpace 𝕜 G] {f : E → F} {s : Set E} (g : F →L[𝕜] G) (h : CPolynomialOn 𝕜 f s)
Goal: CPolynomialOn 𝕜 (⇑g ∘ f) s
-- HasFiniteFPowerSeriesOnBall.eq_partialSum
theorem HasFiniteFPowerSeriesOnBall.eq_partialSum
    (hf : HasFiniteFPowerSeriesOnBall f p x n r) :
    ∀ y ∈ EMetric.ball (0 : E) r, ∀ m, n ≤ m →
    f (x + y) = p.partialSum m y 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x : E} {r : ℝ≥0∞} {n : ℕ} (hf : HasFiniteFPowerSeriesOnBall f p x n r) (y : E)
Goal: y ∈ EMetric.ball 0 r → ∀ (m : ℕ), n ≤ m → f (x + y) = p.partialSum m y
-- HasFiniteFPowerSeriesOnBall.eq_partialSum'
theorem HasFiniteFPowerSeriesOnBall.eq_partialSum'
    (hf : HasFiniteFPowerSeriesOnBall f p x n r) :
    ∀ y ∈ EMetric.ball x r, ∀ m, n ≤ m →
    f y = p.partialSum m (y - x) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x : E} {r : ℝ≥0∞} {n : ℕ} (hf : HasFiniteFPowerSeriesOnBall f p x n r) (y : E)
Goal: y ∈ EMetric.ball x r → ∀ (m : ℕ), n ≤ m → f y = p.partialSum m (y - x)
-- HasFiniteFPowerSeriesOnBall.eq_zero_of_bound_zero
theorem HasFiniteFPowerSeriesOnBall.eq_zero_of_bound_zero
    (hf : HasFiniteFPowerSeriesOnBall f pf x 0 r) : ∀ y ∈ EMetric.ball x r, f y = 0 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {pf : FormalMultilinearSeries 𝕜 E F} {x : E} {r : ℝ≥0∞} (hf : HasFiniteFPowerSeriesOnBall f pf x 0 r) (y : E)
Goal: y ∈ EMetric.ball x r → f y = 0
-- HasFiniteFPowerSeriesOnBall.bound_zero_of_eq_zero
theorem HasFiniteFPowerSeriesOnBall.bound_zero_of_eq_zero (hf : ∀ y ∈ EMetric.ball x r, f y = 0)
    (r_pos : 0 < r) (hp : ∀ n, p n = 0) : HasFiniteFPowerSeriesOnBall f p x 0 r 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x : E} {r : ℝ≥0∞} (hf : ∀ y ∈ EMetric.ball x r, f y = 0) (r_pos : 0 < r) (hp : ∀ (n : ℕ), p n = 0)
Goal: HasFiniteFPowerSeriesOnBall f p x 0 r
-- HasFiniteFPowerSeriesAt.eventually_zero_of_bound_zero
theorem HasFiniteFPowerSeriesAt.eventually_zero_of_bound_zero
    (hf : HasFiniteFPowerSeriesAt f pf x 0) : f =ᶠ[𝓝 x] 0 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {pf : FormalMultilinearSeries 𝕜 E F} {x : E} (hf : HasFiniteFPowerSeriesAt f pf x 0)
Goal: f =ᶠ[𝓝 x] 0
-- HasFiniteFPowerSeriesOnBall.eq_const_of_bound_one
theorem HasFiniteFPowerSeriesOnBall.eq_const_of_bound_one
    (hf : HasFiniteFPowerSeriesOnBall f pf x 1 r) : ∀ y ∈ EMetric.ball x r, f y = f x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {pf : FormalMultilinearSeries 𝕜 E F} {x : E} {r : ℝ≥0∞} (hf : HasFiniteFPowerSeriesOnBall f pf x 1 r) (y : E)
Goal: y ∈ EMetric.ball x r → f y = f x
-- HasFiniteFPowerSeriesAt.eventually_const_of_bound_one
theorem HasFiniteFPowerSeriesAt.eventually_const_of_bound_one
    (hf : HasFiniteFPowerSeriesAt f pf x 1) : f =ᶠ[𝓝 x] (fun _ => f x) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {pf : FormalMultilinearSeries 𝕜 E F} {x : E} (hf : HasFiniteFPowerSeriesAt f pf x 1)
Goal: f =ᶠ[𝓝 x] fun x_1 => f x
-- HasFiniteFPowerSeriesOnBall.continuousOn
protected theorem HasFiniteFPowerSeriesOnBall.continuousOn
    (hf : HasFiniteFPowerSeriesOnBall f p x n r) :
    ContinuousOn f (EMetric.ball x r) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x : E} {r : ℝ≥0∞} {n : ℕ} (hf : HasFiniteFPowerSeriesOnBall f p x n r)
Goal: ContinuousOn f (EMetric.ball x r)
-- HasFiniteFPowerSeriesAt.continuousAt
protected theorem HasFiniteFPowerSeriesAt.continuousAt (hf : HasFiniteFPowerSeriesAt f p x n) :
    ContinuousAt f x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x : E} {n : ℕ} (hf : HasFiniteFPowerSeriesAt f p x n)
Goal: ContinuousAt f x
-- CPolynomialAt.continuousAt
protected theorem CPolynomialAt.continuousAt (hf : CPolynomialAt 𝕜 f x) : ContinuousAt f x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {x : E} (hf : CPolynomialAt 𝕜 f x)
Goal: ContinuousAt f x
-- CPolynomialOn.continuousOn
protected theorem CPolynomialOn.continuousOn {s : Set E} (hf : CPolynomialOn 𝕜 f s) :
    ContinuousOn f s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {s : Set E} (hf : CPolynomialOn 𝕜 f s)
Goal: ContinuousOn f s
-- CPolynomialOn.continuous
theorem CPolynomialOn.continuous {f : E → F} (fa : CPolynomialOn 𝕜 f univ) : Continuous f 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} (fa : CPolynomialOn 𝕜 f Set.univ)
Goal: Continuous f
-- FormalMultilinearSeries.sum_of_finite
protected theorem FormalMultilinearSeries.sum_of_finite (p : FormalMultilinearSeries 𝕜 E F)
    {n : ℕ} (hn : ∀ m, n ≤ m → p m = 0) (x : E) :
    p.sum x = p.partialSum n x 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ} (hn : ∀ (m : ℕ), n ≤ m → p m = 0) (x : E)
Goal: p.sum x = p.partialSum n x
-- FormalMultilinearSeries.hasSum_of_finite
protected theorem FormalMultilinearSeries.hasSum_of_finite (p : FormalMultilinearSeries 𝕜 E F)
    {n : ℕ} (hn : ∀ m, n ≤ m → p m = 0) (x : E) :
    HasSum (fun n : ℕ => p n fun _ => x) (p.sum x) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ} (hn : ∀ (m : ℕ), n ≤ m → p m = 0) (x : E)
Goal: HasSum (fun n => (p n) fun x_1 => x) (p.sum x)
-- FormalMultilinearSeries.hasFiniteFPowerSeriesOnBall_of_finite
protected theorem FormalMultilinearSeries.hasFiniteFPowerSeriesOnBall_of_finite
    (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ} (hn : ∀ m, n ≤ m → p m = 0) :
    HasFiniteFPowerSeriesOnBall p.sum p 0 n ⊤ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ} (hn : ∀ (m : ℕ), n ≤ m → p m = 0)
Goal: HasFiniteFPowerSeriesOnBall p.sum p 0 n ⊤
-- HasFiniteFPowerSeriesOnBall.sum
theorem HasFiniteFPowerSeriesOnBall.sum (h : HasFiniteFPowerSeriesOnBall f p x n r) {y : E}
    (hy : y ∈ EMetric.ball (0 : E) r) : f (x + y) = p.sum y 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {x : E} {r : ℝ≥0∞} {n : ℕ} (h : HasFiniteFPowerSeriesOnBall f p x n r) {y : E} (hy : y ∈ EMetric.ball 0 r)
Goal: f (x + y) = p.sum y
-- FormalMultilinearSeries.continuousOn_of_finite
protected theorem FormalMultilinearSeries.continuousOn_of_finite
    (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ} (hn : ∀ m, n ≤ m → p m = 0) :
    Continuous p.sum 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ} (hn : ∀ (m : ℕ), n ≤ m → p m = 0)
Goal: Continuous p.sum
-- FormalMultilinearSeries.changeOriginSeriesTerm_bound
lemma changeOriginSeriesTerm_bound (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ}
    (hn : ∀ (m : ℕ), n ≤ m → p m = 0) (k l : ℕ) {s : Finset (Fin (k + l))}
    (hs : s.card = l) (hkl : n ≤ k + l) :
    p.changeOriginSeriesTerm k l s hs = 0 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ} (hn : ∀ (m : ℕ), n ≤ m → p m = 0) (k l : ℕ) {s : Finset (Fin (k + l))} (hs : s.card = l) (hkl : n ≤ k + l)
Goal: p.changeOriginSeriesTerm k l s hs = 0
-- FormalMultilinearSeries.changeOriginSeries_finite_of_finite
lemma changeOriginSeries_finite_of_finite (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ}
    (hn : ∀ (m : ℕ), n ≤ m → p m = 0) (k : ℕ) : ∀ {m : ℕ}, n ≤ k + m →
    p.changeOriginSeries k m = 0 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ} (hn : ∀ (m : ℕ), n ≤ m → p m = 0) (k : ℕ) {m : ℕ}
Goal: n ≤ k + m → p.changeOriginSeries k m = 0
-- FormalMultilinearSeries.changeOriginSeries_sum_eq_partialSum_of_finite
lemma changeOriginSeries_sum_eq_partialSum_of_finite (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ}
    (hn : ∀ (m : ℕ), n ≤ m → p m = 0) (k : ℕ) :
    (p.changeOriginSeries k).sum = (p.changeOriginSeries k).partialSum (n - k) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ} (hn : ∀ (m : ℕ), n ≤ m → p m = 0) (k : ℕ)
Goal: (p.changeOriginSeries k).sum = (p.changeOriginSeries k).partialSum (n - k)
-- FormalMultilinearSeries.changeOrigin_finite_of_finite
lemma changeOrigin_finite_of_finite (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ}
    (hn : ∀ (m : ℕ), n ≤ m → p m = 0) {k : ℕ} (hk : n ≤ k) :
    p.changeOrigin x k = 0 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {x : E} (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ} (hn : ∀ (m : ℕ), n ≤ m → p m = 0) {k : ℕ} (hk : n ≤ k)
Goal: p.changeOrigin x k = 0
-- FormalMultilinearSeries.hasFiniteFPowerSeriesOnBall_changeOrigin
theorem hasFiniteFPowerSeriesOnBall_changeOrigin (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ}
    (k : ℕ) (hn : ∀ (m : ℕ), n + k ≤ m → p m = 0) :
    HasFiniteFPowerSeriesOnBall (p.changeOrigin · k) (p.changeOriginSeries k) 0 n ⊤ 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ} (k : ℕ) (hn : ∀ (m : ℕ), n + k ≤ m → p m = 0)
Goal: HasFiniteFPowerSeriesOnBall (fun x => p.changeOrigin x k) (p.changeOriginSeries k) 0 n ⊤
-- FormalMultilinearSeries.changeOrigin_eval_of_finite
theorem changeOrigin_eval_of_finite (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ}
    (hn : ∀ (m : ℕ), n ≤ m → p m = 0) (x y : E) :
    (p.changeOrigin x).sum y = p.sum (x + y) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ} (hn : ∀ (m : ℕ), n ≤ m → p m = 0) (x y : E)
Goal: (p.changeOrigin x).sum y = p.sum (x + y)
-- FormalMultilinearSeries.cPolynomialAt_changeOrigin_of_finite
theorem cPolynomialAt_changeOrigin_of_finite (p : FormalMultilinearSeries 𝕜 E F)
    {n : ℕ} (hn : ∀ (m : ℕ), n ≤ m → p m = 0) (k : ℕ) :
    CPolynomialAt 𝕜 (p.changeOrigin · k) 0 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ} (hn : ∀ (m : ℕ), n ≤ m → p m = 0) (k : ℕ)
Goal: CPolynomialAt 𝕜 (fun x => p.changeOrigin x k) 0
-- HasFiniteFPowerSeriesOnBall.changeOrigin
theorem HasFiniteFPowerSeriesOnBall.changeOrigin (hf : HasFiniteFPowerSeriesOnBall f p x n r)
    (h : (‖y‖₊ : ℝ≥0∞) < r) :
    HasFiniteFPowerSeriesOnBall f (p.changeOrigin y) (x + y) n (r - ‖y‖₊) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {r : ℝ≥0∞} {n : ℕ} {x y : E} (hf : HasFiniteFPowerSeriesOnBall f p x n r) (h : ↑‖y‖₊ < r)
Goal: HasFiniteFPowerSeriesOnBall f (p.changeOrigin y) (x + y) n (r - ↑‖y‖₊)
-- HasFiniteFPowerSeriesOnBall.cPolynomialAt_of_mem
theorem HasFiniteFPowerSeriesOnBall.cPolynomialAt_of_mem
    (hf : HasFiniteFPowerSeriesOnBall f p x n r) (h : y ∈ EMetric.ball x r) :
    CPolynomialAt 𝕜 f y 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {r : ℝ≥0∞} {n : ℕ} {x y : E} (hf : HasFiniteFPowerSeriesOnBall f p x n r) (h : y ∈ EMetric.ball x r)
Goal: CPolynomialAt 𝕜 f y
-- HasFiniteFPowerSeriesOnBall.cPolynomialOn
theorem HasFiniteFPowerSeriesOnBall.cPolynomialOn (hf : HasFiniteFPowerSeriesOnBall f p x n r) :
    CPolynomialOn 𝕜 f (EMetric.ball x r) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {p : FormalMultilinearSeries 𝕜 E F} {r : ℝ≥0∞} {n : ℕ} {x : E} (hf : HasFiniteFPowerSeriesOnBall f p x n r)
Goal: CPolynomialOn 𝕜 f (EMetric.ball x r)
-- isOpen_cPolynomialAt
theorem isOpen_cPolynomialAt : IsOpen { x | CPolynomialAt 𝕜 f x } 
Arguments: (𝕜 : Type u_1) {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] (f : E → F)
Goal: IsOpen {x | CPolynomialAt 𝕜 f x}
-- CPolynomialAt.eventually_cPolynomialAt
theorem CPolynomialAt.eventually_cPolynomialAt {f : E → F} {x : E} (h : CPolynomialAt 𝕜 f x) :
    ∀ᶠ y in 𝓝 x, CPolynomialAt 𝕜 f y 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {x : E} (h : CPolynomialAt 𝕜 f x)
Goal: ∀ᶠ (y : E) in 𝓝 x, CPolynomialAt 𝕜 f y
-- CPolynomialAt.exists_mem_nhds_cPolynomialOn
theorem CPolynomialAt.exists_mem_nhds_cPolynomialOn {f : E → F} {x : E} (h : CPolynomialAt 𝕜 f x) :
    ∃ s ∈ 𝓝 x, CPolynomialOn 𝕜 f s 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {x : E} (h : CPolynomialAt 𝕜 f x)
Goal: ∃ s ∈ 𝓝 x, CPolynomialOn 𝕜 f s
-- CPolynomialAt.exists_ball_cPolynomialOn
theorem CPolynomialAt.exists_ball_cPolynomialOn {f : E → F} {x : E} (h : CPolynomialAt 𝕜 f x) :
    ∃ r : ℝ, 0 < r ∧ CPolynomialOn 𝕜 f (Metric.ball x r) 
Arguments: {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E] [NormedAddCommGroup F] [NormedSpace 𝕜 F] {f : E → F} {x : E} (h : CPolynomialAt 𝕜 f x)
Goal: ∃ r, 0 < r ∧ CPolynomialOn 𝕜 f (Metric.ball x r)
-- BoxIntegral.BoxAdditiveMap.sum_partition_boxes'
structure BoxAdditiveMap (ι M : Type*) [AddCommMonoid M] (I : WithTop (Box ι)) where
    toFun : Box ι → M
  sum_partition_boxes' : ∀ J : Box ι, ↑J ≤ I → ∀ π : Prepartition J, π.IsPartition →
    ∑ Ji ∈ π.boxes, toFun Ji = toFun J
Arguments: {ι : Type u_3} {M : Type u_4} [AddCommMonoid M] {I : WithTop (BoxIntegral.Box ι)} (self : BoxIntegral.BoxAdditiveMap ι M I) (J : BoxIntegral.Box ι)
Goal: ↑J ≤ I → ∀ (π : BoxIntegral.Prepartition J), π.IsPartition → ∑ Ji ∈ π.boxes, self.toFun Ji = self.toFun J
-- BoxIntegral.BoxAdditiveMap.coe_mk
@[simp]
theorem coe_mk (f h) : ⇑(mk f h : ι →ᵇᵃ[I₀] M) = f 
Arguments: {ι : Type u_1} {M : Type u_2} [AddCommMonoid M] {I₀ : WithTop (BoxIntegral.Box ι)} (f : BoxIntegral.Box ι → M) (h :
  ∀ (J : BoxIntegral.Box ι), ↑J ≤ I₀ → ∀ (π : BoxIntegral.Prepartition J), π.IsPartition → ∑ Ji ∈ π.boxes, f Ji = f J)
Goal: ⇑{ toFun := f, sum_partition_boxes' := h } = f
-- BoxIntegral.BoxAdditiveMap.coe_injective
theorem coe_injective : Injective fun (f : ι →ᵇᵃ[I₀] M) x => f x 
Arguments: {ι : Type u_1} {M : Type u_2} [AddCommMonoid M] {I₀ : WithTop (BoxIntegral.Box ι)}
Goal: Function.Injective fun f x => f x
-- BoxIntegral.BoxAdditiveMap.coe_inj
theorem coe_inj {f g : ι →ᵇᵃ[I₀] M} : (f : Box ι → M) = g ↔ f = g 
Arguments: {ι : Type u_1} {M : Type u_2} [AddCommMonoid M] {I₀ : WithTop (BoxIntegral.Box ι)} {f g : BoxIntegral.BoxAdditiveMap ι M I₀}
Goal: ⇑f = ⇑g ↔ f = g
-- BoxIntegral.BoxAdditiveMap.sum_partition_boxes
theorem sum_partition_boxes (f : ι →ᵇᵃ[I₀] M) (hI : ↑I ≤ I₀) {π : Prepartition I}
    (h : π.IsPartition) : ∑ J ∈ π.boxes, f J = f I 
Arguments: {ι : Type u_1} {M : Type u_2} [AddCommMonoid M] {I₀ : WithTop (BoxIntegral.Box ι)} {I : BoxIntegral.Box ι} (f : BoxIntegral.BoxAdditiveMap ι M I₀) (hI : ↑I ≤ I₀) {π : BoxIntegral.Prepartition I} (h : π.IsPartition)
Goal: ∑ J ∈ π.boxes, f J = f I
-- BoxIntegral.BoxAdditiveMap.instZero_zero_apply
@[simps (config := .asFn)]
instance : Zero (ι →ᵇᵃ[I₀] M) 
Arguments: {ι : Type u_1} {M : Type u_2} [AddCommMonoid M] {I₀ : WithTop (BoxIntegral.Box ι)}
Goal: ⇑0 = 0
-- BoxIntegral.BoxAdditiveMap.map_split_add
@[simp]
theorem map_split_add (f : ι →ᵇᵃ[I₀] M) (hI : ↑I ≤ I₀) (i : ι) (x : ℝ) :
    (I.splitLower i x).elim' 0 f + (I.splitUpper i x).elim' 0 f = f I 
Arguments: {ι : Type u_1} {M : Type u_2} [AddCommMonoid M] {I₀ : WithTop (BoxIntegral.Box ι)} {I : BoxIntegral.Box ι} (f : BoxIntegral.BoxAdditiveMap ι M I₀) (hI : ↑I ≤ I₀) (i : ι) (x : ℝ)
Goal: Option.elim' 0 (⇑f) (I.splitLower i x) + Option.elim' 0 (⇑f) (I.splitUpper i x) = f I
-- BoxIntegral.BoxAdditiveMap.restrict_apply
@[simps]
def restrict (f : ι →ᵇᵃ[I₀] M) (I : WithTop (Box ι)) (hI : I ≤ I₀) : ι →ᵇᵃ[I] M 
Arguments: {ι : Type u_1} {M : Type u_2} [AddCommMonoid M] {I₀ : WithTop (BoxIntegral.Box ι)} (f : BoxIntegral.BoxAdditiveMap ι M I₀) (I : WithTop (BoxIntegral.Box ι)) (hI : I ≤ I₀) (a : BoxIntegral.Box ι)
Goal: (f.restrict I hI) a = f a
-- BoxIntegral.BoxAdditiveMap.map_apply
@[simps (config := .asFn)]
def map (f : ι →ᵇᵃ[I₀] M) (g : M →+ N) : ι →ᵇᵃ[I₀] N 
Arguments: {ι : Type u_1} {M : Type u_2} {N : Type u_3} [AddCommMonoid M] [AddCommMonoid N] {I₀ : WithTop (BoxIntegral.Box ι)} (f : BoxIntegral.BoxAdditiveMap ι M I₀) (g : M →+ N)
Goal: ⇑(f.map g) = ⇑g ∘ ⇑f
-- BoxIntegral.BoxAdditiveMap.sum_boxes_congr
theorem sum_boxes_congr [Finite ι] (f : ι →ᵇᵃ[I₀] M) (hI : ↑I ≤ I₀) {π₁ π₂ : Prepartition I}
    (h : π₁.iUnion = π₂.iUnion) : ∑ J ∈ π₁.boxes, f J = ∑ J ∈ π₂.boxes, f J 
Arguments: {ι : Type u_1} {M : Type u_2} [AddCommMonoid M] {I₀ : WithTop (BoxIntegral.Box ι)} {I : BoxIntegral.Box ι} [Finite ι] (f : BoxIntegral.BoxAdditiveMap ι M I₀) (hI : ↑I ≤ I₀) {π₁ π₂ : BoxIntegral.Prepartition I} (h : π₁.iUnion = π₂.iUnion)
Goal: ∑ J ∈ π₁.boxes, f J = ∑ J ∈ π₂.boxes, f J
-- BoxIntegral.BoxAdditiveMap.toSMul_apply
@[simp]
theorem toSMul_apply (f : ι →ᵇᵃ[I₀] ℝ) (I : Box ι) (x : E) : f.toSMul I x = f I • x 
Arguments: {ι : Type u_1} {I₀ : WithTop (BoxIntegral.Box ι)} {E : Type u_4} [NormedAddCommGroup E] [NormedSpace ℝ E] (f : BoxIntegral.BoxAdditiveMap ι ℝ I₀) (I : BoxIntegral.Box ι) (x : E)
Goal: (f.toSMul I) x = f I • x
-- BoxIntegral.BoxAdditiveMap.upperSubLower_apply
@[simps!]
def upperSubLower.{u} {G : Type u} [AddCommGroup G] (I₀ : Box (Fin (n + 1))) (i : Fin (n + 1))
    (f : ℝ → Box (Fin n) → G) (fb : Icc (I₀.lower i) (I₀.upper i) → Fin n →ᵇᵃ[I₀.face i] G)
    (hf : ∀ (x) (hx : x ∈ Icc (I₀.lower i) (I₀.upper i)) (J), f x J = fb ⟨x, hx⟩ J) :
    Fin (n + 1) →ᵇᵃ[I₀] G 
Arguments: {n : ℕ} {G : Type u} [AddCommGroup G] (I₀ : BoxIntegral.Box (Fin (n + 1))) (i : Fin (n + 1)) (f : ℝ → BoxIntegral.Box (Fin n) → G) (fb : ↑(Set.Icc (I₀.lower i) (I₀.upper i)) → BoxIntegral.BoxAdditiveMap (Fin n) G ↑(I₀.face i)) (hf : ∀ (x : ℝ) (hx : x ∈ Set.Icc (I₀.lower i) (I₀.upper i)) (J : BoxIntegral.Box (Fin n)), f x J = (fb ⟨x, hx⟩) J) (J : BoxIntegral.Box (Fin (n + 1)))
Goal: (BoxIntegral.BoxAdditiveMap.upperSubLower I₀ i f fb hf) J = f (J.upper i) (J.face i) - f (J.lower i) (J.face i)
-- BoxIntegral.IntegrationParams.henstock_le_riemann
theorem henstock_le_riemann : Henstock ≤ Riemann 
Goal: BoxIntegral.IntegrationParams.Henstock ≤ BoxIntegral.IntegrationParams.Riemann