File: Mathlib/Analysis/Calculus/SmoothSeries.lean

full_name: summable_of_summable_hasFDerivAt_of_isPreconnected
code: /-- Consider a series of functions `∑' n, f n x` on a preconnected open set. If the series converges
at a point, and all functions in the series are differentiable with a summable bound on the
derivatives, then the series converges everywhere on the set. -/
theorem summable_of_summable_hasFDerivAt_of_isPreconnected (hu : Summable u) (hs : IsOpen s)
    (h's : IsPreconnected s) (hf : ∀ n x, x ∈ s → HasFDerivAt (f n) (f' n x) x)
    (hf' : ∀ n x, x ∈ s → ‖f' n x‖ ≤ u n) (hx₀ : x₀ ∈ s) (hf0 : Summable (f · x₀))
    (hx : x ∈ s) : Summable fun n => f n x 
goal: Summable fun n => f n x

full_name: summable_of_summable_hasDerivAt_of_isPreconnected
code: /-- Consider a series of functions `∑' n, f n x` on a preconnected open set. If the series converges
at a point, and all functions in the series are differentiable with a summable bound on the
derivatives, then the series converges everywhere on the set. -/
theorem summable_of_summable_hasDerivAt_of_isPreconnected (hu : Summable u) (ht : IsOpen t)
    (h't : IsPreconnected t) (hg : ∀ n y, y ∈ t → HasDerivAt (g n) (g' n y) y)
    (hg' : ∀ n y, y ∈ t → ‖g' n y‖ ≤ u n) (hy₀ : y₀ ∈ t) (hg0 : Summable (g · y₀))
    (hy : y ∈ t) : Summable fun n => g n y 
goal: Summable fun n => g n y

full_name: hasFDerivAt_tsum_of_isPreconnected
code: /-- Consider a series of functions `∑' n, f n x` on a preconnected open set. If the series converges
at a point, and all functions in the series are differentiable with a summable bound on the
derivatives, then the series is differentiable on the set and its derivative is the sum of the
derivatives. -/
theorem hasFDerivAt_tsum_of_isPreconnected (hu : Summable u) (hs : IsOpen s)
    (h's : IsPreconnected s) (hf : ∀ n x, x ∈ s → HasFDerivAt (f n) (f' n x) x)
    (hf' : ∀ n x, x ∈ s → ‖f' n x‖ ≤ u n) (hx₀ : x₀ ∈ s) (hf0 : Summable fun n => f n x₀)
    (hx : x ∈ s) : HasFDerivAt (fun y => ∑' n, f n y) (∑' n, f' n x) x 
goal: HasFDerivAt (fun y => ∑' (n : α), f n y) (∑' (n : α), f' n x) x

full_name: hasDerivAt_tsum_of_isPreconnected
code: /-- Consider a series of functions `∑' n, f n x` on a preconnected open set. If the series converges
at a point, and all functions in the series are differentiable with a summable bound on the
derivatives, then the series is differentiable on the set and its derivative is the sum of the
derivatives. -/
theorem hasDerivAt_tsum_of_isPreconnected (hu : Summable u) (ht : IsOpen t)
    (h't : IsPreconnected t) (hg : ∀ n y, y ∈ t → HasDerivAt (g n) (g' n y) y)
    (hg' : ∀ n y, y ∈ t → ‖g' n y‖ ≤ u n) (hy₀ : y₀ ∈ t) (hg0 : Summable fun n => g n y₀)
    (hy : y ∈ t) : HasDerivAt (fun z => ∑' n, g n z) (∑' n, g' n y) y 
goal: HasDerivAt (fun z => ∑' (n : α), g n z) (∑' (n : α), g' n y) y

full_name: summable_of_summable_hasFDerivAt
code: /-- Consider a series of functions `∑' n, f n x`. If the series converges at a
point, and all functions in the series are differentiable with a summable bound on the derivatives,
then the series converges everywhere. -/
theorem summable_of_summable_hasFDerivAt (hu : Summable u)
    (hf : ∀ n x, HasFDerivAt (f n) (f' n x) x) (hf' : ∀ n x, ‖f' n x‖ ≤ u n)
    (hf0 : Summable fun n => f n x₀) (x : E) : Summable fun n => f n x 
goal: Summable fun n => f n x

full_name: summable_of_summable_hasDerivAt
code: /-- Consider a series of functions `∑' n, f n x`. If the series converges at a
point, and all functions in the series are differentiable with a summable bound on the derivatives,
then the series converges everywhere. -/
theorem summable_of_summable_hasDerivAt (hu : Summable u)
    (hg : ∀ n y, HasDerivAt (g n) (g' n y) y) (hg' : ∀ n y, ‖g' n y‖ ≤ u n)
    (hg0 : Summable fun n => g n y₀) (y : 𝕜) : Summable fun n => g n y 
goal: Summable fun n => g n y

full_name: hasFDerivAt_tsum
code: /-- Consider a series of functions `∑' n, f n x`. If the series converges at a
point, and all functions in the series are differentiable with a summable bound on the derivatives,
then the series is differentiable and its derivative is the sum of the derivatives. -/
theorem hasFDerivAt_tsum (hu : Summable u) (hf : ∀ n x, HasFDerivAt (f n) (f' n x) x)
    (hf' : ∀ n x, ‖f' n x‖ ≤ u n) (hf0 : Summable fun n => f n x₀) (x : E) :
    HasFDerivAt (fun y => ∑' n, f n y) (∑' n, f' n x) x 
goal: HasFDerivAt (fun y => ∑' (n : α), f n y) (∑' (n : α), f' n x) x

full_name: hasDerivAt_tsum
code: /-- Consider a series of functions `∑' n, f n x`. If the series converges at a
point, and all functions in the series are differentiable with a summable bound on the derivatives,
then the series is differentiable and its derivative is the sum of the derivatives. -/
theorem hasDerivAt_tsum (hu : Summable u) (hg : ∀ n y, HasDerivAt (g n) (g' n y) y)
    (hg' : ∀ n y, ‖g' n y‖ ≤ u n) (hg0 : Summable fun n => g n y₀) (y : 𝕜) :
    HasDerivAt (fun z => ∑' n, g n z) (∑' n, g' n y) y 
goal: HasDerivAt (fun z => ∑' (n : α), g n z) (∑' (n : α), g' n y) y

full_name: differentiable_tsum
code: /-- Consider a series of functions `∑' n, f n x`. If all functions in the series are differentiable
with a summable bound on the derivatives, then the series is differentiable.
Note that our assumptions do not ensure the pointwise convergence, but if there is no pointwise
convergence then the series is zero everywhere so the result still holds. -/
theorem differentiable_tsum (hu : Summable u) (hf : ∀ n x, HasFDerivAt (f n) (f' n x) x)
    (hf' : ∀ n x, ‖f' n x‖ ≤ u n) : Differentiable 𝕜 fun y => ∑' n, f n y 
goal: Differentiable 𝕜 fun y => ∑' (n : α), f n y

full_name: differentiable_tsum'
code: /-- Consider a series of functions `∑' n, f n x`. If all functions in the series are differentiable
with a summable bound on the derivatives, then the series is differentiable.
Note that our assumptions do not ensure the pointwise convergence, but if there is no pointwise
convergence then the series is zero everywhere so the result still holds. -/
theorem differentiable_tsum' (hu : Summable u) (hg : ∀ n y, HasDerivAt (g n) (g' n y) y)
    (hg' : ∀ n y, ‖g' n y‖ ≤ u n) : Differentiable 𝕜 fun z => ∑' n, g n z 
goal: Differentiable 𝕜 fun z => ∑' (n : α), g n z

full_name: fderiv_tsum_apply
code: theorem fderiv_tsum_apply (hu : Summable u) (hf : ∀ n, Differentiable 𝕜 (f n))
    (hf' : ∀ n x, ‖fderiv 𝕜 (f n) x‖ ≤ u n) (hf0 : Summable fun n => f n x₀) (x : E) :
    fderiv 𝕜 (fun y => ∑' n, f n y) x = ∑' n, fderiv 𝕜 (f n) x 
goal: fderiv 𝕜 (fun y => ∑' (n : α), f n y) x = ∑' (n : α), fderiv 𝕜 (f n) x

full_name: deriv_tsum_apply
code: theorem deriv_tsum_apply (hu : Summable u) (hg : ∀ n, Differentiable 𝕜 (g n))
    (hg' : ∀ n y, ‖deriv (g n) y‖ ≤ u n) (hg0 : Summable fun n => g n y₀) (y : 𝕜) :
    deriv (fun z => ∑' n, g n z) y = ∑' n, deriv (g n) y 
goal: deriv (fun z => ∑' (n : α), g n z) y = ∑' (n : α), deriv (g n) y

full_name: fderiv_tsum
code: theorem fderiv_tsum (hu : Summable u) (hf : ∀ n, Differentiable 𝕜 (f n))
    (hf' : ∀ n x, ‖fderiv 𝕜 (f n) x‖ ≤ u n) (hf0 : Summable fun n => f n x₀) :
    (fderiv 𝕜 fun y => ∑' n, f n y) = fun x => ∑' n, fderiv 𝕜 (f n) x 
goal: (fderiv 𝕜 fun y => ∑' (n : α), f n y) = fun x => ∑' (n : α), fderiv 𝕜 (f n) x

full_name: deriv_tsum
code: theorem deriv_tsum (hu : Summable u) (hg : ∀ n, Differentiable 𝕜 (g n))
    (hg' : ∀ n y, ‖deriv (g n) y‖ ≤ u n) (hg0 : Summable fun n => g n y₀) :
    (deriv fun y => ∑' n, g n y) = fun y => ∑' n, deriv (g n) y 
goal: (deriv fun y => ∑' (n : α), g n y) = fun y => ∑' (n : α), deriv (g n) y

full_name: iteratedFDeriv_tsum
code: /-- Consider a series of smooth functions, with summable uniform bounds on the successive
derivatives. Then the iterated derivative of the sum is the sum of the iterated derivative. -/
theorem iteratedFDeriv_tsum (hf : ∀ i, ContDiff 𝕜 N (f i))
    (hv : ∀ k : ℕ, (k : ℕ∞) ≤ N → Summable (v k))
    (h'f : ∀ (k : ℕ) (i : α) (x : E), (k : ℕ∞) ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i) {k : ℕ}
    (hk : (k : ℕ∞) ≤ N) :
    (iteratedFDeriv 𝕜 k fun y => ∑' n, f n y) = fun x => ∑' n, iteratedFDeriv 𝕜 k (f n) x 
goal: (iteratedFDeriv 𝕜 k fun y => ∑' (n : α), f n y) = fun x => ∑' (n : α), iteratedFDeriv 𝕜 k (f n) x

full_name: iteratedFDeriv_tsum_apply
code: /-- Consider a series of smooth functions, with summable uniform bounds on the successive
derivatives. Then the iterated derivative of the sum is the sum of the iterated derivative. -/
theorem iteratedFDeriv_tsum_apply (hf : ∀ i, ContDiff 𝕜 N (f i))
    (hv : ∀ k : ℕ, (k : ℕ∞) ≤ N → Summable (v k))
    (h'f : ∀ (k : ℕ) (i : α) (x : E), (k : ℕ∞) ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i) {k : ℕ}
    (hk : (k : ℕ∞) ≤ N) (x : E) :
    iteratedFDeriv 𝕜 k (fun y => ∑' n, f n y) x = ∑' n, iteratedFDeriv 𝕜 k (f n) x 
goal: iteratedFDeriv 𝕜 k (fun y => ∑' (n : α), f n y) x = ∑' (n : α), iteratedFDeriv 𝕜 k (f n) x

full_name: contDiff_tsum
code: /-- Consider a series of functions `∑' i, f i x`. Assume that each individual function `f i` is of
class `C^N`, and moreover there is a uniform summable upper bound on the `k`-th derivative
for each `k ≤ N`. Then the series is also `C^N`. -/
theorem contDiff_tsum (hf : ∀ i, ContDiff 𝕜 N (f i)) (hv : ∀ k : ℕ, (k : ℕ∞) ≤ N → Summable (v k))
    (h'f : ∀ (k : ℕ) (i : α) (x : E), (k : ℕ∞) ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i) :
    ContDiff 𝕜 N fun x => ∑' i, f i x 
goal: ContDiff 𝕜 N fun x => ∑' (i : α), f i x

full_name: contDiff_tsum_of_eventually
code: /-- Consider a series of functions `∑' i, f i x`. Assume that each individual function `f i` is of
class `C^N`, and moreover there is a uniform summable upper bound on the `k`-th derivative
for each `k ≤ N` (except maybe for finitely many `i`s). Then the series is also `C^N`. -/
theorem contDiff_tsum_of_eventually (hf : ∀ i, ContDiff 𝕜 N (f i))
    (hv : ∀ k : ℕ, (k : ℕ∞) ≤ N → Summable (v k))
    (h'f :
      ∀ k : ℕ,
        (k : ℕ∞) ≤ N →
          ∀ᶠ i in (Filter.cofinite : Filter α), ∀ x : E, ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i) :
    ContDiff 𝕜 N fun x => ∑' i, f i x 
goal: ContDiff 𝕜 N fun x => ∑' (i : α), f i x