mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Calculus.ContDiff.WithLp

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
H : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : (i : ι) → NormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : NormedSpace 𝕜 H
inst✝¹ : Fintype ι
p : ℝ≥0∞
inst✝ : Fact (1 ≤ p)
n : WithTop ℕ∞
f : H → PiLp p E
⊢ ContDiff 𝕜 n f ↔ ∀ (i : ι), ContDiff 𝕜 n fun x ↦ (f x).ofLp i

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.UniformLimitsDeriv

Goal state before:
pp: ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : ι → E → G
f' : ι → E → E →L[𝕜] G
x : E
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2
hfg : Cauchy (map (fun n ↦ f n x) l)
this✝ : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
hf' : TendstoUniformlyOnFilter (fun n z ↦ -f' n.1 z + f' n.2 z) 0 (l ×ˢ l) (𝓝 x)
this : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
ε : ℝ
hε : ε > 0
t : Set G
ht : t ∈ map (fun n ↦ f n x) l
ht' : ∀ x ∈ t, ∀ y ∈ t, dist x y < ε
n : ι × ι
hn : (fun n ↦ f n.1 x ∈ t ∧ f n.2 x ∈ t) n
y : E
x✝ : (fun x ↦ True) y
⊢ dist (0 (n, y).2) (f (n, y).1.1 x - f (n, y).1.2 x) < ε

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.WithLp

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
H : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : (i : ι) → NormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : NormedSpace 𝕜 H
inst✝¹ : Finite ι
p : ENNReal
inst✝ : Fact (1 ≤ p)
f : H → PiLp p E
t : Set H
y : H
⊢ DifferentiableWithinAt 𝕜 f t y ↔ ∀ (i : ι), DifferentiableWithinAt 𝕜 (fun x ↦ (f x).ofLp i) t y

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.WithLp

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
H : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : (i : ι) → NormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : NormedSpace 𝕜 H
inst✝¹ : Finite ι
p : ENNReal
inst✝ : Fact (1 ≤ p)
f : H → PiLp p E
y : H
⊢ DifferentiableAt 𝕜 f y ↔ ∀ (i : ι), DifferentiableAt 𝕜 (fun x ↦ (f x).ofLp i) y

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.WithLp

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
H : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : (i : ι) → NormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : NormedSpace 𝕜 H
inst✝¹ : Finite ι
p : ENNReal
inst✝ : Fact (1 ≤ p)
f : H → PiLp p E
t : Set H
⊢ DifferentiableOn 𝕜 f t ↔ ∀ (i : ι), DifferentiableOn 𝕜 (fun x ↦ (f x).ofLp i) t

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.UniformLimitsDeriv

Goal state before:
pp: ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : ι → E → G
f' : ι → E → E →L[𝕜] G
x : E
hf' : UniformCauchySeqOnFilter f' l (𝓝 x)
hf : ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2
hfg : Cauchy (map (fun n ↦ f n x) l)
⊢ UniformCauchySeqOnFilter f l (𝓝 x)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝² : (i : α) → NormedAddCommGroup (E i)
inst✝¹ : DecidableEq α
inst✝ : Fact (1 ≤ p)
hp : p ≠ ∞
f : ↥(lp E p)
hp₀ : 0 < p
hp' : 0 < p.toReal
this :
  ∀ ε > 0,
    ∀ᶠ (x : Finset α) in (SummationFilter.unconditional α).filter,
      dist (∑ b ∈ x, ‖↑f b‖ ^ p.toReal) (‖f‖ ^ p.toReal) < ε
ε : ℝ
hε : ε > 0
s : Finset α
hs : |‖f‖ ^ p.toReal - ∑ b ∈ s, ‖↑f b‖ ^ p.toReal| < ε ^ p.toReal
⊢ ‖∑ i ∈ s, lp.single p i (↑f i) - f‖ ^ p.toReal = ‖f‖ ^ p.toReal - ∑ i ∈ s, ‖↑f i‖ ^ p.toReal

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.WithLp

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
H : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : (i : ι) → NormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : NormedSpace 𝕜 H
inst✝¹ : Finite ι
p : ENNReal
inst✝ : Fact (1 ≤ p)
f : H → PiLp p E
⊢ Differentiable 𝕜 f ↔ ∀ (i : ι), Differentiable 𝕜 fun x ↦ (f x).ofLp i

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.WithLp

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
H : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : (i : ι) → NormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : NormedSpace 𝕜 H
inst✝¹ : Finite ι
p : ENNReal
inst✝ : Fact (1 ≤ p)
f : H → PiLp p E
f' : H →L[𝕜] PiLp p E
y : H
⊢ HasStrictFDerivAt f f' y ↔ ∀ (i : ι), HasStrictFDerivAt (fun x ↦ (f x).ofLp i) (PiLp.proj p E i ∘SL f') y

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝² : (i : α) → NormedAddCommGroup (E i)
inst✝¹ : DecidableEq α
inst✝ : Fact (1 ≤ p)
hp : p ≠ ∞
f : ↥(lp E p)
hp₀ : 0 < p
hp' : 0 < p.toReal
this :
  ∀ ε > 0,
    ∀ᶠ (x : Finset α) in (SummationFilter.unconditional α).filter,
      dist (∑ b ∈ x, ‖↑f b‖ ^ p.toReal) (‖f‖ ^ p.toReal) < ε
ε : ℝ
hε : ε > 0
s : Finset α
hs : |‖∑ i ∈ s, lp.single p i (↑f i) - f‖ ^ p.toReal| < ε ^ p.toReal
H : ‖∑ i ∈ s, lp.single p i (↑f i) - f‖ ^ p.toReal = ‖f‖ ^ p.toReal - ∑ i ∈ s, ‖↑f i‖ ^ p.toReal
⊢ |‖∑ i ∈ s, lp.single p i (↑f i) - f‖ ^ p.toReal| = ‖∑ i ∈ s, lp.single p i (↑f i) - f‖ ^ p.toReal

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝² : (i : α) → NormedAddCommGroup (E i)
inst✝¹ : DecidableEq α
inst✝ : Fact (1 ≤ p)
hp : p ≠ ∞
f : ↥(lp E p)
⊢ HasSum (fun i ↦ lp.single p i (↑f i)) f

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.FDeriv.WithLp

Goal state before:
pp: 𝕜 : Type u_1
ι : Type u_2
E : ι → Type u_3
H : Type u_4
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedAddCommGroup H
inst✝⁴ : (i : ι) → NormedAddCommGroup (E i)
inst✝³ : (i : ι) → NormedSpace 𝕜 (E i)
inst✝² : NormedSpace 𝕜 H
inst✝¹ : Finite ι
p : ENNReal
inst✝ : Fact (1 ≤ p)
f : H → PiLp p E
f' : H →L[𝕜] PiLp p E
t : Set H
y : H
⊢ HasFDerivWithinAt f f' t y ↔ ∀ (i : ι), HasFDerivWithinAt (fun x ↦ (f x).ofLp i) (PiLp.proj p E i ∘SL f') t y

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝⁵ : (i : α) → NormedAddCommGroup (E i)
inst✝⁴ : DecidableEq α
F : Type u_5
inst✝³ : AddCommMonoid F
inst✝² : TopologicalSpace F
inst✝¹ : T2Space F
inst✝ : Fact (1 ≤ p)
hp : p ≠ ∞
f g : ↥(lp E p) →ₜ+ F
h : ∀ (i : α), f.comp (singleContinuousAddMonoidHom E p i) = g.comp (singleContinuousAddMonoidHom E p i)
⊢ f = g

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
α : Type u_3
E : α → Type u_4
p✝ q✝ : ℝ≥0∞
inst✝³ : (i : α) → NormedAddCommGroup (E i)
inst✝² : NormedRing 𝕜
inst✝¹ : (i : α) → Module 𝕜 (E i)
inst✝ : ∀ (i : α), IsBoundedSMul 𝕜 (E i)
p q r : ℝ≥0∞
h : p ≤ q
x✝¹ x✝ : ↥(lp E p)
⊢ ⟨↑(x✝¹ + x✝), ⋯⟩ = ⟨↑x✝¹, ⋯⟩ + ⟨↑x✝, ⋯⟩

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
α : Type u_3
E : α → Type u_4
p✝ q✝ : ℝ≥0∞
inst✝³ : (i : α) → NormedAddCommGroup (E i)
inst✝² : NormedRing 𝕜
inst✝¹ : (i : α) → Module 𝕜 (E i)
inst✝ : ∀ (i : α), IsBoundedSMul 𝕜 (E i)
p q r : ℝ≥0∞
h : p ≤ q
x✝¹ : 𝕜
x✝ : ↥(lp E p)
⊢ ⟨↑(x✝¹ • x✝), ⋯⟩ = (RingHom.id 𝕜) x✝¹ • ⟨↑x✝, ⋯⟩

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.UniformLimitsDeriv

Goal state before:
pp: ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : ι → E → G
f' : ι → E → E →L[𝕜] G
x : E
r : ℝ
hf : ∀ (n : ι), ∀ y ∈ Metric.ball x r, HasFDerivAt (f n) (f' n y) y
hfg : Cauchy (map (fun n ↦ f n x) l)
this✝³ : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
hf' : TendstoUniformlyOn (fun n z ↦ -f' n.1 z + f' n.2 z) 0 (l ×ˢ l) (Metric.ball x r)
this✝² : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
this✝¹ : l.NeBot
hr : 0 < r
this✝ :
  TendstoUniformlyOn (fun n z ↦ f n.1 z - f n.2 z - (f n.1 x - f n.2 x)) 0 (l ×ˢ l) (Metric.ball x r) ∧
    TendstoUniformlyOn (fun n x_1 ↦ f n.1 x - f n.2 x) 0 (l ×ˢ l) (Metric.ball x r)
this :
  TendstoUniformlyOn ((-fun n z ↦ f n.1 z - f n.2 z - (f n.1 x - f n.2 x)) + -fun n x_1 ↦ f n.1 x - f n.2 x) 0 (l ×ˢ l)
    (Metric.ball x r)
n : ι × ι
z : E
a✝ : z ∈ Metric.ball x r
⊢ ((-fun n z ↦ f n.1 z - f n.2 z - (f n.1 x - f n.2 x)) + -fun n x_1 ↦ f n.1 x - f n.2 x) n z =
    (fun z ↦ -f n.1 z + f n.2 z) z

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: 𝕜 : Type u_1
α : Type u_3
E : α → Type u_4
inst✝³ : (i : α) → NormedAddCommGroup (E i)
inst✝² : NormedRing 𝕜
inst✝¹ : (i : α) → Module 𝕜 (E i)
inst✝ : ∀ (i : α), IsBoundedSMul 𝕜 (E i)
p q : ℝ≥0∞
h : p ≤ q
f : ↥(lp E p)
⊢ ↑((linearMapOfLE 𝕜 E h) f) = ↑f

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: 𝕜 : Type u_1
α : Type u_3
E : α → Type u_4
inst✝³ : (i : α) → NormedAddCommGroup (E i)
inst✝² : NormedRing 𝕜
inst✝¹ : (i : α) → Module 𝕜 (E i)
inst✝ : ∀ (i : α), IsBoundedSMul 𝕜 (E i)
p q : ℝ≥0∞
h : p ≤ q
⊢ (linearMapOfLE 𝕜 E h).toAddMonoidHom = AddSubgroup.inclusion ⋯

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.UniformLimitsDeriv

Goal state before:
pp: ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : ι → E → G
f' : ι → E → E →L[𝕜] G
x : E
r : ℝ
hf : ∀ (n : ι), ∀ y ∈ Metric.ball x r, HasFDerivAt (f n) (f' n y) y
hfg : Cauchy (map (fun n ↦ f n x) l)
this✝² : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
hf' : TendstoUniformlyOn (fun n z ↦ -f' n.1 z + f' n.2 z) 0 (l ×ˢ l) (Metric.ball x r)
this✝¹ : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
this✝ : l.NeBot
hr : 0 < r
this :
  TendstoUniformlyOn (fun n z ↦ f n.1 z - f n.2 z - (f n.1 x - f n.2 x)) 0 (l ×ˢ l) (Metric.ball x r) ∧
    TendstoUniformlyOn (fun n x_1 ↦ f n.1 x - f n.2 x) 0 (l ×ˢ l) (Metric.ball x r)
⊢ TendstoUniformlyOn (fun n z ↦ -f n.1 z + f n.2 z) 0 (l ×ˢ l) (Metric.ball x r)

Tactic:
have := this.1.neg.add this.2.neg
    rw [neg_zero, add_zero] at this
    refine this.congr ?_
    filter_upwards with n z _ using (by simp; abel)

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: 𝕜 : Type u_1
α : Type u_3
E : α → Type u_4
inst✝³ : (i : α) → NormedAddCommGroup (E i)
inst✝² : NormedRing 𝕜
inst✝¹ : (i : α) → Module 𝕜 (E i)
inst✝ : ∀ (i : α), IsBoundedSMul 𝕜 (E i)
p q r : ℝ≥0∞
hpq : p ≤ q
hqr : q ≤ r
⊢ linearMapOfLE 𝕜 E hqr ∘ₗ linearMapOfLE 𝕜 E hpq = linearMapOfLE 𝕜 E ⋯

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.UniformLimitsDeriv

Goal state before:
pp: ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : ι → E → G
f' : ι → E → E →L[𝕜] G
x : E
r : ℝ
hf : ∀ (n : ι), ∀ y ∈ Metric.ball x r, HasFDerivAt (f n) (f' n y) y
hfg : Cauchy (map (fun n ↦ f n x) l)
this✝² : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
hf' : TendstoUniformlyOn (fun n z ↦ -f' n.1 z + f' n.2 z) 0 (l ×ˢ l) (Metric.ball x r)
this✝¹ : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
this✝ : l.NeBot
hr : 0 < r
this :
  TendstoUniformlyOn (fun n z ↦ f n.1 z - f n.2 z - (f n.1 x - f n.2 x)) 0 (l ×ˢ l) (Metric.ball x r) ∧
    TendstoUniformlyOn (fun n x_1 ↦ f n.1 x - f n.2 x) 0 (l ×ˢ l) (Metric.ball x r)
⊢ TendstoUniformlyOn (fun n z ↦ -f n.1 z + f n.2 z) 0 (l ×ˢ l) (Metric.ball x r)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
α : Type u_3
E : α → Type u_4
p✝ q✝ : ℝ≥0∞
inst✝⁴ : (i : α) → NormedAddCommGroup (E i)
inst✝³ : NormedRing 𝕜
inst✝² : (i : α) → Module 𝕜 (E i)
inst✝¹ : ∀ (i : α), IsBoundedSMul 𝕜 (E i)
p q r : ℝ≥0∞
inst✝ : Fact (1 ≤ p)
i : α
x : ↥(lp E p)
⊢ ‖(evalₗ E p i) x‖ ≤ 1 * ‖x‖

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
E : α → Type u_4
inst✝¹ : (i : α) → NormedAddCommGroup (E i)
ι : Type u_5
l : Filter ι
inst✝ : l.NeBot
C : ℝ
F : ι → ↥(lp E ∞)
hCF : ∀ᶠ (k : ι) in l, ‖F k‖ ≤ C
f : (a : α) → E a
hf : Tendsto (id fun i ↦ ↑(F i)) l (𝓝 f)
a : α
⊢ ‖f a‖ ≤ C

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝¹ : (i : α) → NormedAddCommGroup (E i)
ι : Type u_5
l : Filter ι
inst✝ : l.NeBot
_i : Fact (1 ≤ p)
hp : p ≠ ∞
C : ℝ
F : ι → ↥(lp E p)
hCF : ∀ᶠ (k : ι) in l, ‖F k‖ ≤ C
f : (a : α) → E a
hf : Tendsto (id fun i ↦ ↑(F i)) l (𝓝 f)
s : Finset α
hp' : p ≠ 0
hp'' : 0 < p.toReal
G : ((a : α) → E a) → ℝ := fun f ↦ ∑ a ∈ s, ‖f a‖ ^ p.toReal
⊢ Continuous[Pi.topologicalSpace, _] G

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.UniformLimitsDeriv

Goal state before:
pp: ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : ι → E → G
f' : ι → E → E →L[𝕜] G
x : E
r : ℝ
hf : ∀ (n : ι), ∀ y ∈ Metric.ball x r, HasFDerivAt (f n) (f' n y) y
hfg : Cauchy (map (fun n ↦ f n x) l)
this✝¹ : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
hf' : ∀ ε > 0, ∀ᶠ (n : ι × ι) in l ×ˢ l, ∀ x_1 ∈ Metric.ball x r, dist (0 x_1) (-f' n.1 x_1 + f' n.2 x_1) < ε
this✝ : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
this : l.NeBot
hr : 0 < r
ε : ℝ
hε : ε > 0
⊢ ∃ q, 0 < q ∧ q * r < ε

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝¹ : (i : α) → NormedAddCommGroup (E i)
ι : Type u_5
l : Filter ι
inst✝ : l.NeBot
_i : Fact (1 ≤ p)
hp : p ≠ ∞
C : ℝ
F : ι → ↥(lp E p)
hCF : ∀ᶠ (k : ι) in l, ‖F k‖ ≤ C
f : (a : α) → E a
hf : Tendsto (id fun i ↦ ↑(F i)) l (𝓝 f)
s : Finset α
⊢ ∑ i ∈ s, ‖f i‖ ^ p.toReal ≤ C ^ p.toReal

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝¹ : (i : α) → NormedAddCommGroup (E i)
ι : Type u_5
l : Filter ι
inst✝ : l.NeBot
_i : Fact (1 ≤ p)
C : ℝ
F : ι → ↥(lp E p)
hCF : ∀ᶠ (k : ι) in l, ‖F k‖ ≤ C
f : ↥(lp E p)
hf : Tendsto (id fun i ↦ ↑(F i)) l (𝓝 ↑f)
⊢ ‖f‖ ≤ C

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝¹ : (i : α) → NormedAddCommGroup (E i)
ι : Type u_5
l : Filter ι
inst✝ : l.NeBot
_i : Fact (1 ≤ p)
F : ι → ↥(lp E p)
hF : Bornology.IsBounded (Set.range F)
f : (a : α) → E a
hf : Tendsto (id fun i ↦ ↑(F i)) l (𝓝 f)
⊢ Memℓp f p

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.UniformLimitsDeriv

Goal state before:
pp: ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : ι → E → G
f' : ι → E → E →L[𝕜] G
x : E
r : ℝ
hf : ∀ (n : ι), ∀ y ∈ Metric.ball x r, HasFDerivAt (f n) (f' n y) y
hfg : Cauchy (map (fun n ↦ f n x) l)
this✝¹ : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
hf' : ∀ ε > 0, ∀ᶠ (n : ι × ι) in l ×ˢ l, ∀ x_1 ∈ Metric.ball x r, dist (0 x_1) (-f' n.1 x_1 + f' n.2 x_1) < ε
this✝ : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
this : l.NeBot
hr : 0 < r
ε : ℝ
hε : ε > 0
q : ℝ
hqpos : 0 < q
hq : q * r < ε
n : ι × ι
y : E
hy : y ∈ Metric.ball x r
hn : ∀ x_1 ∈ Metric.ball x r, ‖f' n.1 x_1 - f' n.2 x_1‖ < q
mvt : ‖(f n.1 - f n.2) y - (f n.1 - f n.2) x‖ ≤ q * ‖y - x‖
⊢ ‖y - x‖ < r

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
_i : Fact (1 ≤ p)
F : ℕ → ↥(lp E p)
hF : CauchySeq F
f : ↥(lp E p)
hf : Tendsto (id fun i ↦ ↑(F i)) atTop (𝓝 ↑f)
⊢ Tendsto F atTop (𝓝 f)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.UniformLimitsDeriv

Goal state before:
pp: ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : ι → E → G
f' : ι → E → E →L[𝕜] G
x : E
r : ℝ
hf' : UniformCauchySeqOn f' l (Metric.ball x r)
hf : ∀ (n : ι), ∀ y ∈ Metric.ball x r, HasFDerivAt (f n) (f' n y) y
hfg : Cauchy (map (fun n ↦ f n x) l)
⊢ UniformCauchySeqOn f l (Metric.ball x r)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: 𝕜 : Type u_1
𝕜' : Type u_2
α : Type u_3
E : α → Type u_4
p q : ℝ≥0∞
inst✝² : (i : α) → NormedAddCommGroup (E i)
ι : Type u_5
l : Filter ι
inst✝¹ : l.NeBot
_i : Fact (1 ≤ p)
inst✝ : ∀ (a : α), CompleteSpace (E a)
⊢ ∀ (u : ℕ → ↥(lp E p)), CauchySeq u → ∃ a, Tendsto u atTop (𝓝 a)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
ι : Type u_5
inst✝ : PseudoMetricSpace α
g : α → ι → ℝ
K : ℝ≥0
hg : ∀ (i : ι), LipschitzWith K fun x ↦ g x i
a₀ a : α
M : ℝ
hM : M ∈ upperBounds (Set.range fun i ↦ ‖g a₀ i‖)
i : ι
⊢ |g a i| = |g a i - g a₀ i + g a₀ i|

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
ι : Type u_5
inst✝ : PseudoMetricSpace α
g : α → ι → ℝ
K : ℝ≥0
hg : ∀ (i : ι), LipschitzWith K fun x ↦ g x i
a₀ a : α
M : ℝ
hM : M ∈ upperBounds (Set.range fun i ↦ ‖g a₀ i‖)
i : ι
⊢ |g a i - g a₀ i| + |g a₀ i| ≤ ↑K * dist a a₀ + M

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
ι : Type u_5
inst✝ : PseudoMetricSpace α
g : α → ι → ℝ
K : ℝ≥0
hg : ∀ (i : ι), LipschitzWith K fun x ↦ g x i
a₀ : α
hga₀b : Memℓp (g a₀) ∞
a : α
⊢ Memℓp (g a) ∞

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
ι : Type u_5
inst✝ : PseudoMetricSpace α
f : α → ↥(lp (fun x ↦ ℝ) ∞)
s : Set α
K : ℝ≥0
hfl : ∀ x ∈ s, ∀ y ∈ s, dist (f x) (f y) ≤ ↑K * dist x y
i : ι
x : α
hx : x ∈ s
y : α
hy : y ∈ s
⊢ dist (↑(f x) i) (↑(f y) i) ≤ dist (f x) (f y)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.UniformLimitsDeriv

Goal state before:
pp: ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : ι → E → G
f' : ι → E → E →L[𝕜] G
s : Set E
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
h's : IsPreconnected s
hf' : UniformCauchySeqOn f' l s
hf : ∀ (n : ι), ∀ y ∈ s, HasFDerivAt (f n) (f' n y) y
x₀ x : E
hx₀ : x₀ ∈ s
hx : x ∈ s
hfg : Cauchy (map (fun n ↦ f n x₀) l)
this : l.NeBot
t : Set E := {y | y ∈ s ∧ Cauchy (map (fun n ↦ f n y) l)}
A : ∀ (x : E) (ε : ℝ), x ∈ t → Metric.ball x ε ⊆ s → Metric.ball x ε ⊆ t
⊢ IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
ι : Type u_5
inst✝ : PseudoMetricSpace α
f : α → ↥(lp (fun x ↦ ℝ) ∞)
s : Set α
K : ℝ≥0
⊢ LipschitzOnWith K f s ↔ ∀ (i : ι), LipschitzOnWith K (fun a ↦ ↑(f a) i) s

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: α : Type u_3
ι : Type u_5
inst✝ : PseudoMetricSpace α
f : α → ↥(lp (fun x ↦ ℝ) ∞)
K : ℝ≥0
⊢ LipschitzWith K f ↔ ∀ (i : ι), LipschitzWith K fun a ↦ ↑(f a) i

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.PolynomialExp

Goal state before:
pp: p : ℝ[X]
⊢ Tendsto (fun x ↦ eval x p / rexp x) atTop (𝓝 0)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.SmoothTransition

Goal state before:
pp: x : ℝ
hx : x ≤ 0
⊢ expNegInvGlue x = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.SmoothTransition

Goal state before:
pp: x : ℝ
hx : 0 < x
⊢ 0 < expNegInvGlue x

Tactic:
simp [expNegInvGlue, not_le.2 hx, exp_pos]

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.SmoothTransition

Goal state before:
pp: x : ℝ
hx : 0 < x
⊢ 0 < expNegInvGlue x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.SmoothTransition

Goal state before:
pp: x : ℝ
⊢ 0 ≤ expNegInvGlue x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.SmoothTransition

Goal state before:
pp: ⊢ Monotone expNegInvGlue

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.SmoothTransition

Goal state before:
pp: h✝ : AnalyticAt ℝ expNegInvGlue 0
r : ℝ
hr : 0 < r
h : AnalyticOnNhd ℝ expNegInvGlue (Metric.ball 0 r)
this : expNegInvGlue (r / 2) = 0
⊢ False

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.SmoothTransition

Goal state before:
pp: h✝ : AnalyticAt ℝ expNegInvGlue 0
r : ℝ
hr : 0 < r
h : AnalyticOnNhd ℝ expNegInvGlue (Metric.ball 0 r)
⊢ 0 ∈ Metric.ball 0 r

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.SmoothTransition

Goal state before:
pp: h✝ : AnalyticAt ℝ expNegInvGlue 0
r : ℝ
hr : 0 < r
h : AnalyticOnNhd ℝ expNegInvGlue (Metric.ball 0 r)
⊢ 0 ∈ closure ({z | expNegInvGlue z = 0} \ {0})

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.SmoothTransition

Goal state before:
pp: h✝ : AnalyticAt ℝ expNegInvGlue 0
r : ℝ
hr : 0 < r
h : AnalyticOnNhd ℝ expNegInvGlue (Metric.ball 0 r)
⊢ r / 2 ∈ Metric.ball 0 r

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.SmoothTransition

Goal state before:
pp: ⊢ ¬AnalyticAt ℝ expNegInvGlue 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.SmoothTransition

Goal state before:
pp: p : ℝ[X]
⊢ Tendsto (fun x ↦ Polynomial.eval x⁻¹ p * expNegInvGlue x) (𝓝 0) (𝓝 0)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.Calculus.UniformLimitsDeriv

Goal state before:
pp: ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedSpace 𝕜 E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
f : ι → E → G
f' : ι → E → E →L[𝕜] G
s : Set E
hs : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
h's : IsPreconnected s
hf' : UniformCauchySeqOn f' l s
hf : ∀ (n : ι), ∀ y ∈ s, HasFDerivAt (f n) (f' n y) y
x₀ x✝ : E
hx₀ : x₀ ∈ s
hx : x✝ ∈ s
hfg : Cauchy (map (fun n ↦ f n x₀) l)
this : l.NeBot
t : Set E := {y | y ∈ s ∧ Cauchy (map (fun n ↦ f n y) l)}
A : ∀ (x : E) (ε : ℝ), x ∈ t → Metric.ball x ε ⊆ s → Metric.ball x ε ⊆ t
open_t : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t
st_nonempty : (s ∩ t).Nonempty
x : E
xt : x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t
xs : x ∈ s
ε : ℝ
εpos : ε > 0
hε : Metric.ball x ε ⊆ s
y : E
yt : y ∈ t
hxy : dist x y < ε / 2
⊢ Metric.ball y (ε / 2) ⊆ Metric.ball x ε

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.SmoothTransition

Goal state before:
pp: p : ℝ[X]
x : ℝ
⊢ HasDerivAt (fun x ↦ Polynomial.eval x⁻¹ p * expNegInvGlue x)
    (Polynomial.eval x⁻¹ (X ^ 2 * (p - derivative p)) * expNegInvGlue x) x

Tactic:
by

Goal state after: