mathlib-initiative/mathlib-tactics huggingface dataset:
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✝² : 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✝¹ : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
this✝ :
  TendstoUniformlyOnFilter (fun n z ↦ f n.1 z - f n.2 z - (f n.1 x✝ - f n.2 x✝)) 0 (l ×ˢ l) (𝓝 x✝) ∧
    TendstoUniformlyOnFilter (fun n x ↦ f n.1 x✝ - f n.2 x✝) 0 (l ×ˢ l) (𝓝 x✝)
this :
  TendstoUniformlyOnFilter ((-fun n z ↦ f n.1 z - f n.2 z - (f n.1 x✝ - f n.2 x✝)) + -fun n x ↦ f n.1 x✝ - f n.2 x✝) 0
    (l ×ˢ l) (𝓝 x✝)
x : (ι × ι) × E
⊢ ((-fun n z ↦ f n.1 z - f n.2 z - (f n.1 x✝ - f n.2 x✝)) + -fun n x ↦ f n.1 x✝ - f n.2 x✝) x.1 x.2 =
    -f x.1.1 x.2 + f x.1.2 x.2

Tactic:
simp; abel

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)
hp : 0 < p.toReal
C : ℝ
hC : 0 ≤ C
f : ↥(lp E p)
hf : ∑' (i : α), ‖↑f i‖ ^ p.toReal ≤ C ^ p.toReal
⊢ ‖f‖ ≤ C

Tactic:
← Real.rpow_le_rpow_iff (norm_nonneg' _) hC hp,

Goal state after:
α : Type u_3
E : α → Type u_4
p : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
hp : 0 < p.toReal
C : ℝ
hC : 0 ≤ C
f : ↥(lp E p)
hf : ∑' (i : α), ‖↑f i‖ ^ p.toReal ≤ C ^ p.toReal
⊢ ‖f‖ ^ p.toReal ≤ C ^ p.toReal
Module: Mathlib.Analysis.Normed.Module.Ball.Homeomorph

Goal state before:
pp: E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
y : E
hy : y ∈ ball 0 1
⊢ (√(1 + ‖(√(1 - ‖y‖ ^ 2))⁻¹ • y‖ ^ 2))⁻¹ • (√(1 - ‖y‖ ^ 2))⁻¹ • y = y

Tactic:
by
    have : 0 < 1 - ‖y‖ ^ 2 := by nlinarith [norm_nonneg y, mem_ball_zero_iff.1 hy]
    match_scalars
    simp [norm_smul]
    field_simp
    simp [field, sq_abs, Real.sq_sqrt this.le]

Goal state after:
Module: Mathlib.Analysis.Normed.Module.Ball.Homeomorph

Goal state before:
pp: E : Type u_1
inst✝³ : SeminormedAddCommGroup E
inst✝² : NormedSpace ℝ E
P : Type u_2
inst✝¹ : PseudoMetricSpace P
inst✝ : NormedAddTorsor E P
c : P
r : ℝ
hr : 0 < r
⊢ (univBall c r).target = ball c r

Tactic:
rw [univBall, dif_pos hr]; rfl

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:
simp; abel

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : NormedSpace ℝ E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
f g : G → E
s : Set G
x : G
n : WithTop ℕ∞
hf : ContDiffWithinAt ℝ n f s x
hg : ContDiffWithinAt ℝ n g s x
hne : f x ≠ g x
⊢ ContDiffWithinAt ℝ n (fun y ↦ Dist.dist (f y) (g y)) s x

Tactic:
by
  simp only [dist_eq_norm]; exact (hf.sub hg).norm 𝕜 (sub_ne_zero.2 hne)

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁵ : RCLike 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : InnerProductSpace 𝕜 E
inst✝² : NormedSpace ℝ E
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace ℝ G
f g : G → E
x : G
hf : DifferentiableAt ℝ f x
hg : DifferentiableAt ℝ g x
hne : f x ≠ g x
⊢ DifferentiableAt ℝ (fun y ↦ Dist.dist (f y) (g y)) x

Tactic:
by
  simp only [dist_eq_norm]; exact (hf.sub hg).norm 𝕜 (sub_ne_zero.2 hne)

Goal state after:
Module: Mathlib.Analysis.Convolution

Goal state before:
pp: 𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
inst✝⁹ : NormedAddCommGroup E
inst✝⁸ : NormedAddCommGroup E'
inst✝⁷ : NormedAddCommGroup F
f : G → E
g g' : G → E'
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : NormedSpace 𝕜 E
inst✝⁴ : NormedSpace 𝕜 E'
inst✝³ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
inst✝² : MeasurableSpace G
μ : Measure G
inst✝¹ : NormedSpace ℝ F
inst✝ : AddGroup G
hfg : ConvolutionExists f g L μ
hfg' : ConvolutionExists f g' L μ
x : G
⊢ (f ⋆[L, μ] (g + g')) x = (f ⋆[L, μ] g + f ⋆[L, μ] g') x

Tactic:
exact (hfg x).distrib_add (hfg' x)

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

Goal state before:
pp: case refine_1
ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : IsRCLikeNormedField 𝕜
inst✝³ : NormedSpace 𝕜 E
G : Type u_4
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
f : ι → E → G
g : E → G
f' : ι → E → E →L[𝕜] G
g' : E → E →L[𝕜] G
x : E
inst✝ : l.NeBot
hf' : TendstoUniformlyOnFilter f' g' 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 : ∀ᶠ (y : E) in 𝓝 x, Tendsto (fun n ↦ f n y) l (𝓝 (g y))
this✝² : RCLike 𝕜 := ⋯
this✝¹ :
  (fun a ↦ (↑‖a.2 - x‖)⁻¹ • (g a.2 - g x - (g' x) (a.2 - x))) =
    ((fun a ↦ (↑‖a.2 - x‖)⁻¹ • (g a.2 - g x - (f a.1 a.2 - f a.1 x))) + fun a ↦
        (↑‖a.2 - x‖)⁻¹ • (f a.1 a.2 - f a.1 x - ((f' a.1 x) a.2 - (f' a.1 x) x))) +
      fun a ↦ (↑‖a.2 - x‖)⁻¹ • (f' a.1 x - g' x) (a.2 - x)
this✝ : 𝓝 0 = 𝓝 (0 + 0 + 0)
this :
  ∀ ε > 0,
    ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, dist ((↑‖n.2 - x‖)⁻¹ • (g n.2 - g x)) ((↑‖n.2 - x‖)⁻¹ • (f n.1 n.2 - f n.1 x)) < ε
ε : ℝ
hε : ε > 0
⊢ ∀ (x_1 : ι × E),
    dist ((↑‖x_1.2 - x‖)⁻¹ • (g x_1.2 - g x)) ((↑‖x_1.2 - x‖)⁻¹ • (f x_1.1 x_1.2 - f x_1.1 x)) < ε →
      dist ((↑‖x_1.2 - x‖)⁻¹ • (g x_1.2 - g x - (f x_1.1 x_1.2 - f x_1.1 x))) 0 < ε

Tactic:
intro n hn

Goal state after:
case refine_1
ι : Type u_1
l : Filter ι
E : Type u_2
inst✝⁶ : NormedAddCommGroup E
𝕜 : Type u_3
inst✝⁵ : NontriviallyNormedField 𝕜
inst✝⁴ : IsRCLikeNormedField 𝕜
inst✝³ : NormedSpace 𝕜 E
G : Type u_4
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
f : ι → E → G
g : E → G
f' : ι → E → E →L[𝕜] G
g' : E → E →L[𝕜] G
x : E
inst✝ : l.NeBot
hf' : TendstoUniformlyOnFilter f' g' 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 : ∀ᶠ (y : E) in 𝓝 x, Tendsto (fun n ↦ f n y) l (𝓝 (g y))
this✝² : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
this✝¹ :
  (fun a ↦ (↑‖a.2 - x‖)⁻¹ • (g a.2 - g x - (g' x) (a.2 - x))) =
    ((fun a ↦ (↑‖a.2 - x‖)⁻¹ • (g a.2 - g x - (f a.1 a.2 - f a.1 x))) + fun a ↦
        (↑‖a.2 - x‖)⁻¹ • (f a.1 a.2 - f a.1 x - ((f' a.1 x) a.2 - (f' a.1 x) x))) +
      fun a ↦ (↑‖a.2 - x‖)⁻¹ • (f' a.1 x - g' x) (a.2 - x)
this✝ : 𝓝 0 = 𝓝 (0 + 0 + 0)
this :
  ∀ ε > 0,
    ∀ᶠ (n : ι × E) in l ×ˢ 𝓝 x, dist ((↑‖n.2 - x‖)⁻¹ • (g n.2 - g x)) ((↑‖n.2 - x‖)⁻¹ • (f n.1 n.2 - f n.1 x)) < ε
ε : ℝ
hε : ε > 0
n : ι × E
hn : dist ((↑‖n.2 - x‖)⁻¹ • (g n.2 - g x)) ((↑‖n.2 - x‖)⁻¹ • (f n.1 n.2 - f n.1 x)) < ε
⊢ dist ((↑‖n.2 - x‖)⁻¹ • (g n.2 - g x - (f n.1 n.2 - f n.1 x))) 0 < ε
Module: Mathlib.Analysis.Convolution

Goal state before:
pp: 𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedAddCommGroup E'
inst✝¹¹ : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝¹⁰ : NontriviallyNormedField 𝕜
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : NormedSpace 𝕜 E'
inst✝⁷ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
inst✝⁶ : MeasurableSpace G
μ : Measure G
inst✝⁵ : NormedSpace ℝ F
inst✝⁴ : AddCommGroup G
inst✝³ : μ.IsAddLeftInvariant
inst✝² : μ.IsNegInvariant
inst✝¹ : MeasurableNeg G
inst✝ : MeasurableAdd G
x : G
⊢ ∫ (x_1 : G), (L.flip (g (x - x_1))) (f (x - (x - x_1))) ∂μ = ∫ (t : G), (L (f t)) (g (x - t)) ∂μ

Tactic:
simp_rw [sub_sub_self, flip_apply]

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

Goal state before:
pp: ι : Type u_1
l : Filter ι
𝕜 : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
G : Type u_3
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
f f' : ι → 𝕜 → G
x : 𝕜
inst✝ : IsRCLikeNormedField 𝕜
hf' : UniformCauchySeqOnFilter f' l (𝓝 x)
hf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasDerivAt (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:
hasDerivAt_iff_hasFDerivAt

Goal state after:
ι : Type u_1
l : Filter ι
𝕜 : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
G : Type u_3
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
f f' : ι → 𝕜 → G
x : 𝕜
inst✝ : IsRCLikeNormedField 𝕜
hf' : UniformCauchySeqOnFilter f' l (𝓝 x)
hfg : Cauchy (map (fun n ↦ f n x) l)
hf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (ContinuousLinearMap.toSpanSingleton 𝕜 (f' n.1 n.2)) n.2
⊢ UniformCauchySeqOnFilter f l (𝓝 x)
Module: Mathlib.Analysis.Calculus.UniformLimitsDeriv

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

Tactic:
hasDerivAt_iff_hasFDerivAt

Goal state after:
ι : Type u_1
l : Filter ι
𝕜 : Type u_2
inst✝³ : NontriviallyNormedField 𝕜
G : Type u_3
inst✝² : NormedAddCommGroup G
inst✝¹ : NormedSpace 𝕜 G
f f' : ι → 𝕜 → G
x : 𝕜
inst✝ : IsRCLikeNormedField 𝕜
r : ℝ
hf' : UniformCauchySeqOn f' l (Metric.ball x r)
hfg : Cauchy (map (fun n ↦ f n x) l)
hf : ∀ (n : ι), ∀ y ∈ Metric.ball x r, HasFDerivAt (f n) (ContinuousLinearMap.toSpanSingleton 𝕜 (f' n y)) y
⊢ UniformCauchySeqOn f l (Metric.ball x r)
Module: Mathlib.Analysis.Calculus.UniformLimitsDeriv

Goal state before:
pp: ι : Type u_1
l : Filter ι
𝕜 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
G : Type u_3
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
f : ι → 𝕜 → G
g : 𝕜 → G
f' : ι → 𝕜 → G
g' : 𝕜 → G
x : 𝕜
inst✝¹ : IsRCLikeNormedField 𝕜
inst✝ : l.NeBot
hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x)
hf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasDerivAt (f n.1) (f' n.1 n.2) n.2
hfg : ∀ᶠ (y : 𝕜) in 𝓝 x, Tendsto (fun n ↦ f n y) l (𝓝 (g y))
F' : ι → 𝕜 → 𝕜 →L[𝕜] G := fun n z ↦ ContinuousLinearMap.smulRight 1 (f' n z)
G' : 𝕜 → 𝕜 →L[𝕜] G := fun z ↦ ContinuousLinearMap.smulRight 1 (g' z)
⊢ HasDerivAt g (g' x) x

Tactic:
hasDerivAt_iff_hasFDerivAt

Goal state after:
ι : Type u_1
l : Filter ι
𝕜 : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
G : Type u_3
inst✝³ : NormedAddCommGroup G
inst✝² : NormedSpace 𝕜 G
f : ι → 𝕜 → G
g : 𝕜 → G
f' : ι → 𝕜 → G
g' : 𝕜 → G
x : 𝕜
inst✝¹ : IsRCLikeNormedField 𝕜
inst✝ : l.NeBot
hf' : TendstoUniformlyOnFilter f' g' l (𝓝 x)
hfg : ∀ᶠ (y : 𝕜) in 𝓝 x, Tendsto (fun n ↦ f n y) l (𝓝 (g y))
F' : ι → 𝕜 → 𝕜 →L[𝕜] G := fun n z ↦ ContinuousLinearMap.smulRight 1 (f' n z)
G' : 𝕜 → 𝕜 →L[𝕜] G := fun z ↦ ContinuousLinearMap.smulRight 1 (g' z)
hf : ∀ᶠ (n : ι × 𝕜) in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (ContinuousLinearMap.toSpanSingleton 𝕜 (f' n.1 n.2)) n.2
⊢ HasFDerivAt g (ContinuousLinearMap.toSpanSingleton 𝕜 (g' x)) x
Module: Mathlib.Analysis.Calculus.ParametricIntegral

Goal state before:
pp: case pos
α : Type u_1
inst✝⁴ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace 𝕜 E
bound : α → ℝ
F : 𝕜 → α → E
x₀ : 𝕜
s : Set 𝕜
F' : α → E
hs : s ∈ 𝓝 x₀
hF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) μ
hF_int : Integrable (F x₀) μ
hF'_meas : AEStronglyMeasurable F' μ
h_lipsch : ∀ᵐ (a : α) ∂μ, LipschitzOnWith (Real.nnabs (bound a)) (fun x ↦ F x a) s
bound_integrable : Integrable bound μ
L : E →L[𝕜] 𝕜 →L[𝕜] E := (ContinuousLinearMap.smulRightL 𝕜 𝕜 E) 1
h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x ↦ F x a) (L (F' a)) x₀
hm : AEStronglyMeasurable (⇑L ∘ F') μ
key : HasFDerivAt (fun x ↦ ∫ (a : α), F x a ∂μ) (∫ (a : α), (⇑L ∘ F') a ∂μ) x₀
hF'_int : Integrable F' μ
hE : CompleteSpace E
⊢ HasDerivAt (fun x ↦ ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀

Tactic:
hasDerivAt_iff_hasFDerivAt

Goal state after:
case pos
α : Type u_1
inst✝⁴ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝³ : RCLike 𝕜
E : Type u_3
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
inst✝ : NormedSpace 𝕜 E
bound : α → ℝ
F : 𝕜 → α → E
x₀ : 𝕜
s : Set 𝕜
F' : α → E
hs : s ∈ 𝓝 x₀
hF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) μ
hF_int : Integrable (F x₀) μ
hF'_meas : AEStronglyMeasurable F' μ
h_lipsch : ∀ᵐ (a : α) ∂μ, LipschitzOnWith (Real.nnabs (bound a)) (fun x ↦ F x a) s
bound_integrable : Integrable bound μ
L : E →L[𝕜] 𝕜 →L[𝕜] E := (ContinuousLinearMap.smulRightL 𝕜 𝕜 E) 1
h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x ↦ F x a) (L (F' a)) x₀
hm : AEStronglyMeasurable (⇑L ∘ F') μ
key : HasFDerivAt (fun x ↦ ∫ (a : α), F x a ∂μ) (∫ (a : α), (⇑L ∘ F') a ∂μ) x₀
hF'_int : Integrable F' μ
hE : CompleteSpace E
⊢ HasFDerivAt (fun x ↦ ∫ (a : α), F x a ∂μ) (ContinuousLinearMap.toSpanSingleton 𝕜 (∫ (a : α), F' a ∂μ)) x₀
Module: Mathlib.MeasureTheory.Measure.Haar.NormedSpace

Goal state before:
pp: E : Type u_1
inst✝⁷ : NormedAddCommGroup E
inst✝⁶ : NormedSpace ℝ E
inst✝⁵ : MeasurableSpace E
inst✝⁴ : BorelSpace E
inst✝³ : FiniteDimensional ℝ E
μ : Measure E
inst✝² : μ.IsAddHaarMeasure
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
R : ℝ
s : Set E
hR : R ≠ 0
e : E ≃ᵐ E := (Homeomorph.smul (Units.mk0 R hR)).toMeasurableEquiv
⊢ ∫ (x : E) in s, f (R • x) ∂μ = |(R ^ finrank ℝ E)⁻¹| • ∫ (x : E) in R • s, f x ∂μ

Tactic:
calc
  ∫ x in s, f (R • x) ∂μ
    = ∫ x in e ⁻¹' e.symm ⁻¹' s, f (e x) ∂μ := by simp [← preimage_comp]; rfl
  _ = ∫ y in e.symm ⁻¹' s, f y ∂map (fun x ↦ R • x) μ := (setIntegral_map_equiv _ _ _).symm
  _ = |(R ^ finrank ℝ E)⁻¹| • ∫ y in e.symm ⁻¹' s, f y ∂μ := by
    simp [map_addHaar_smul μ hR, integral_smul_measure, ENNReal.toReal_ofReal, abs_nonneg]
  _ = |(R ^ finrank ℝ E)⁻¹| • ∫ x in R • s, f x ∂μ := by
    congr 3
    ext y
    rw [mem_smul_set_iff_inv_smul_mem₀ hR]
    rfl

Goal state after:
Module: Mathlib.MeasureTheory.Measure.EverywherePos

Goal state before:
pp: α : Type u_1
inst✝¹ : TopologicalSpace α
inst✝ : MeasurableSpace α
μ : Measure α
s : Set α
⊢ μ.everywherePosSubset s = s \ {x | ∃ n ∈ 𝓝[s] x, μ n = 0}

Tactic:
ext x; simp [everywherePosSubset, pos_iff_ne_zero]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.EverywherePos

Goal state before:
pp: G : Type u_2
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : LocallyCompactSpace G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : μ.InnerRegularCompactLTTop
⊢ μ.InnerRegularWRT (fun s ↦ ∃ f, Continuous[inst✝⁷, _] f ∧ HasCompactSupport f ∧ s = f ⁻¹' {1}) fun s ↦
    IsCompact s ∧ IsClosed[inst✝⁷] s

Tactic:
intro K

Goal state after:
G : Type u_2
inst✝⁸ : Group G
inst✝⁷ : TopologicalSpace G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : LocallyCompactSpace G
inst✝⁴ : MeasurableSpace G
inst✝³ : BorelSpace G
μ : Measure G
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : IsFiniteMeasureOnCompacts μ
inst✝ : μ.InnerRegularCompactLTTop
K : Set G
⊢ (fun s ↦ IsCompact s ∧ IsClosed[inst✝⁷] s) K →
    ∀ r < μ K, ∃ K_1 ⊆ K, (fun s ↦ ∃ f, Continuous[inst✝⁷, _] f ∧ HasCompactSupport f ∧ s = f ⁻¹' {1}) K_1 ∧ r < μ K_1
Module: Mathlib.Analysis.Calculus.BumpFunction.Normed

Goal state before:
pp: E : Type u_1
inst✝⁶ : NormedAddCommGroup E
inst✝⁵ : NormedSpace ℝ E
inst✝⁴ : HasContDiffBump E
inst✝³ : MeasurableSpace E
c : E
f : ContDiffBump c
μ : Measure E
inst✝² : BorelSpace E
inst✝¹ : FiniteDimensional ℝ E
inst✝ : IsLocallyFiniteMeasure μ
⊢ ∫ (x : E), ↑f x ∂μ ≤ μ.real (closedBall c f.rOut)

Tactic:
calc
  ∫ x, f x ∂μ = ∫ x in closedBall c f.rOut, f x ∂μ := by
    apply (setIntegral_eq_integral_of_forall_compl_eq_zero (fun x hx ↦ ?_)).symm
    apply f.zero_of_le_dist (le_of_lt _)
    simpa using hx
  _ ≤ ∫ x in closedBall c f.rOut, 1 ∂μ := by
    apply setIntegral_mono f.integrable.integrableOn _ (fun x ↦ f.le_one)
    simp [measure_closedBall_lt_top]
  _ = μ.real (closedBall c f.rOut) := by simp

Goal state after:
Module: Mathlib.MeasureTheory.Covering.Besicovitch

Goal state before:
pp: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ y < i, y < p.lastStep → p.color y < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n ↦ if n = N then i else g n
⊢ ∀ n ≤ N, p.color (G n) = n

Tactic:
intro n hn

Goal state after:
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ y < i, y < p.lastStep → p.color y < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n ↦ if n = N then i else g n
n : ℕ
hn : n ≤ N
⊢ p.color (G n) = n
Module: Mathlib.MeasureTheory.Covering.Besicovitch

Goal state before:
pp: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ y < i, y < p.lastStep → p.color y < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n ↦ if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
⊢ ∀ n ≤ N, G n < p.lastStep

Tactic:
intro n hn

Goal state after:
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ y < i, y < p.lastStep → p.color y < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n ↦ if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
n : ℕ
hn : n ≤ N
⊢ G n < p.lastStep
Module: Mathlib.MeasureTheory.Covering.Besicovitch

Goal state before:
pp: α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ y < i, y < p.lastStep → p.color y < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n ↦ if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
⊢ ∀ n ≤ N, p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))

Tactic:
intro n hn

Goal state after:
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
N : ℕ
hN : IsEmpty (SatelliteConfig α N p.τ)
i : Ordinal.{u}
IH : ∀ y < i, y < p.lastStep → p.color y < N
hi : i < p.lastStep
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty),
      {p.color ↑j}
color_i : p.color i = sInf (univ \ A)
N_mem : N ∈ univ \ A
Inf_eq_N : sInf (univ \ A) = N
g : ℕ → Ordinal.{u}
hg :
  ∀ k < N,
    g k < i ∧
      (closedBall (p.c (p.index (g k))) (p.r (p.index (g k))) ∩
            closedBall (p.c (p.index i)) (p.r (p.index i))).Nonempty ∧
        k = p.color (g k)
G : ℕ → Ordinal.{u} := fun n ↦ if n = N then i else g n
color_G : ∀ n ≤ N, p.color (G n) = n
G_lt_last : ∀ n ≤ N, G n < p.lastStep
n : ℕ
hn : n ≤ N
⊢ p.c (p.index (G n)) ∉ p.iUnionUpTo (G n) ∧ p.R (G n) ≤ p.τ * p.r (p.index (G n))
Module: Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension

Goal state before:
pp: E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x : E
Dpos : 0 < D
D_lt_one : D < 1
hx : ‖x‖ < 1 + D
⊢ 0 < y D x

Tactic:
refine (integral_pos_iff_support_of_nonneg (w_mul_φ_nonneg D x) ?_).2 ?_

Goal state after:
case refine_1
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x : E
Dpos : 0 < D
D_lt_one : D < 1
hx : ‖x‖ < 1 + D
⊢ Integrable (fun i ↦ w D i * φ (x - i)) μ

case refine_2
E : Type u_1
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : FiniteDimensional ℝ E
inst✝¹ : MeasurableSpace E
inst✝ : BorelSpace E
D : ℝ
x : E
Dpos : 0 < D
D_lt_one : D < 1
hx : ‖x‖ < 1 + D
⊢ 0 < μ (support fun i ↦ w D i * φ (x - i))
Module: Mathlib.Analysis.Calculus.BumpFunction.SmoothApprox

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
f : C(E, F)
⊢ ∀ (a : Set E) (b : ℝ), IsCompact a → 0 < b → ∃ y, ContDiff ℝ ∞ ⇑y ∧ ∀ x ∈ a, dist (y x) (f x) < b

Tactic:
intro K

Goal state after:
E : Type u_1
F : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace ℝ E
inst✝³ : FiniteDimensional ℝ E
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace ℝ F
inst✝ : CompleteSpace F
f : C(E, F)
K : Set E
⊢ ∀ (b : ℝ), IsCompact K → 0 < b → ∃ y, ContDiff ℝ ∞ ⇑y ∧ ∀ x ∈ K, dist (y x) (f x) < b