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

Goal state before:
pp: α : Type u_1
inst✝⁶ : MeasurableSpace α
μ : Measure α
𝕜 : Type u_2
inst✝⁵ : RCLike 𝕜
E : Type u_3
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace ℝ E
inst✝² : NormedSpace 𝕜 E
H : Type u_4
inst✝¹ : NormedAddCommGroup H
inst✝ : NormedSpace 𝕜 H
F : H → α → E
x₀ : H
bound : α → ℝ
s : Set H
F' : α → H →L[𝕜] E
hs : s ∈ 𝓝 x₀
hF_meas : ∀ x ∈ s, AEStronglyMeasurable (F x) μ
hF_int : Integrable (F x₀) μ
hF'_meas : AEStronglyMeasurable F' μ
bound_integrable : Integrable bound μ
h_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x ↦ F x a) (F' a) x₀
ε : ℝ
ε_pos : ε > 0
hε : ball x₀ ε ⊆ s
x₀_in : x₀ ∈ ball x₀ ε
nneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹
b : α → ℝ := fun a ↦ |bound a|
b_int : Integrable b μ
b_nonneg : ∀ (a : α), 0 ≤ b a
h_lipsch : ∀ᵐ (a : α) ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
hF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ
hF'_int : Integrable F' μ
hE : CompleteSpace E
h_ball : ball x₀ ε ∈ 𝓝 x₀
this :
  ∀ᶠ (x : H) in 𝓝 x₀,
    ‖x - x₀‖⁻¹ * ‖∫ (a : α), F x a ∂μ - ∫ (a : α), F x₀ a ∂μ - (∫ (a : α), F' a ∂μ) (x - x₀)‖ =
      ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - (F' a) (x - x₀)) ∂μ‖
x : H
hx : x ∈ ball x₀ ε
a : α
ha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖
⊢ ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a) - ‖x - x₀‖⁻¹ • (F' a) (x - x₀)‖ =
    ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a) - ‖x - x₀‖⁻¹ • (F' a) (x - x₀)‖

Tactic:
]

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 := 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
⊢ Tendsto (fun a ↦ (↑‖a.2 - x‖)⁻¹ • (g a.2 - g x - (f a.1 a.2 - f a.1 x))) (l.curry (𝓝 x)) (𝓝 0)

Tactic:
]

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
⊢ Tendsto (fun a ↦ (↑‖a.2 - x‖)⁻¹ • (g a.2 - g x - (f a.1 a.2 - f a.1 x))) (l.curry (𝓝 x)) (𝓝 0)
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 := 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 > 0,
    ∀ᶠ (x_1 : ι × E) in l.curry (𝓝 x), 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:
]

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 > 0,
    ∀ᶠ (x_1 : ι × E) in l.curry (𝓝 x), 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
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 → 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
x : G
h2x : x ∈ support (f ⋆[L, μ] g)
hx : ∀ (x_1 x_2 : G), g x_1 = 0 ∨ f x_2 = 0 ∨ ¬x_1 + x_2 = x
⊢ ∫ (t : G), (L (f t)) (g (x - t)) ∂μ = 0

Tactic:
]

Goal state after:
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 → 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
x : G
h2x : x ∈ support (f ⋆[L, μ] g)
hx : ∀ (x_1 x_2 : G), g x_1 = 0 ∨ f x_2 = 0 ∨ ¬x_1 + x_2 = x
⊢ ∫ (t : G), (L (f t)) (g (x - t)) ∂μ = 0
Module: Mathlib.Analysis.Convolution

Goal state before:
pp: case e'_7.inl
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 → 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
x : G
h2x : x ∈ support (f ⋆[L, μ] g)
hx : ∀ (x_1 x_2 : G), g x_1 = 0 ∨ f x_2 = 0 ∨ ¬x_1 + x_2 = x
t : G
h : g (x - t) = 0
⊢ 0 = 0

Tactic:
]

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 := 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 : ‖(↑‖n.2 - x‖)⁻¹ • (g n.2 - g x) - (↑‖n.2 - x‖)⁻¹ • (f n.1 n.2 - f n.1 x)‖ < ε
⊢ ‖(↑‖n.2 - x‖)⁻¹ • (g n.2 - g x - (f n.1 n.2 - f n.1 x)) - 0‖ < ε

Tactic:
]

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 : ‖(↑‖n.2 - x‖)⁻¹ • (g n.2 - g x) - (↑‖n.2 - x‖)⁻¹ • (f n.1 n.2 - f n.1 x)‖ < ε
⊢ ‖(↑‖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: case e'_7.inr.inl
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 → 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
x : G
h2x : x ∈ support (f ⋆[L, μ] g)
hx : ∀ (x_1 x_2 : G), g x_1 = 0 ∨ f x_2 = 0 ∨ ¬x_1 + x_2 = x
t : G
h : f t = 0
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: case refine_2
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✝¹ : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
this✝ :
  (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)
⊢ ∀ ε > 0,
    ∀ᶠ (x_1 : ι × E) in l.curry (𝓝 x),
      dist ((↑‖x_1.2 - x‖)⁻¹ • (f x_1.1 x_1.2 - f x_1.1 x - ((f' x_1.1 x) x_1.2 - (f' x_1.1 x) x))) 0 < ε

Tactic:
]

Goal state after:
case refine_2
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✝¹ : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
this✝ :
  (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)
⊢ ∀ ε > 0,
    ∀ᶠ (x_1 : ι × E) in l.curry (𝓝 x),
      dist ((↑‖x_1.2 - x‖)⁻¹ • (f x_1.1 x_1.2 - f x_1.1 x - ((f' x_1.1 x) x_1.2 - (f' x_1.1 x) x))) 0 < ε
Module: Mathlib.Analysis.Calculus.UniformLimitsDeriv

Goal state before:
pp: case refine_2
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✝¹ : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
this✝ :
  (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)
ε : ℝ
hε : ε > 0
⊢ ∀ᶠ (x_1 : ι) in l,
    ∀ᶠ (y : E) in 𝓝 x,
      dist
          ((↑‖(x_1, y).2 - x‖)⁻¹ •
            (f (x_1, y).1 (x_1, y).2 - f (x_1, y).1 x - ((f' (x_1, y).1 x) (x_1, y).2 - (f' (x_1, y).1 x) x)))
          0 <
        ε

Tactic:
]

Goal state after:
case refine_2
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✝¹ : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
this✝ :
  (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)
ε : ℝ
hε : ε > 0
⊢ ∀ᶠ (x_1 : ι) in l,
    ∀ᶠ (y : E) in 𝓝 x,
      dist
          ((↑‖(x_1, y).2 - x‖)⁻¹ •
            (f (x_1, y).1 (x_1, y).2 - f (x_1, y).1 x - ((f' (x_1, y).1 x) (x_1, y).2 - (f' (x_1, y).1 x) x)))
          0 <
        ε
Module: Mathlib.Analysis.Calculus.UniformLimitsDeriv

Goal state before:
pp: case refine_2
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)
ε : ℝ
hε : ε > 0
n : ι
hn : ∀ᶠ (y : E) in 𝓝 x, HasFDerivAt (f (n, y).1) (f' (n, y).1 (n, y).2) (n, y).2
this :
  ∀ ε > 0,
    ∀ᶠ (x_1 : E) in 𝓝 (n, x).2,
      dist (‖x_1 - (n, x).2‖⁻¹ * ‖f (n, x).1 x_1 - f (n, x).1 (n, x).2 - (f' (n, x).1 (n, x).2) (x_1 - (n, x).2)‖) 0 < ε
⊢ ∀ᶠ (y : E) in 𝓝 x,
    dist ((↑‖(n, y).2 - x‖)⁻¹ • (f (n, y).1 (n, y).2 - f (n, y).1 x - ((f' (n, y).1 x) (n, y).2 - (f' (n, y).1 x) x)))
        0 <
      ε

Tactic:
]

Goal state after:
case refine_2
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)
ε : ℝ
hε : ε > 0
n : ι
hn : ∀ᶠ (y : E) in 𝓝 x, HasFDerivAt (f (n, y).1) (f' (n, y).1 (n, y).2) (n, y).2
this :
  ∀ ε > 0,
    ∀ᶠ (x_1 : E) in 𝓝 (n, x).2,
      dist (‖x_1 - (n, x).2‖⁻¹ * ‖f (n, x).1 x_1 - f (n, x).1 (n, x).2 - (f' (n, x).1 (n, x).2) (x_1 - (n, x).2)‖) 0 < ε
⊢ ∀ᶠ (y : E) in 𝓝 x,
    dist ((↑‖(n, y).2 - x‖)⁻¹ • (f (n, y).1 (n, y).2 - f (n, y).1 x - ((f' (n, y).1 x) (n, y).2 - (f' (n, y).1 x) x)))
        0 <
      ε
Module: Mathlib.Analysis.Calculus.UniformLimitsDeriv

Goal state before:
pp: case refine_2
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)
ε : ℝ
hε : ε > 0
n : ι
hn : ∀ᶠ (y : E) in 𝓝 x, HasFDerivAt (f (n, y).1) (f' (n, y).1 (n, y).2) (n, y).2
this :
  ∀ ε > 0,
    ∀ᶠ (x_1 : E) in 𝓝 (n, x).2,
      dist (‖x_1 - (n, x).2‖⁻¹ * ‖f (n, x).1 x_1 - f (n, x).1 (n, x).2 - (f' (n, x).1 (n, x).2) (x_1 - (n, x).2)‖) 0 < ε
y : E
hy : ‖‖y - (n, x).2‖⁻¹ * ‖f (n, x).1 y - f (n, x).1 (n, x).2 - (f' (n, x).1 (n, x).2) (y - (n, x).2)‖ - 0‖ < ε
⊢ ‖(↑‖(n, y).2 - x‖)⁻¹ • (f (n, y).1 (n, y).2 - f (n, y).1 x - ((f' (n, y).1 x) (n, y).2 - (f' (n, y).1 x) x)) - 0‖ < ε

Tactic:
]

Goal state after:
case refine_2
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)
ε : ℝ
hε : ε > 0
n : ι
hn : ∀ᶠ (y : E) in 𝓝 x, HasFDerivAt (f (n, y).1) (f' (n, y).1 (n, y).2) (n, y).2
this :
  ∀ ε > 0,
    ∀ᶠ (x_1 : E) in 𝓝 (n, x).2,
      dist (‖x_1 - (n, x).2‖⁻¹ * ‖f (n, x).1 x_1 - f (n, x).1 (n, x).2 - (f' (n, x).1 (n, x).2) (x_1 - (n, x).2)‖) 0 < ε
y : E
hy : ‖‖y - (n, x).2‖⁻¹ * ‖f (n, x).1 y - f (n, x).1 (n, x).2 - (f' (n, x).1 (n, x).2) (y - (n, x).2)‖ - 0‖ < ε
⊢ ‖(↑‖(n, y).2 - x‖)⁻¹ • (f (n, y).1 (n, y).2 - f (n, y).1 x - ((f' (n, y).1 x) (n, y).2 - (f' (n, y).1 x) x)) - 0‖ < ε
Module: Mathlib.Analysis.Calculus.ContDiff.Convolution

Goal state before:
pp: case inr.bound_integrable
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 : RCLike 𝕜
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : NormedSpace 𝕜 E'
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : MeasurableSpace G
μ : Measure G
L : E →L[𝕜] E' →L[𝕜] F
inst✝⁴ : NormedAddCommGroup G
inst✝³ : BorelSpace G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : SFinite μ
inst✝ : μ.IsAddLeftInvariant
hcg : HasCompactSupport g
hf : LocallyIntegrable f μ
hg : ContDiff 𝕜 1 g
x₀ : G
fin_dim : FiniteDimensional 𝕜 G
this : ProperSpace G
L' : E →L[𝕜] (G →L[𝕜] E') →L[𝕜] G →L[𝕜] F := precompR G L
h1 : ∀ᶠ (x : G) in 𝓝 x₀, AEStronglyMeasurable (fun t ↦ (L (f t)) (g (x - t))) μ
h2 : ∀ (x : G), AEStronglyMeasurable (fun t ↦ (L' (f t)) (fderiv 𝕜 g (x - t))) μ
h3 : ∀ (x t : G), HasFDerivAt (fun x ↦ g (x - t)) (fderiv 𝕜 g (x - t)) x
K' : Set G := -tsupport (fderiv 𝕜 g) + closedBall x₀ 1
hK' : IsCompact K'
⊢ IntegrableOn (fun t ↦ ‖L'‖ * ‖f t‖ * ⨆ i, ‖fderiv 𝕜 g i‖) K' μ

Tactic:
]

Goal state after:
case inr.bound_integrable
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 : RCLike 𝕜
inst✝⁹ : NormedSpace 𝕜 E
inst✝⁸ : NormedSpace 𝕜 E'
inst✝⁷ : NormedSpace ℝ F
inst✝⁶ : NormedSpace 𝕜 F
inst✝⁵ : MeasurableSpace G
μ : Measure G
L : E →L[𝕜] E' →L[𝕜] F
inst✝⁴ : NormedAddCommGroup G
inst✝³ : BorelSpace G
inst✝² : NormedSpace 𝕜 G
inst✝¹ : SFinite μ
inst✝ : μ.IsAddLeftInvariant
hcg : HasCompactSupport g
hf : LocallyIntegrable f μ
hg : ContDiff 𝕜 1 g
x₀ : G
fin_dim : FiniteDimensional 𝕜 G
this : ProperSpace G
L' : E →L[𝕜] (G →L[𝕜] E') →L[𝕜] G →L[𝕜] F := precompR G L
h1 : ∀ᶠ (x : G) in 𝓝 x₀, AEStronglyMeasurable (fun t ↦ (L (f t)) (g (x - t))) μ
h2 : ∀ (x : G), AEStronglyMeasurable (fun t ↦ (L' (f t)) (fderiv 𝕜 g (x - t))) μ
h3 : ∀ (x t : G), HasFDerivAt (fun x ↦ g (x - t)) (fderiv 𝕜 g (x - t)) x
K' : Set G := -tsupport (fderiv 𝕜 g) + closedBall x₀ 1
hK' : IsCompact K'
⊢ IntegrableOn (fun t ↦ ‖L'‖ * ‖f t‖ * ⨆ i, ‖fderiv 𝕜 g i‖) K' μ