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

Goal state before:
pp: α : Type u_1
𝕜 : Type u_3
F : Type u_5
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : IsRCLikeNormedField 𝕜
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
g g' : α → 𝕜 → F
y₀ : 𝕜
hu : Summable u
hg : ∀ (n : α) (y : 𝕜), HasDerivAt (g n) (g' n y) y
hg' : ∀ (n : α) (y : 𝕜), ‖g' n y‖ ≤ u n
hg0 : Summable fun n ↦ g n y₀
y : 𝕜
⊢ HasDerivAt (fun z ↦ ∑' (n : α), g n z) (∑' (n : α), g' n y) y

Tactic:
by
  exact hasDerivAt_tsum_of_isPreconnected hu isOpen_univ isPreconnected_univ
    (fun n y _ => hg n y) (fun n y _ => hg' n y) (mem_univ _) hg0 (mem_univ _)

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)
I : Type u_5
B : I → Type u_6
inst✝³ : (i : I) → NonUnitalNormedRing (B i)
inst✝² : (i : I) → StarRing (B i)
inst✝¹ : ∀ (i : I), NormedStarGroup (B i)
inst✝ : ∀ (i : I), CStarRing (B i)
f : ↥(lp B ∞)
i : I
⊢ ‖star (↑f i) * ↑f i‖ ≤ ‖star f * f‖

Tactic:
exact lp.norm_apply_le_norm ENNReal.top_ne_zero (star f * f) i

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

Goal state before:
pp: case inr.right
x : ℝ
hx✝ : x ≠ 0
hx : 0 < x
this : 2 * √x ^ (2 - 1) ≠ 0
⊢ ∀ (n : WithTop ℕ∞), ContDiffAt ℝ n (fun x ↦ √x) x

Tactic:
· exact fun n => sqPartialHomeomorph.contDiffAt_symm_deriv this hx (hasDerivAt_pow 2 (√x))
        (contDiffAt_id.pow 2)

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

Goal state before:
pp: case inr
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)
this✝¹ : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
this✝ : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
this : l.NeBot
hr : 0 < r
⊢ UniformCauchySeqOn f l (Metric.ball x r)

Tactic:
rw [SeminormedAddGroup.uniformCauchySeqOn_iff_tendstoUniformlyOn_zero] at hf' ⊢

Goal state after:
case inr
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' : TendstoUniformlyOn (fun n z ↦ -f' n.1 z + f' n.2 z) 0 (l ×ˢ l) (Metric.ball x r)
this✝ : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 E
this : l.NeBot
hr : 0 < r
⊢ TendstoUniformlyOn (fun n z ↦ -f n.1 z + f n.2 z) 0 (l ×ˢ l) (Metric.ball x r)
Module: Mathlib.Analysis.Calculus.UniformLimitsDeriv

Goal state before:
pp: case inr.left
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
hn : ∀ x_1 ∈ Metric.ball x r, dist (0 x_1) (-f' n.1 x_1 + f' n.2 x_1) < q
⊢ ∀ x_1 ∈ Metric.ball x r, dist (0 x_1) (f n.1 x_1 - f n.2 x_1 - (f n.1 x - f n.2 x)) < ε

Tactic:
y

Goal state after:
case inr.left
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
hn : ∀ x_1 ∈ Metric.ball x r, dist (0 x_1) (-f' n.1 x_1 + f' n.2 x_1) < q
y : E
⊢ y ∈ Metric.ball x r → dist (0 y) (f n.1 y - f n.2 y - (f n.1 x - f n.2 x)) < ε
Module: Mathlib.Analysis.Normed.Lp.lpSpace

Goal state before:
pp: case mp
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 ∈ s, dist (↑(f x) i) (↑(f y) i) ≤ ↑K * dist x y

Tactic:
y

Goal state after:
case mp
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
⊢ y ∈ s → dist (↑(f x) i) (↑(f y) i) ≤ ↑K * dist x y
Module: Mathlib.Analysis.Normed.Lp.lpSpace

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

Tactic:
y

Goal state after:
case mpr
α : Type u_3
ι : Type u_5
inst✝ : PseudoMetricSpace α
f : α → ↥(lp (fun x ↦ ℝ) ∞)
s : Set α
K : ℝ≥0
hgl : ∀ (i : ι), ∀ x ∈ s, ∀ y ∈ s, dist (↑(f x) i) (↑(f y) i) ≤ ↑K * dist x y
x : α
hx : x ∈ s
y : α
⊢ y ∈ s → dist (f x) (f y) ≤ ↑K * dist x y
Module: Mathlib.Analysis.SpecialFunctions.SmoothTransition

Goal state before:
pp: x : ℝ
⊢ ∀ ⦃b : ℝ⦄, x ≤ b → expNegInvGlue x ≤ expNegInvGlue b

Tactic:
y

Goal state after:
x y : ℝ
⊢ x ≤ y → expNegInvGlue x ≤ expNegInvGlue y
Module: Mathlib.Analysis.SpecialFunctions.SmoothTransition

Goal state before:
pp: x : ℝ
⊢ ∀ ⦃b : ℝ⦄, x ≤ b → x.smoothTransition ≤ b.smoothTransition

Tactic:
y

Goal state after:
x y : ℝ
⊢ x ≤ y → x.smoothTransition ≤ y.smoothTransition
Module: Mathlib.Analysis.Calculus.UniformLimitsDeriv

Goal state before:
pp: 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 : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
| Tendsto (fun y ↦ ‖y.2 - x‖⁻¹ * ‖g y.2 - g x - (g' x) (y.2 - x)‖) (l.curry (𝓝 x)) (𝓝 0)

Tactic:
congr

Goal state after:
case f
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 : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
| fun y ↦ ‖y.2 - x‖⁻¹ * ‖g y.2 - g x - (g' x) (y.2 - x)‖

case l₁
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 : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
| l.curry (𝓝 x)

case l₂
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 : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
| 𝓝 0
Module: Mathlib.Analysis.Calculus.UniformLimitsDeriv

Goal state before:
pp: 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 : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
x✝ : ι × E
⊢ (↑‖x✝.2 - x‖)⁻¹ • (g x✝.2 - g x - (g' x) (x✝.2 - x)) =
    (↑‖x✝.2 - x‖)⁻¹ •
      (g x✝.2 - g x - (f x✝.1 x✝.2 - f x✝.1 x) + (f x✝.1 x✝.2 - f x✝.1 x - ((f' x✝.1 x) x✝.2 - (f' x✝.1 x) x)) +
        (f' x✝.1 x - g' x) (x✝.2 - x))

Tactic:
congr

Goal state after:
case e_a
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 : RCLike 𝕜 := IsRCLikeNormedField.rclike 𝕜
x✝ : ι × E
⊢ g x✝.2 - g x - (g' x) (x✝.2 - x) =
    g x✝.2 - g x - (f x✝.1 x✝.2 - f x✝.1 x) + (f x✝.1 x✝.2 - f x✝.1 x - ((f' x✝.1 x) x✝.2 - (f' x✝.1 x) x)) +
      (f' x✝.1 x - g' x) (x✝.2 - x)
Module: Mathlib.Analysis.Convolution

Goal state before:
pp: case refine_1
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
inst✝¹⁴ : NormedAddCommGroup E
inst✝¹³ : NormedAddCommGroup E'
inst✝¹² : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝¹¹ : NontriviallyNormedField 𝕜
inst✝¹⁰ : NormedSpace 𝕜 E
inst✝⁹ : NormedSpace 𝕜 E'
inst✝⁸ : NormedSpace 𝕜 F
L : E →L[𝕜] E' →L[𝕜] F
inst✝⁷ : MeasurableSpace G
μ : Measure G
inst✝⁶ : NormedSpace ℝ F
inst✝⁵ : AddGroup G
inst✝⁴ : TopologicalSpace G
inst✝³ : IsTopologicalAddGroup G
inst✝² : BorelSpace G
inst✝¹ : FirstCountableTopology G
inst✝ : SecondCountableTopologyEither G E'
hbg : BddAbove (range fun x ↦ ‖g x‖)
hf : Integrable f μ
hg : Continuous[inst✝⁴, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
x₀ : G
this : ∀ᶠ (x : G) in 𝓝 x₀, ∀ᵐ (t : G) ∂μ, ‖(L (f t)) (g (x - t))‖ ≤ ‖L‖ * ‖f t‖ * ⨆ i, ‖g i‖
⊢ ∀ᶠ (x : G) in 𝓝 x₀, AEStronglyMeasurable (fun a ↦ (L (f a)) (g (x - a))) μ

Tactic:
exact Eventually.of_forall fun x =>
      hf.aestronglyMeasurable.convolution_integrand_snd' L hg.aestronglyMeasurable

Goal state after:
Module: Mathlib.Data.Set.Pointwise.Support

Goal state before:
pp: α : Type u_1
β : Type u_2
γ : Type u_3
inst✝² : GroupWithZero α
inst✝¹ : MulAction α β
inst✝ : Zero γ
c : α
hc : c ≠ 0
f : β → γ
⊢ (support fun x ↦ f (c⁻¹ • x)) = c • support f

Tactic:
ext x
  simp only [mem_smul_set_iff_inv_smul_mem₀ hc, mem_support]

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

Goal state before:
pp: 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
hs : s ∈ 𝓝 x₀
hF_meas : ∀ᶠ (x : 𝕜) in 𝓝 x₀, AEStronglyMeasurable (F x) μ
hF_int : Integrable (F x₀) μ
F' : 𝕜 → α → E
hF'_meas : AEStronglyMeasurable (F' x₀) μ
h_bound : ∀ᵐ (a : α) ∂μ, ∀ x ∈ s, ‖F' x a‖ ≤ bound a
bound_integrable : Integrable bound μ
h_diff : ∀ᵐ (a : α) ∂μ, ∀ x ∈ s, HasDerivAt (fun x ↦ F x a) (F' x a) x
⊢ Integrable (F' x₀) μ ∧ HasDerivAt (fun n ↦ ∫ (a : α), F n a ∂μ) (∫ (a : α), F' x₀ a ∂μ) x₀

Tactic:
rcases Metric.mem_nhds_iff.1 hs with ⟨ε, ε_pos, hε⟩
  have x₀_in : x₀ ∈ ball x₀ ε := mem_ball_self ε_pos
  have diff_x₀ : ∀ᵐ a ∂μ, HasDerivAt (F · a) (F' x₀ a) x₀ :=
    h_diff.mono fun a ha ↦ ha x₀ (hε x₀_in)
  have : ∀ᵐ a ∂μ, LipschitzOnWith (Real.nnabs (bound a)) (fun x : 𝕜 ↦ F x a) (ball x₀ ε) := by
    apply (h_diff.and h_bound).mono
    rintro a ⟨ha_deriv, ha_bound⟩
    refine (convex_ball _ _).lipschitzOnWith_of_nnnorm_hasDerivWithin_le
      (fun x x_in ↦ (ha_deriv x (hε x_in)).hasDerivWithinAt) fun x x_in ↦ ?_
    rw [← NNReal.coe_le_coe, coe_nnnorm, Real.coe_nnabs]
    exact (ha_bound x (hε x_in)).trans (le_abs_self _)
  exact hasDerivAt_integral_of_dominated_loc_of_lip (ball_mem_nhds x₀ ε_pos) hF_meas hF_int
    hF'_meas this bound_integrable diff_x₀

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
hs : μ.IsEverywherePos s
c : ℝ≥0
hc : c ≠ 0
⊢ ↑c ≠ 0

Tactic:
by simpa using hc

Goal state after:
Module: Mathlib.Analysis.Convolution

Goal state before:
pp: 𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
E'' : Type uE''
F : Type uF
F' : Type uF'
F'' : Type uF''
inst✝²⁶ : NormedAddCommGroup E
inst✝²⁵ : NormedAddCommGroup E'
inst✝²⁴ : NormedAddCommGroup E''
inst✝²³ : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝²² : RCLike 𝕜
inst✝²¹ : NormedSpace 𝕜 E
inst✝²⁰ : NormedSpace 𝕜 E'
inst✝¹⁹ : NormedSpace 𝕜 E''
inst✝¹⁸ : NormedSpace ℝ F
inst✝¹⁷ : NormedSpace 𝕜 F
inst✝¹⁶ : MeasurableSpace G
μ ν : Measure G
L : E →L[𝕜] E' →L[𝕜] F
inst✝¹⁵ : CompleteSpace F
inst✝¹⁴ : NormedAddCommGroup F'
inst✝¹³ : NormedSpace ℝ F'
inst✝¹² : NormedSpace 𝕜 F'
inst✝¹¹ : CompleteSpace F'
inst✝¹⁰ : NormedAddCommGroup F''
inst✝⁹ : NormedSpace ℝ F''
inst✝⁸ : NormedSpace 𝕜 F''
inst✝⁷ : CompleteSpace F''
k : G → E''
L₂ : F →L[𝕜] E'' →L[𝕜] F'
L₃ : E →L[𝕜] F'' →L[𝕜] F'
L₄ : E' →L[𝕜] E'' →L[𝕜] F''
inst✝⁶ : AddGroup G
inst✝⁵ : SFinite μ
inst✝⁴ : SFinite ν
inst✝³ : μ.IsAddRightInvariant
inst✝² : MeasurableAdd₂ G
inst✝¹ : ν.IsAddRightInvariant
inst✝ : MeasurableNeg G
hL : ∀ (x : E) (y : E') (z : E''), (L₂ ((L x) y)) z = (L₃ x) ((L₄ y) z)
x₀ : G
hfg : ∀ᵐ (y : G) ∂μ, ConvolutionExistsAt f g y L ν
hgk : ∀ᵐ (x : G) ∂ν, ConvolutionExistsAt g k x L₄ μ
hi : Integrable (uncurry fun x y ↦ (L₃ (f y)) ((L₄ (g (x - y))) (k (x₀ - x)))) (μ.prod ν)
⊢ ∫ (s : G), ∫ (t : G), (L₃ (f s)) ((L₄ (g (t - s))) (k (x₀ - t))) ∂μ ∂ν =
    ∫ (s : G), ∫ (u : G), (L₃ (f s)) ((L₄ (g u)) (k (x₀ - s - u))) ∂μ ∂ν

Tactic:
congr

Goal state after:
case e_f
𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
E'' : Type uE''
F : Type uF
F' : Type uF'
F'' : Type uF''
inst✝²⁶ : NormedAddCommGroup E
inst✝²⁵ : NormedAddCommGroup E'
inst✝²⁴ : NormedAddCommGroup E''
inst✝²³ : NormedAddCommGroup F
f : G → E
g : G → E'
inst✝²² : RCLike 𝕜
inst✝²¹ : NormedSpace 𝕜 E
inst✝²⁰ : NormedSpace 𝕜 E'
inst✝¹⁹ : NormedSpace 𝕜 E''
inst✝¹⁸ : NormedSpace ℝ F
inst✝¹⁷ : NormedSpace 𝕜 F
inst✝¹⁶ : MeasurableSpace G
μ ν : Measure G
L : E →L[𝕜] E' →L[𝕜] F
inst✝¹⁵ : CompleteSpace F
inst✝¹⁴ : NormedAddCommGroup F'
inst✝¹³ : NormedSpace ℝ F'
inst✝¹² : NormedSpace 𝕜 F'
inst✝¹¹ : CompleteSpace F'
inst✝¹⁰ : NormedAddCommGroup F''
inst✝⁹ : NormedSpace ℝ F''
inst✝⁸ : NormedSpace 𝕜 F''
inst✝⁷ : CompleteSpace F''
k : G → E''
L₂ : F →L[𝕜] E'' →L[𝕜] F'
L₃ : E →L[𝕜] F'' →L[𝕜] F'
L₄ : E' →L[𝕜] E'' →L[𝕜] F''
inst✝⁶ : AddGroup G
inst✝⁵ : SFinite μ
inst✝⁴ : SFinite ν
inst✝³ : μ.IsAddRightInvariant
inst✝² : MeasurableAdd₂ G
inst✝¹ : ν.IsAddRightInvariant
inst✝ : MeasurableNeg G
hL : ∀ (x : E) (y : E') (z : E''), (L₂ ((L x) y)) z = (L₃ x) ((L₄ y) z)
x₀ : G
hfg : ∀ᵐ (y : G) ∂μ, ConvolutionExistsAt f g y L ν
hgk : ∀ᵐ (x : G) ∂ν, ConvolutionExistsAt g k x L₄ μ
hi : Integrable (uncurry fun x y ↦ (L₃ (f y)) ((L₄ (g (x - y))) (k (x₀ - x)))) (μ.prod ν)
⊢ (fun s ↦ ∫ (t : G), (L₃ (f s)) ((L₄ (g (t - s))) (k (x₀ - t))) ∂μ) = fun s ↦
    ∫ (u : G), (L₃ (f s)) ((L₄ (g u)) (k (x₀ - s - u))) ∂μ
Module: Mathlib.MeasureTheory.Measure.Haar.Unique

Goal state before:
pp: G : Type u_1
inst✝⁹ : TopologicalSpace G
inst✝⁸ : Group G
inst✝⁷ : IsTopologicalGroup G
inst✝⁶ : MeasurableSpace G
inst✝⁵ : BorelSpace G
μ ν : Measure G
inst✝⁴ : IsFiniteMeasureOnCompacts μ
inst✝³ : IsFiniteMeasureOnCompacts ν
inst✝² : μ.IsMulLeftInvariant
inst✝¹ : ν.IsMulRightInvariant
inst✝ : ν.IsOpenPosMeasure
f g : G → ℝ
hf : Continuous[inst✝⁹, _] f
h'f : HasCompactSupport f
hg : Continuous[inst✝⁹, _] g
h'g : HasCompactSupport g
g_nonneg : 0 ≤ g
x₀ : G
g_pos : g x₀ ≠ 0
Hf : LocallyCompactSpace G
D : G → ℝ := ⋯
D_cont : Continuous[inst✝⁹, _] D
x : G
C : Continuous[inst✝⁹, _] fun y ↦ g (y⁻¹ * x)
⊢ 0 ≤ fun x_1 ↦ g (x_1⁻¹ * x)

Tactic:
exact fun y ↦ g_nonneg (y⁻¹ * x)

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
h : ¬{i | ¬∃ b, p.c b ∉ p.iUnionUpTo i ∧ p.R i ≤ p.τ * p.r b}.Nonempty
x : Ordinal.{u}
⊢ ∀ ⦃a₂ : Ordinal.{u}⦄, p.index x = p.index a₂ → x = a₂

Tactic:
y

Goal state after:
α : Type u_1
inst✝¹ : MetricSpace α
β : Type u
inst✝ : Nonempty β
p : TauPackage β α
h : ¬{i | ¬∃ b, p.c b ∉ p.iUnionUpTo i ∧ p.R i ≤ p.τ * p.r b}.Nonempty
x y : Ordinal.{u}
⊢ p.index x = p.index y → x = y
Module: Mathlib.MeasureTheory.Covering.Besicovitch

Goal state before:
pp: case ind
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)
⊢ p.color i < N

Tactic:
color_i

Goal state after:
case ind
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)
⊢ sInf (univ \ A) < N
Module: Mathlib.MeasureTheory.Measure.Haar.Unique

Goal state before:
pp: G : Type u_1
inst✝⁸ : TopologicalSpace G
inst✝⁷ : Group G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : LocallyCompactSpace G
μ' μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : IsFiniteMeasureOnCompacts μ'
inst✝ : μ'.IsMulLeftInvariant
c : ℝ≥0
g : C(G, ℝ)
hg : HasCompactSupport ⇑g ∧ 0 ≤ ⇑g ∧ g 1 ≠ 0
⊢ ↑((c • μ').haarScalarFactor μ) = ↑(c • μ'.haarScalarFactor μ)

Tactic:
calc
    haarScalarFactor (c • μ') μ = (∫ x, g x ∂(c • μ')) / ∫ x, g x ∂μ :=
      haarScalarFactor_eq_integral_div_of_continuous_nonneg_pos _ _ hg
    _ = (c • (∫ x, g x ∂μ')) / ∫ x, g x ∂μ := by simp
    _ = c • ((∫ x, g x ∂μ') / ∫ x, g x ∂μ) := smul_div_assoc c _ _
    _ = c • haarScalarFactor μ' μ := by
      rw [← haarScalarFactor_eq_integral_div_of_continuous_nonneg_pos _ _ hg]

Goal state after:
Module: Mathlib.MeasureTheory.Measure.Haar.Unique

Goal state before:
pp: G : Type u_1
inst✝⁸ : TopologicalSpace G
inst✝⁷ : Group G
inst✝⁶ : IsTopologicalGroup G
inst✝⁵ : MeasurableSpace G
inst✝⁴ : BorelSpace G
inst✝³ : LocallyCompactSpace G
μ' μ : Measure G
inst✝² : μ.IsHaarMeasure
inst✝¹ : IsFiniteMeasureOnCompacts μ'
inst✝ : μ'.IsMulLeftInvariant
c : ℝ≥0
hc : c ≠ 0
this : (c • μ).IsHaarMeasure
g : C(G, ℝ)
hg : HasCompactSupport ⇑g ∧ 0 ≤ ⇑g ∧ g 1 ≠ 0
⊢ ↑(c * μ'.haarScalarFactor (c • μ)) = ↑(μ'.haarScalarFactor μ)

Tactic:
calc
    c * haarScalarFactor μ' (c • μ) = c * ((∫ x, g x ∂μ') / ∫ x, g x ∂(c • μ)) := by
      rw [haarScalarFactor_eq_integral_div_of_continuous_nonneg_pos _ _ hg]
    _ = c * ((∫ x, g x ∂μ') / (c • ∫ x, g x ∂μ)) := by simp
    _ = (∫ x, g x ∂μ') / (∫ x, g x ∂μ) := by
      rw [NNReal.smul_def, smul_eq_mul, ← mul_div_assoc]
      exact mul_div_mul_left (∫ (x : G), g x ∂μ') (∫ (x : G), g x ∂μ) (by simp [hc])
    _ = μ'.haarScalarFactor μ :=
      (haarScalarFactor_eq_integral_div_of_continuous_nonneg_pos _ _ hg).symm

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

Goal state before:
pp: case inr.refine_1
α : Type u_1
inst✝ : MetricSpace α
β : Type u
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
q : BallPackage β α
h✝ : Nonempty β
p : TauPackage β α := { toBallPackage := q, τ := τ, one_lt_tau := hτ }
s : Fin N → Set β := fun i ↦ ⋃ k, ⋃ (_ : k < p.lastStep), ⋃ (_ : p.color k = ↑i), {p.index k}
i : Fin N
x : β
hx : x ∈ s i
⊢ ∀ ⦃y : β⦄, y ∈ s i → x ≠ y → Function.onFun Disjoint (fun j ↦ closedBall (q.c j) (q.r j)) x y

Tactic:
y

Goal state after:
case inr.refine_1
α : Type u_1
inst✝ : MetricSpace α
β : Type u
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
q : BallPackage β α
h✝ : Nonempty β
p : TauPackage β α := { toBallPackage := q, τ := τ, one_lt_tau := hτ }
s : Fin N → Set β := fun i ↦ ⋃ k, ⋃ (_ : k < p.lastStep), ⋃ (_ : p.color k = ↑i), {p.index k}
i : Fin N
x : β
hx : x ∈ s i
y : β
⊢ y ∈ s i → x ≠ y → Function.onFun Disjoint (fun j ↦ closedBall (q.c j) (q.r j)) x y
Module: Mathlib.MeasureTheory.Covering.Besicovitch

Goal state before:
pp: α : Type u_1
inst✝ : MetricSpace α
β : Type u
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
q : BallPackage β α
h✝ : Nonempty β
p : TauPackage β α := { toBallPackage := q, τ := τ, one_lt_tau := hτ }
s : Fin N → Set β := fun i ↦ ⋃ k, ⋃ (_ : k < p.lastStep), ⋃ (_ : p.color k = ↑i), {p.index k}
i : Fin N
jx : Ordinal.{u}
jx_lt : jx < p.lastStep
jxi : p.color jx = ↑i
hx : p.index jx ∈ s i
jy : Ordinal.{u}
jy_lt : jy < p.lastStep
jyi : p.color jy = ↑i
hy : p.index jy ∈ s i
x_ne_y : p.index jx ≠ p.index jy
jxy : jx < jy
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index jy)) (p.r (p.index jy))).Nonempty),
      {p.color ↑j}
color_j : p.color jy = sInf (univ \ A)
h : p.color jy ∈ univ \ A
⊢ Function.onFun Disjoint (fun j ↦ closedBall (q.c j) (q.r j)) (p.index jx) (p.index jy)

Tactic:
simp only [A, not_exists, true_and, exists_prop, mem_iUnion, mem_singleton_iff, not_and,
      mem_univ, Set.mem_sdiff, Subtype.exists] at h

Goal state after:
α : Type u_1
inst✝ : MetricSpace α
β : Type u
N : ℕ
τ : ℝ
hτ : 1 < τ
hN : IsEmpty (SatelliteConfig α N τ)
q : BallPackage β α
h✝ : Nonempty β
p : TauPackage β α := { toBallPackage := q, τ := τ, one_lt_tau := hτ }
s : Fin N → Set β := fun i ↦ ⋃ k, ⋃ (_ : k < p.lastStep), ⋃ (_ : p.color k = ↑i), {p.index k}
i : Fin N
jx : Ordinal.{u}
jx_lt : jx < p.lastStep
jxi : p.color jx = ↑i
hx : p.index jx ∈ s i
jy : Ordinal.{u}
jy_lt : jy < p.lastStep
jyi : p.color jy = ↑i
hy : p.index jy ∈ s i
x_ne_y : p.index jx ≠ p.index jy
jxy : jx < jy
A : Set ℕ :=
  ⋃ j,
    ⋃ (_ :
      (closedBall (p.c (p.index ↑j)) (p.r (p.index ↑j)) ∩ closedBall (p.c (p.index jy)) (p.r (p.index jy))).Nonempty),
      {p.color ↑j}
color_j : p.color jy = sInf (univ \ A)
h :
  ∀ x < jy,
    (closedBall (p.c (p.index x)) (p.r (p.index x)) ∩ closedBall (p.c (p.index jy)) (p.r (p.index jy))).Nonempty →
      ¬p.color jy = p.color x
⊢ Function.onFun Disjoint (fun j ↦ closedBall (q.c j) (q.r j)) (p.index jx) (p.index jy)