mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.CStarAlgebra.Unitary.Connected

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
u : ↥(unitary A)
hu : dist u 1 < 2
ε : ℝ
huε : dist u 1 ^ 2 < ε
hε2 : ε < 2 ^ 2
hε : 0 < ε
huε' : dist u 1 < √ε
⊢ Continuous fun z ↦ 2 * (1 - z.re)

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.Unitary.Connected

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
u : ↥(unitary A)
hu : dist u 1 < 2
ε : ℝ
huε : dist u 1 ^ 2 < ε
hε2 : ε < 2 ^ 2
hε : 0 < ε
huε' : dist u 1 < √ε
⊢ Continuous fun z ↦ ε

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.Unitary.Connected

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
⊢ ContinuousOn argSelfAdjoint (ball 1 2)

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.Unitary.Connected

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
⊢ ContinuousOn expUnitary (ball 0 π)

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.Unitary.Connected

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
x : ↥(selfAdjoint A)
⊢ Continuous fun t ↦ expUnitary (↑t • x)

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.Unitary.Connected

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
u v : ↥(unitary A)
huv : ‖↑v - ↑u‖ < 2
⊢ Continuous fun t ↦ expUnitary (↑t • argSelfAdjoint (v * star u)) * u

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.Analysis.CStarAlgebra.Unitary.Connected

Goal state before:
pp: A : Type u_1
inst✝ : CStarAlgebra A
u : ↥(unitary A)
δ : ℝ
hδ₀ : 0 < δ
hδ₂ : δ < 2
this : IsPathConnected (ball 1 δ)
⊢ Continuous fun x ↦ u * x

Tactic:
by fun_prop

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : DivisionRing k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : AffineSpace V P
n : ℕ
inst✝¹ : NeZero n
inst✝ : CharZero k
s : Simplex k P n
i : Fin (n + 1)
⊢ s.faceOppositeCentroid i -ᵥ s.points i = (↑n)⁻¹ • ∑ x, (s.points x -ᵥ s.points i)

Tactic:
by
  rw [faceOppositeCentroid_eq_affineCombination,
    affineCombination_eq_weightedVSubOfPoint_vadd_of_sum_eq_one _ _ _ ?_ (s.points i)]
  · simp only [weightedVSubOfPoint_apply, vadd_vsub]
    have h (i : Fin (n + 1)) : ∑ j ∈ {i}ᶜ, (n : k)⁻¹ • (s.points j -ᵥ s.points i) =
      ∑ j : (Fin (n + 1)), ((n : k)⁻¹ • (s.points j -ᵥ s.points i)) := by
      rw [← Finset.sum_compl_add_sum {i}]
      simp
    rw [h i, smul_sum]
  · simp only [sum_const, card_compl, Fintype.card_fin, card_singleton, add_tsub_cancel_right,
      nsmul_eq_mul]
    rw [mul_inv_cancel₀ (NeZero.ne (n : k))]

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : DivisionRing k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : AffineSpace V P
n : ℕ
inst✝¹ : NeZero n
inst✝ : CharZero k
s : Simplex k P n
i j : Fin (n + 1)
h1 : ∀ (i : Fin (n + 1)), ∑ x, (s.points x -ᵥ s.points i) = ∑ x, (s.points x -ᵥ s.points 0 - (s.points i -ᵥ s.points 0))
this : s.points i -ᵥ s.points j = -(s.points j -ᵥ s.points i)
⊢ (↑n)⁻¹ • n • (s.points j -ᵥ s.points i) = (↑n)⁻¹ • n • (s.points j -ᵥ s.points i)

Tactic:
congr 1

Goal state after:
Module: Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : DivisionRing k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : AffineSpace V P
n : ℕ
inst✝¹ : NeZero n
inst✝ : CharZero k
s : Simplex k P n
i : Fin (n + 1)
⊢ s.faceOppositeCentroid i -ᵥ s.centroid = ((↑n)⁻¹ * ↑n) • (s.faceOppositeCentroid i -ᵥ s.centroid)

Tactic:
inv_mul_cancel₀ (NeZero.ne (n : k)),

Goal state after:
k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : DivisionRing k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : AffineSpace V P
n : ℕ
inst✝¹ : NeZero n
inst✝ : CharZero k
s : Simplex k P n
i : Fin (n + 1)
⊢ s.faceOppositeCentroid i -ᵥ s.centroid = 1 • (s.faceOppositeCentroid i -ᵥ s.centroid)
Module: Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : DivisionRing k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : AffineSpace V P
n : ℕ
inst✝¹ : NeZero n
inst✝ : CharZero k
s : Simplex k P n
i : Fin (n + 1)
⊢ s.centroid -ᵥ s.faceOppositeCentroid i = ((↑n)⁻¹ * ↑n) • (s.centroid -ᵥ s.faceOppositeCentroid i)

Tactic:
inv_mul_cancel₀ (NeZero.ne (n : k)),

Goal state after:
k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : DivisionRing k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : AffineSpace V P
n : ℕ
inst✝¹ : NeZero n
inst✝ : CharZero k
s : Simplex k P n
i : Fin (n + 1)
⊢ s.centroid -ᵥ s.faceOppositeCentroid i = 1 • (s.centroid -ᵥ s.faceOppositeCentroid i)
Module: Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : DivisionRing k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : AffineSpace V P
n : ℕ
inst✝¹ : NeZero n
inst✝ : CharZero k
s : Simplex k P n
i : Fin (n + 1)
this : -(s.points i -ᵥ s.centroid) = -1 • (s.points i -ᵥ s.centroid)
⊢ ((↑n)⁻¹ * -1) • (s.points i -ᵥ s.centroid) = (-1 / ↑n) • (s.points i -ᵥ s.centroid)

Tactic:
congr 1

Goal state after:
case e_a
k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : DivisionRing k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : AffineSpace V P
n : ℕ
inst✝¹ : NeZero n
inst✝ : CharZero k
s : Simplex k P n
i : Fin (n + 1)
this : -(s.points i -ᵥ s.centroid) = -1 • (s.points i -ᵥ s.centroid)
⊢ (↑n)⁻¹ * -1 = -1 / ↑n
Module: Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid

Goal state before:
pp: k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : DivisionRing k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : AffineSpace V P
n : ℕ
inst✝¹ : NeZero n
inst✝ : CharZero k
s : Simplex k P n
hn : 1 < n
p : P
h : ∀ (i : Fin (n + 1)), p ∈ s.median i
i₀ : Fin (n + 1) := 0
hp : p = (p -ᵥ s.centroid) +ᵥ s.centroid
s' : Finset (Fin (n + 1)) := {i₀}ᶜ
u : ↥s' → V := fun i ↦ s.points ↑i -ᵥ s.centroid
h_span : ∀ (i : ↥s'), p -ᵥ s.centroid ∈ k ∙ u i
hi : LinearIndependent k u
⊢ n + 1 - 1 = n

Tactic:
add_tsub_cancel_right

Goal state after:
k : Type u_1
V : Type u_2
P : Type u_3
inst✝⁵ : DivisionRing k
inst✝⁴ : AddCommGroup V
inst✝³ : Module k V
inst✝² : AffineSpace V P
n : ℕ
inst✝¹ : NeZero n
inst✝ : CharZero k
s : Simplex k P n
hn : 1 < n
p : P
h : ∀ (i : Fin (n + 1)), p ∈ s.median i
i₀ : Fin (n + 1) := 0
hp : p = (p -ᵥ s.centroid) +ᵥ s.centroid
s' : Finset (Fin (n + 1)) := {i₀}ᶜ
u : ↥s' → V := fun i ↦ s.points ↑i -ᵥ s.centroid
h_span : ∀ (i : ↥s'), p -ᵥ s.centroid ∈ k ∙ u i
hi : LinearIndependent k u
⊢ n = n
Module: Mathlib.MeasureTheory.Group.Integral

Goal state before:
pp: G : Type u_4
F : Type u_6
inst✝⁴ : MeasurableSpace G
inst✝³ : NormedAddCommGroup F
μ : Measure G
inst✝² : Group G
inst✝¹ : MeasurableInv G
inst✝ : μ.IsInvInvariant
f : G → F
s : Set G
hf : IntegrableOn f s μ
this : s⁻¹ = ⇑(MeasurableEquiv.inv G) ⁻¹' s
⊢ Integrable f (map (⇑(MeasurableEquiv.inv G)) (μ.restrict s⁻¹))

Tactic:
rw [this, ← MeasurableEquiv.restrict_map]

Goal state after:
G : Type u_4
F : Type u_6
inst✝⁴ : MeasurableSpace G
inst✝³ : NormedAddCommGroup F
μ : Measure G
inst✝² : Group G
inst✝¹ : MeasurableInv G
inst✝ : μ.IsInvInvariant
f : G → F
s : Set G
hf : IntegrableOn f s μ
this : s⁻¹ = ⇑(MeasurableEquiv.inv G) ⁻¹' s
⊢ Integrable f ((map (⇑(MeasurableEquiv.inv G)) μ).restrict s)
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✝ : (i : α) → NormedAddCommGroup (E i)
hp : Fact (1 ≤ p)
f g : ↥(lp E p)
⊢ ‖f + g‖ ≤ ‖f‖ + ‖g‖

Tactic:
rcases p.dichotomy with (rfl | hp')

Goal state after:
case inl
𝕜 : Type u_1
𝕜' : Type u_2
α : Type u_3
E : α → Type u_4
q : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
hp : Fact (1 ≤ ∞)
f g : ↥(lp E ∞)
⊢ ‖f + g‖ ≤ ‖f‖ + ‖g‖

case inr
𝕜 : Type u_1
𝕜' : Type u_2
α : Type u_3
E : α → Type u_4
p q : ℝ≥0∞
inst✝ : (i : α) → NormedAddCommGroup (E i)
hp : Fact (1 ≤ p)
f g : ↥(lp E p)
hp' : 1 ≤ p.toReal
⊢ ‖f + g‖ ≤ ‖f‖ + ‖g‖
Module: Mathlib.Analysis.Calculus.SmoothSeries

Goal state before:
pp: α : Type u_1
𝕜 : Type u_3
E : Type u_4
F : Type u_5
inst✝⁶ : NontriviallyNormedField 𝕜
inst✝⁵ : IsRCLikeNormedField 𝕜
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
inst✝² : NormedAddCommGroup F
inst✝¹ : CompleteSpace F
u : α → ℝ
inst✝ : NormedSpace 𝕜 F
f : α → E → F
f' : α → E → E →L[𝕜] F
hu : Summable u
hf : ∀ (n : α) (x : E), HasFDerivAt (f n) (f' n x) x
hf' : ∀ (n : α) (x : E), ‖f' n x‖ ≤ u n
⊢ Differentiable 𝕜 fun y ↦ ∑' (n : α), f n y

Tactic:
by
  by_cases! h : ∃ x₀, Summable fun n => f n x₀
  · rcases h with ⟨x₀, hf0⟩
    intro x
    exact (hasFDerivAt_tsum hu hf hf' hf0 x).differentiableAt
  · have : (fun x => ∑' n, f n x) = 0 := by ext1 x; exact tsum_eq_zero_of_not_summable (h x)
    rw [this]
    exact differentiable_const 0

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✝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_1 ↦ 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_1 ↦ f n.1 x - f n.2 x)
    (0 + 0) (l ×ˢ l) (𝓝 x)
⊢ TendstoUniformlyOnFilter (fun n z ↦ -f n.1 z + f n.2 z) 0 (l ×ˢ l) (𝓝 x)

Tactic:
add_zero

Goal state after:
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_1 ↦ 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_1 ↦ f n.1 x - f n.2 x) 0
    (l ×ˢ l) (𝓝 x)
⊢ TendstoUniformlyOnFilter (fun n z ↦ -f n.1 z + f n.2 z) 0 (l ×ˢ l) (𝓝 x)
Module: Mathlib.Analysis.Normed.Module.Ball.Homeomorph

Goal state before:
pp: E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
this : Continuous fun x ↦ (√(1 + ‖x‖ ^ 2))⁻¹
⊢ ContinuousOn (fun x ↦ (√(1 + ‖x‖ ^ 2))⁻¹ • x) univ

Tactic:
by fun_prop

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

Goal state before:
pp: E : Type u_1
inst✝¹ : SeminormedAddCommGroup E
inst✝ : NormedSpace ℝ E
⊢ Continuous fun x ↦ √(1 + ‖x‖ ^ 2)

Tactic:
by fun_prop

Goal state after:
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
t : Set H
y : H
⊢ ContDiffWithinAt 𝕜 n f t y ↔ ∀ (i : ι), ContDiffWithinAt 𝕜 n (fun x ↦ (f x).ofLp i) t y

Tactic:
rw [← (PiLp.continuousLinearEquiv p 𝕜 E).comp_contDiffWithinAt_iff, contDiffWithinAt_pi]
  rfl

Goal state after:
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
t : Set H
⊢ ContDiffOn 𝕜 n (⇑(PiLp.continuousLinearEquiv p 𝕜 E) ∘ f) t ↔ ∀ (i : ι), ContDiffOn 𝕜 n (fun x ↦ (f x).ofLp i) t

Tactic:
contDiffOn_pi

Goal state after:
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
t : Set H
⊢ (∀ (i : ι), ContDiffOn 𝕜 n (fun x ↦ (⇑(PiLp.continuousLinearEquiv p 𝕜 E) ∘ f) x i) t) ↔
    ∀ (i : ι), ContDiffOn 𝕜 n (fun x ↦ (f x).ofLp i) t
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 + 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:
add_zero

Goal state after:
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)
⊢ 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: 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
⊢ s ⊆ t

Tactic:
have open_t : IsOpen t := by
    rw [Metric.isOpen_iff]
    intro x hx
    rcases Metric.isOpen_iff.1 hs x hx.1 with ⟨ε, εpos, hε⟩
    exact ⟨ε, εpos, A x ε hx hε⟩

Goal state after:
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
⊢ s ⊆ t
Module: Mathlib.Analysis.SpecialFunctions.SmoothTransition

Goal state before:
pp: case inr.inl
p : ℝ[X]
⊢ HasDerivAt (fun x ↦ Polynomial.eval x⁻¹ p * expNegInvGlue x)
    (Polynomial.eval 0⁻¹ (X ^ 2 * (p - derivative p)) * expNegInvGlue 0) 0

Tactic:
rw [expNegInvGlue.zero, mul_zero, hasDerivAt_iff_tendsto_slope]
    refine ((tendsto_polynomial_inv_mul_zero (p * X)).mono_left inf_le_left).congr fun x ↦ ?_
    simp [slope_def_field, div_eq_mul_inv, mul_right_comm]

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

Goal state before:
pp: x : ℝ
h : 1 ≤ x
⊢ expNegInvGlue x = expNegInvGlue x + 0

Tactic:
add_zero

Goal state after:
x : ℝ
h : 1 ≤ x
⊢ expNegInvGlue x = expNegInvGlue x
Module: Mathlib.Analysis.Calculus.BumpFunction.InnerProduct

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : InnerProductSpace ℝ E
R : ℝ
x : E
hR : 0 < R - 1
hx : x ≠ 0
⊢ ContDiffAt ℝ (↑⊤) (fun a ↦ a.1 - 1) (R, x)

Tactic:
by fun_prop

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 → 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 : AddGroup G
inst✝¹ : TopologicalSpace G
inst✝ : ContinuousSub G
hg : Continuous[inst✝¹, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] g
t : G
⊢ Continuous[inst✝¹, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] fun x ↦ g (x - t)

Tactic:
by fun_prop

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✝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 ν : Measure G
inst✝⁵ : AddGroup G
inst✝⁴ : MeasurableAdd₂ G
inst✝³ : MeasurableNeg G
inst✝² : SFinite μ
inst✝¹ : μ.IsAddRightInvariant
inst✝ : SFinite ν
hf : Integrable f ν
hg : Integrable g μ
h_meas : AEStronglyMeasurable (fun p ↦ (L (f p.2)) (g (p.1 - p.2))) (μ.prod ν)
h2_meas : AEStronglyMeasurable (fun y ↦ ∫ (x : G), ‖(L (f y)) (g (x - y))‖ ∂μ) ν
⊢ Integrable (fun y ↦ ∫ (x : G), ‖(L (f y)) (g (x - y))‖ ∂μ) ν

Tactic:
refine Integrable.mono' ?_ h2_meas
      (Eventually.of_forall fun t => (?_ : _ ≤ ‖L‖ * ‖f t‖ * ∫ x, ‖g (x - t)‖ ∂μ))

Goal state after:
case refine_1
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 ν : Measure G
inst✝⁵ : AddGroup G
inst✝⁴ : MeasurableAdd₂ G
inst✝³ : MeasurableNeg G
inst✝² : SFinite μ
inst✝¹ : μ.IsAddRightInvariant
inst✝ : SFinite ν
hf : Integrable f ν
hg : Integrable g μ
h_meas : AEStronglyMeasurable (fun p ↦ (L (f p.2)) (g (p.1 - p.2))) (μ.prod ν)
h2_meas : AEStronglyMeasurable (fun y ↦ ∫ (x : G), ‖(L (f y)) (g (x - y))‖ ∂μ) ν
⊢ Integrable (fun t ↦ ‖L‖ * ‖f t‖ * ∫ (x : G), ‖g (x - t)‖ ∂μ) ν

case refine_2
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 ν : Measure G
inst✝⁵ : AddGroup G
inst✝⁴ : MeasurableAdd₂ G
inst✝³ : MeasurableNeg G
inst✝² : SFinite μ
inst✝¹ : μ.IsAddRightInvariant
inst✝ : SFinite ν
hf : Integrable f ν
hg : Integrable g μ
h_meas : AEStronglyMeasurable (fun p ↦ (L (f p.2)) (g (p.1 - p.2))) (μ.prod ν)
h2_meas : AEStronglyMeasurable (fun y ↦ ∫ (x : G), ‖(L (f y)) (g (x - y))‖ ∂μ) ν
t : G
⊢ ‖∫ (x : G), ‖(L (f t)) (g (x - t))‖ ∂μ‖ ≤ ‖L‖ * ‖f t‖ * ∫ (x : G), ‖g (x - t)‖ ∂μ
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
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
⊢ ∫ (t : G), (L 0) (g (x✝ - t)) ∂μ = 0

Tactic:
L.map_zero₂,

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
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
⊢ ∫ (t : G), 0 ∂μ = 0
Module: Mathlib.Analysis.Convolution

Goal state before:
pp: 𝕜 : Type u𝕜
G : Type uG
E : Type uE
E' : Type uE'
F : Type uF
P : Type uP
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedAddCommGroup E'
inst✝¹¹ : NormedAddCommGroup F
f : 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 : AddGroup G
inst✝³ : TopologicalSpace G
inst✝² : IsTopologicalAddGroup G
inst✝¹ : BorelSpace G
inst✝ : TopologicalSpace P
g : P → G → E'
s : Set P
k : Set G
hk : IsCompact k
hgs : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0
hf : LocallyIntegrable f μ
hg : ContinuousOn (↿g) (s ×ˢ univ)
H : ∃ p ∈ s, ∃ x, g p x ≠ 0
this✝ : LocallyCompactSpace G
q₀ : P
x₀ : G
hq₀ : (q₀, x₀).1 ∈ s
t : Set G
t_comp : IsCompact t
ht : t ∈ 𝓝 x₀
k' : Set G := -k +ᵥ t
k'_comp : IsCompact k'
g' : P × G → G → E' := fun p x ↦ g p.1 (p.2 - x)
s' : Set (P × G) := s ×ˢ t
this : uncurry g' = uncurry g ∘ fun w ↦ (w.1.1, w.1.2 - w.2)
⊢ ContinuousOn (fun w ↦ (w.1.1, w.1.2 - w.2)) (s' ×ˢ univ)

Tactic:
by fun_prop

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✝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‖
⊢ Integrable (fun a ↦ ‖L‖ * ‖f a‖ * ⨆ i, ‖g i‖) μ

Tactic:
by fun_prop

Goal state after: