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

Goal state before:
pp: case hbc
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : E → F
inst✝ : CompleteSpace E
s : Set E
c : ℝ≥0
f' : E →L[𝕜] F
hf : ApproximatesLinearOn f f' s c
f'symm : f'.NonlinearRightInverse
ε : ℝ
b : E
ε0 : 0 ≤ ε
hε : closedBall b ε ⊆ s
y : F
hy : y ∈ closedBall (f b) (((↑f'symm.nnnorm)⁻¹ - ↑c) * ε)
hc : ↑c < (↑f'symm.nnnorm)⁻¹
If' : 0 < ↑f'symm.nnnorm
Icf' : ↑c * ↑f'symm.nnnorm < 1
Jcf' : 1 - ↑c * ↑f'symm.nnnorm ≠ 0
g : E → E := ⋯
hg : g = fun x ↦ x + f'symm.toFun (y - f x)
u : ℕ → E := ⋯
hu : u = fun n ↦ g^[n] b
usucc : ∀ (n : ℕ), u (n + 1) = g (u n)
A : ∀ (z : E), dist (g z) z ≤ ↑f'symm.nnnorm * dist (f z) y
B : ∀ z ∈ closedBall b ε, g z ∈ closedBall b ε → dist (f (g z)) y ≤ ↑c * ↑f'symm.nnnorm * dist (f z) y
n : ℕ
w : E
hw : dist w b ≤ ↑f'symm.nnnorm * (1 - (↑c * ↑f'symm.nnnorm) ^ n) / (1 - ↑c * ↑f'symm.nnnorm) * dist (f b) y
⊢ 1 - (↑c * ↑f'symm.nnnorm) ^ n ≤ 1

Tactic:
rw [sub_le_self_iff]

Goal state after:
case hbc
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : E → F
inst✝ : CompleteSpace E
s : Set E
c : ℝ≥0
f' : E →L[𝕜] F
hf : ApproximatesLinearOn f f' s c
f'symm : f'.NonlinearRightInverse
ε : ℝ
b : E
ε0 : 0 ≤ ε
hε : closedBall b ε ⊆ s
y : F
hy : y ∈ closedBall (f b) (((↑f'symm.nnnorm)⁻¹ - ↑c) * ε)
hc : ↑c < (↑f'symm.nnnorm)⁻¹
If' : 0 < ↑f'symm.nnnorm
Icf' : ↑c * ↑f'symm.nnnorm < 1
Jcf' : 1 - ↑c * ↑f'symm.nnnorm ≠ 0
g : E → E := fun x ↦ x + f'symm.toFun (y - f x)
hg : g = fun x ↦ x + f'symm.toFun (y - f x)
u : ℕ → E := fun n ↦ g^[n] b
hu : u = fun n ↦ g^[n] b
usucc : ∀ (n : ℕ), u (n + 1) = g (u n)
A : ∀ (z : E), dist (g z) z ≤ ↑f'symm.nnnorm * dist (f z) y
B : ∀ z ∈ closedBall b ε, g z ∈ closedBall b ε → dist (f (g z)) y ≤ ↑c * ↑f'symm.nnnorm * dist (f z) y
n : ℕ
w : E
hw : dist w b ≤ ↑f'symm.nnnorm * (1 - (↑c * ↑f'symm.nnnorm) ^ n) / (1 - ↑c * ↑f'symm.nnnorm) * dist (f b) y
⊢ 0 ≤ (↑c * ↑f'symm.nnnorm) ^ n
Module: Mathlib.Analysis.AbsoluteValue.Equivalence

Goal state before:
pp: F : Type u_1
inst✝¹ : Field F
ι : Type u_2
inst✝ : Finite ι
v : ι → AbsoluteValue F ℝ
h : ∀ (i : ι), (v i).IsNontrivial
hv : Pairwise fun i j ↦ ¬(v i).IsEquiv (v j)
this : Fintype ι
z : (i : ι) → WithAbs (v i)
r : ℝ
hr : r > 0
a : ι → F
hx : ∀ (i : ι), 1 < (v i) (a i) ∧ ∀ (j : ι), j ≠ i → (v j) (a i) < 1
y : ℕ → F := fun n ↦ ∑ i, 1 / (1 + (a i)⁻¹ ^ n) * (WithAbs.equiv (v i)) (z i)
u : ι
⊢ Tendsto (fun i ↦ (WithAbs.equiv (v u)).symm (y i)) atTop (𝓝 (z u))

Tactic:
simp_rw [← Fintype.sum_pi_single u z, y, map_sum, map_mul]

Goal state after:
F : Type u_1
inst✝¹ : Field F
ι : Type u_2
inst✝ : Finite ι
v : ι → AbsoluteValue F ℝ
h : ∀ (i : ι), (v i).IsNontrivial
hv : Pairwise fun i j ↦ ¬(v i).IsEquiv (v j)
this : Fintype ι
z : (i : ι) → WithAbs (v i)
r : ℝ
hr : r > 0
a : ι → F
hx : ∀ (i : ι), 1 < (v i) (a i) ∧ ∀ (j : ι), j ≠ i → (v j) (a i) < 1
y : ℕ → F := fun n ↦ ∑ i, 1 / (1 + (a i)⁻¹ ^ n) * (WithAbs.equiv (v i)) (z i)
u : ι
⊢ Tendsto
    (fun i ↦
      ∑ x,
        (WithAbs.equiv (v u)).symm (1 / (1 + (a x)⁻¹ ^ i)) * (WithAbs.equiv (v u)).symm ((WithAbs.equiv (v x)) (z x)))
    atTop (𝓝 (∑ j, Pi.single j (z j) u))
Module: Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : E → F
f' : E ≃L[𝕜] F
c : ℝ≥0
inst✝ : CompleteSpace E
hf : ApproximatesLinearOn f (↑f') univ c
hc : c < ‖↑f'.symm‖₊⁻¹
⊢ 0 < ↑‖↑f'.symm‖₊⁻¹ - ↑c

Tactic:
sub_pos

Goal state after:
𝕜 : Type u_1
inst✝⁵ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁴ : NormedAddCommGroup E
inst✝³ : NormedSpace 𝕜 E
F : Type u_3
inst✝² : NormedAddCommGroup F
inst✝¹ : NormedSpace 𝕜 F
f : E → F
f' : E ≃L[𝕜] F
c : ℝ≥0
inst✝ : CompleteSpace E
hf : ApproximatesLinearOn f (↑f') univ c
hc : c < ‖↑f'.symm‖₊⁻¹
⊢ ↑c < ↑‖↑f'.symm‖₊⁻¹
Module: Mathlib.Analysis.Calculus.FDeriv.Extend

Goal state before:
pp: case neg.refine_1
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
s : Set E
x : E
f' : E →L[ℝ] F
f_diff : DifferentiableOn ℝ f s
s_conv : Convex ℝ s
s_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
f_cont : ∀ y ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s, ContinuousWithinAt f s y
h : Tendsto (fun y ↦ fderiv ℝ f y) (𝓝[s] x) (𝓝 f')
hx : x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
ε : ℝ
ε_pos : 0 < ε
δ : ℝ
δ_pos : δ > 0
hδ : ∀ y ∈ s, dist y x < δ → ‖fderiv ℝ f y - f'‖ < ε
B : Set E := ball x δ
key : ∀ p ∈ (B ∩ s) ×ˢ (B ∩ s), ‖f p.2 - f p.1 - (f' p.2 - f' p.1)‖ ≤ ε * ‖p.2 - p.1‖
u v : E
uv_in : (u, v) ∈ closure[instTopologicalSpaceProd] ((B ∩ s) ×ˢ (B ∩ s))
f_cont' : ∀ y ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s, ContinuousWithinAt (f - ⇑f') s y
this : (B ∩ s) ×ˢ (B ∩ s) ⊆ s ×ˢ s
u_in : u ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
v_in : v ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
⊢ ContinuousWithinAt (fun y ↦ ‖f y.2 - f y.1 - (f' y.2 - f' y.1)‖) ((B ∩ s) ×ˢ (B ∩ s)) (u, v)

Tactic:
apply ContinuousWithinAt.mono _ this

Goal state after:
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
s : Set E
x : E
f' : E →L[ℝ] F
f_diff : DifferentiableOn ℝ f s
s_conv : Convex ℝ s
s_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
f_cont : ∀ y ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s, ContinuousWithinAt f s y
h : Tendsto (fun y ↦ fderiv ℝ f y) (𝓝[s] x) (𝓝 f')
hx : x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
ε : ℝ
ε_pos : 0 < ε
δ : ℝ
δ_pos : δ > 0
hδ : ∀ y ∈ s, dist y x < δ → ‖fderiv ℝ f y - f'‖ < ε
B : Set E := ⋯
key : ∀ p ∈ (B ∩ s) ×ˢ (B ∩ s), ‖f p.2 - f p.1 - (f' p.2 - f' p.1)‖ ≤ ε * ‖p.2 - p.1‖
u v : E
uv_in : (u, v) ∈ closure[instTopologicalSpaceProd] ((B ∩ s) ×ˢ (B ∩ s))
f_cont' : ∀ y ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s, ContinuousWithinAt (f - ⇑f') s y
this : (B ∩ s) ×ˢ (B ∩ s) ⊆ s ×ˢ s
u_in : u ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
v_in : v ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
⊢ ContinuousWithinAt (fun y ↦ ‖f y.2 - f y.1 - (f' y.2 - f' y.1)‖) (s ×ˢ s) (u, v)
Module: Mathlib.RingTheory.Binomial

Goal state before:
pp: R : Type u_2
inst✝³ : NonAssocSemiring R
inst✝² : Pow R ℕ
inst✝¹ : NatPowAssoc R
inst✝ : BinomialRing R
r : R
k : ℕ
⊢ (ascPochhammer ℕ (k + 1)).smeval (r + 1) =
    (k + 1).factorial • multichoose (r + 1) k + (ascPochhammer ℕ (k + 1)).smeval r

Tactic:
ascPochhammer_succ_right,

Goal state after:
R : Type u_2
inst✝³ : NonAssocSemiring R
inst✝² : Pow R ℕ
inst✝¹ : NatPowAssoc R
inst✝ : BinomialRing R
r : R
k : ℕ
⊢ (ascPochhammer ℕ k * (X + ↑k)).smeval (r + 1) =
    (k + 1).factorial • multichoose (r + 1) k + (ascPochhammer ℕ (k + 1)).smeval r
Module: Mathlib.RingTheory.Binomial

Goal state before:
pp: R : Type u_2
inst✝³ : NonAssocSemiring R
inst✝² : Pow R ℕ
inst✝¹ : NatPowAssoc R
inst✝ : BinomialRing R
r : R
k : ℕ
⊢ ((r + 1) ^ 1 + (↑k).smeval (r + 1)) * (ascPochhammer ℕ k).smeval (r + 1) =
    (k + 1).factorial • multichoose (r + 1) k + r ^ 1 * (ascPochhammer ℕ k).smeval (r ^ 1 + smeval 1 r)

Tactic:
Nat.factorial,

Goal state after:
R : Type u_2
inst✝³ : NonAssocSemiring R
inst✝² : Pow R ℕ
inst✝¹ : NatPowAssoc R
inst✝ : BinomialRing R
r : R
k : ℕ
⊢ ((r + 1) ^ 1 + (↑k).smeval (r + 1)) * (ascPochhammer ℕ k).smeval (r + 1) =
    (k.succ * k.factorial) • multichoose (r + 1) k + r ^ 1 * (ascPochhammer ℕ k).smeval (r ^ 1 + smeval 1 r)
Module: Mathlib.Analysis.Calculus.FDeriv.Extend

Goal state before:
pp: case neg.refine_2
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
s : Set E
x : E
f' : E →L[ℝ] F
f_diff : DifferentiableOn ℝ f s
s_conv : Convex ℝ s
s_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
f_cont : ∀ y ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s, ContinuousWithinAt f s y
h : Tendsto (fun y ↦ fderiv ℝ f y) (𝓝[s] x) (𝓝 f')
hx : x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
ε : ℝ
ε_pos : 0 < ε
δ : ℝ
δ_pos : δ > 0
hδ : ∀ y ∈ s, dist y x < δ → ‖fderiv ℝ f y - f'‖ < ε
B : Set E := ball x δ
key : ∀ p ∈ (B ∩ s) ×ˢ (B ∩ s), ‖f p.2 - f p.1 - (f' p.2 - f' p.1)‖ ≤ ε * ‖p.2 - p.1‖
u v : E
uv_in : (u, v) ∈ closure[instTopologicalSpaceProd] ((B ∩ s) ×ˢ (B ∩ s))
f_cont' : ∀ y ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s, ContinuousWithinAt (f - ⇑f') s y
this : (B ∩ s) ×ˢ (B ∩ s) ⊆ s ×ˢ s
u_in : u ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
v_in : v ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
⊢ ContinuousWithinAt (fun y ↦ ε * ‖y.2 - y.1‖) ((B ∩ s) ×ˢ (B ∩ s)) (u, v)

Tactic:
apply ContinuousWithinAt.mono _ this

Goal state after:
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
s : Set E
x : E
f' : E →L[ℝ] F
f_diff : DifferentiableOn ℝ f s
s_conv : Convex ℝ s
s_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
f_cont : ∀ y ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s, ContinuousWithinAt f s y
h : Tendsto (fun y ↦ fderiv ℝ f y) (𝓝[s] x) (𝓝 f')
hx : x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
ε : ℝ
ε_pos : 0 < ε
δ : ℝ
δ_pos : δ > 0
hδ : ∀ y ∈ s, dist y x < δ → ‖fderiv ℝ f y - f'‖ < ε
B : Set E := ⋯
key : ∀ p ∈ (B ∩ s) ×ˢ (B ∩ s), ‖f p.2 - f p.1 - (f' p.2 - f' p.1)‖ ≤ ε * ‖p.2 - p.1‖
u v : E
uv_in : (u, v) ∈ closure[instTopologicalSpaceProd] ((B ∩ s) ×ˢ (B ∩ s))
f_cont' : ∀ y ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s, ContinuousWithinAt (f - ⇑f') s y
this : (B ∩ s) ×ˢ (B ∩ s) ⊆ s ×ˢ s
u_in : u ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
v_in : v ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
⊢ ContinuousWithinAt (fun y ↦ ε * ‖y.2 - y.1‖) (s ×ˢ s) (u, v)
Module: Mathlib.Analysis.Calculus.FDeriv.Extend

Goal state before:
pp: E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
s : Set E
x : E
f' : E →L[ℝ] F
f_diff : DifferentiableOn ℝ f s
s_conv : Convex ℝ s
s_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
f_cont : ∀ y ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s, ContinuousWithinAt f s y
h : Tendsto (fun y ↦ fderiv ℝ f y) (𝓝[s] x) (𝓝 f')
hx : x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
ε : ℝ
ε_pos : 0 < ε
δ : ℝ
δ_pos : δ > 0
hδ : ∀ y ∈ s, dist y x < δ → ‖fderiv ℝ f y - f'‖ < ε
B : Set E := ball x δ
key : ∀ p ∈ (B ∩ s) ×ˢ (B ∩ s), ‖f p.2 - f p.1 - (f' p.2 - f' p.1)‖ ≤ ε * ‖p.2 - p.1‖
u v : E
uv_in : (u, v) ∈ closure[instTopologicalSpaceProd] ((B ∩ s) ×ˢ (B ∩ s))
f_cont' : ∀ y ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s, ContinuousWithinAt (f - ⇑f') s y
this✝ : (B ∩ s) ×ˢ (B ∩ s) ⊆ s ×ˢ s
u_in : u ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
v_in : v ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
this : ∀ (u v : E), f v - f u - (f' v - f' u) = f v - f' v - (f u - f' u)
⊢ Tendsto (fun y ↦ ‖f y.2 - f y.1 - (f' y.2 - f' y.1)‖) (𝓝[s] u ×ˢ 𝓝[s] v) (𝓝 ‖f v - f u - (f' v - f' u)‖)

Tactic:
simp only [this]

Goal state after:
E : Type u_1
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
F : Type u_2
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
f : E → F
s : Set E
x : E
f' : E →L[ℝ] F
f_diff : DifferentiableOn ℝ f s
s_conv : Convex ℝ s
s_open : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
f_cont : ∀ y ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s, ContinuousWithinAt f s y
h : Tendsto (fun y ↦ fderiv ℝ f y) (𝓝[s] x) (𝓝 f')
hx : x ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
ε : ℝ
ε_pos : 0 < ε
δ : ℝ
δ_pos : δ > 0
hδ : ∀ y ∈ s, dist y x < δ → ‖fderiv ℝ f y - f'‖ < ε
B : Set E := ball x δ
key : ∀ p ∈ (B ∩ s) ×ˢ (B ∩ s), ‖f p.2 - f p.1 - (f' p.2 - f' p.1)‖ ≤ ε * ‖p.2 - p.1‖
u v : E
uv_in : (u, v) ∈ closure[instTopologicalSpaceProd] ((B ∩ s) ×ˢ (B ∩ s))
f_cont' : ∀ y ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s, ContinuousWithinAt (f - ⇑f') s y
this✝ : (B ∩ s) ×ˢ (B ∩ s) ⊆ s ×ˢ s
u_in : u ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
v_in : v ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
this : ∀ (u v : E), f v - f u - (f' v - f' u) = f v - f' v - (f u - f' u)
⊢ Tendsto (fun y ↦ ‖f y.2 - f' y.2 - (f y.1 - f' y.1)‖) (𝓝[s] u ×ˢ 𝓝[s] v) (𝓝 ‖f v - f' v - (f u - f' u)‖)
Module: Mathlib.RingTheory.Binomial

Goal state before:
pp: n : ℕ
⊢ (-1) ^ n * -1 * (↑n + 1) * ↑n.factorial = (-1) ^ n * -1 * ↑(n + 1).factorial

Tactic:
Nat.factorial,

Goal state after:
n : ℕ
⊢ (-1) ^ n * -1 * (↑n + 1) * ↑n.factorial = (-1) ^ n * -1 * ↑(n.succ * n.factorial)
Module: Mathlib.RingTheory.Binomial

Goal state before:
pp: n : ℕ
⊢ (ascPochhammer ℕ (n + 1)).smeval (-↑n) = 0

Tactic:
ascPochhammer_succ_right,

Goal state after:
n : ℕ
⊢ (ascPochhammer ℕ n * (X + ↑n)).smeval (-↑n) = 0
Module: Mathlib.RingTheory.Binomial

Goal state before:
pp: n k : ℕ
⊢ (ascPochhammer ℕ (n + (k + 1) + 1)).smeval (-↑n) = 0

Tactic:
ascPochhammer_succ_right,

Goal state after:
n k : ℕ
⊢ (ascPochhammer ℕ (n + (k + 1)) * (X + ↑(n + (k + 1)))).smeval (-↑n) = 0
Module: Mathlib.Analysis.Meromorphic.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
c : 𝕜
R : ℝ
⊢ MeromorphicOn (f ∘ fun x ↦ x - c) (ball c R) ↔ MeromorphicOn f (ball 0 R)

Tactic:
meromorphicOn_comp_sub_const_iff_meromorphicOn,

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
c : 𝕜
R : ℝ
⊢ MeromorphicOn f (ball c R - {c}) ↔ MeromorphicOn f (ball 0 R)
Module: Mathlib.Analysis.Meromorphic.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
c : 𝕜
R : ℝ
⊢ MeromorphicOn (f ∘ fun x ↦ x - c) (closedBall c R) ↔ MeromorphicOn f (closedBall 0 R)

Tactic:
meromorphicOn_comp_sub_const_iff_meromorphicOn,

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
c : 𝕜
R : ℝ
⊢ MeromorphicOn f (closedBall c R - {c}) ↔ MeromorphicOn f (closedBall 0 R)
Module: Mathlib.Analysis.Meromorphic.Basic

Goal state before:
pp: 𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
c : 𝕜
R : ℝ
⊢ MeromorphicOn (f ∘ fun x ↦ x - c) (sphere c R) ↔ MeromorphicOn f (sphere 0 R)

Tactic:
meromorphicOn_comp_sub_const_iff_meromorphicOn,

Goal state after:
𝕜 : Type u_1
inst✝² : NontriviallyNormedField 𝕜
E : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
c : 𝕜
R : ℝ
⊢ MeromorphicOn f (sphere c R - {c}) ↔ MeromorphicOn f (sphere 0 R)
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:
pp: x : ℝ
f g : ℝ → ℝ
f' g' : ℝ
hf : HasStrictDerivAt f f' x
hg : HasStrictDerivAt g g' x
h : 0 < f x
⊢ HasStrictDerivAt (fun x ↦ f x ^ g x) (f' * g x * f x ^ (g x - 1) + g' * f x ^ g x * log (f x)) x

Tactic:
convert!
    (hasStrictFDerivAt_rpow_of_pos ((fun x => (f x, g x)) x) h).comp_hasStrictDerivAt x
      (hf.prodMk hg) using 1
  simp [mul_assoc, mul_comm]

Goal state after:
Module: Mathlib.Analysis.Analytic.IteratedFDeriv

Goal state before:
pp: 𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
h : HasFPowerSeriesWithinOnBall f p s x r
h' : AnalyticOn 𝕜 f s
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
v : Fin n → E
h's : s ⊆ Metric.eball x r
I : insert x s ∩ Metric.eball x r = s
fcont : ContDiffOn 𝕜 (↑n) f s
g : E → F := fun z ↦ (p n) fun x_1 ↦ z - x
⊢ ContDiff 𝕜 ω g

Tactic:
apply (p n).contDiff.comp

Goal state after:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type u_3
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
f : E → F
p : FormalMultilinearSeries 𝕜 E F
s : Set E
x : E
r : ℝ≥0∞
h : HasFPowerSeriesWithinOnBall f p s x r
h' : AnalyticOn 𝕜 f s
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
v : Fin n → E
h's : s ⊆ Metric.eball x r
I : insert x s ∩ Metric.eball x r = s
fcont : ContDiffOn 𝕜 (↑n) f s
g : E → F := ⋯
⊢ ContDiff 𝕜 ω fun z x_1 ↦ z - x
Module: Mathlib.Analysis.SpecialFunctions.Pow.Deriv

Goal state before:
pp: case inl
x p : ℝ
h : x ≠ 0 ∨ 1 ≤ p
hx : x ≠ 0
⊢ HasDerivAt (fun x ↦ x ^ p) (p * x ^ (p - 1)) x

pp: case inr
p : ℝ
h : 0 ≠ 0 ∨ 1 ≤ p
⊢ HasDerivAt (fun x ↦ x ^ p) (p * 0 ^ (p - 1)) 0

Tactic:
· exact (hasStrictDerivAt_rpow_const_of_ne hx _).hasDerivAt

Goal state after:
case inr
p : ℝ
h : 0 ≠ 0 ∨ 1 ≤ p
⊢ HasDerivAt (fun x ↦ x ^ p) (p * 0 ^ (p - 1)) 0
Module: Mathlib.Analysis.SpecialFunctions.Log.ENNRealLog

Goal state before:
pp: case inr.inr.inr.inr
x y : ℝ≥0∞
x_real : 0 < x.toReal
y_real : 0 < y.toReal
⊢ (x * y).log = ↑(Real.log x.toReal) + y.log

Tactic:
log_pos_real'

Goal state after:
case inr.inr.inr.inr
x y : ℝ≥0∞
x_real : 0 < x.toReal
y_real : 0 < y.toReal
⊢ ↑(Real.log (x * y).toReal) = ↑(Real.log x.toReal) + y.log

case inr.inr.inr.inr
x y : ℝ≥0∞
x_real : 0 < x.toReal
y_real : 0 < y.toReal
⊢ 0 < (x * y).toReal
Module: Mathlib.Analysis.Asymptotics.LinearGrowth

Goal state before:
pp: u : ℕ → EReal
a : EReal
⊢ a ≤ liminf (fun n ↦ u n / ↑n) atTop ↔ ∀ b < a, ∀ᶠ (n : ℕ) in atTop, b * ↑n ≤ u n

Tactic:
le_liminf_iff'

Goal state after:
u : ℕ → EReal
a : EReal
⊢ (∀ y < a, ∀ᶠ (a : ℕ) in atTop, y ≤ u a / ↑a) ↔ ∀ b < a, ∀ᶠ (n : ℕ) in atTop, b * ↑n ≤ u n
Module: Mathlib.Analysis.Asymptotics.LinearGrowth

Goal state before:
pp: case inl
u v : ℕ → EReal
hb : ⊥ ≠ ⊤
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + ⊥
⊢ (linearGrowthInf fun x ↦ v x + ⊥) ≤ linearGrowthInf v

pp: case inr
u v : ℕ → EReal
b : EReal
hb : b ≠ ⊤
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + b
b_bot : ⊥ < b
⊢ (linearGrowthInf fun x ↦ v x + b) ≤ linearGrowthInf v

Tactic:
· simp only [add_bot, ← Pi.bot_def, linearGrowthInf_bot, bot_le]

Goal state after:
case inr
u v : ℕ → EReal
b : EReal
hb : b ≠ ⊤
h : ∀ᶠ (n : ℕ) in atTop, u n ≤ v n + b
b_bot : ⊥ < b
⊢ (linearGrowthInf fun x ↦ v x + b) ≤ linearGrowthInf v
Module: Mathlib.Analysis.Asymptotics.ExpGrowth

Goal state before:
pp: u : ℕ → ℝ≥0∞
a : EReal
⊢ a ≤ liminf (fun n ↦ (u n).log / ↑n) atTop ↔ ∀ b < a, ∀ᶠ (n : ℕ) in atTop, (b * ↑n).exp ≤ u n

Tactic:
le_liminf_iff'

Goal state after:
u : ℕ → ℝ≥0∞
a : EReal
⊢ (∀ y < a, ∀ᶠ (a : ℕ) in atTop, y ≤ (u a).log / ↑a) ↔ ∀ b < a, ∀ᶠ (n : ℕ) in atTop, (b * ↑n).exp ≤ u n
Module: Mathlib.Analysis.Asymptotics.LinearGrowth

Goal state before:
pp: case neg.inr.refine_2
u : ℕ → EReal
v : ℕ → ℕ
h : Monotone u
hv₀ : (linearGrowthSup fun n ↦ ↑(v n)) ≠ 0
hv₁ : (linearGrowthSup fun n ↦ ↑(v n)) ≠ ⊤
u_0 : ¬u = ⊥
v_0 : 0 < linearGrowthSup fun n ↦ ↑(v n)
a : EReal
v_a : a > linearGrowthSup fun n ↦ ↑(v n)
b : EReal
a_0 : 0 < a
b_0 : 0 < b
a_top : a ≠ ⊤
a' : EReal
a_a' : a' < a
a_0' : 0 < a'
a_a_inv' : a⁻¹ < a'⁻¹
v_a' : ∀ᶠ (n : ℕ) in atTop, ↑(v n) ≤ a' * ↑n
u_b : ∃ᶠ (x : ℕ) in atTop, (u x ≤ b * ↑x ∧ 0 < x) ∧ ∃ m, a⁻¹ * ↑x ≤ ↑m ∧ ↑m ≤ a'⁻¹ * ↑x
M M' : ℕ
aM_M' : a * ↑M ≤ ↑M'
n : ℕ
n_M' : M' ≤ n
un_bn : u n ≤ b * ↑n
right✝ : 0 < n
k : ℕ
an_k : a⁻¹ * ↑n ≤ ↑k
k_an' : ↑k ≤ ↑n / a'
vk_ak' : ↑(v k) ≤ a' * ↑k
⊢ u (v k) ≤ b * (a * ↑k)

Tactic:
le_div_iff_mul_le a_0' (ne_top_of_lt a_a'),

Goal state after:
case neg.inr.refine_2
u : ℕ → EReal
v : ℕ → ℕ
h : Monotone u
hv₀ : (linearGrowthSup fun n ↦ ↑(v n)) ≠ 0
hv₁ : (linearGrowthSup fun n ↦ ↑(v n)) ≠ ⊤
u_0 : ¬u = ⊥
v_0 : 0 < linearGrowthSup fun n ↦ ↑(v n)
a : EReal
v_a : a > linearGrowthSup fun n ↦ ↑(v n)
b : EReal
a_0 : 0 < a
b_0 : 0 < b
a_top : a ≠ ⊤
a' : EReal
a_a' : a' < a
a_0' : 0 < a'
a_a_inv' : a⁻¹ < a'⁻¹
v_a' : ∀ᶠ (n : ℕ) in atTop, ↑(v n) ≤ a' * ↑n
u_b : ∃ᶠ (x : ℕ) in atTop, (u x ≤ b * ↑x ∧ 0 < x) ∧ ∃ m, a⁻¹ * ↑x ≤ ↑m ∧ ↑m ≤ a'⁻¹ * ↑x
M M' : ℕ
aM_M' : a * ↑M ≤ ↑M'
n : ℕ
n_M' : M' ≤ n
un_bn : u n ≤ b * ↑n
right✝ : 0 < n
k : ℕ
an_k : a⁻¹ * ↑n ≤ ↑k
k_an' : ↑k * a' ≤ ↑n
vk_ak' : ↑(v k) ≤ a' * ↑k
⊢ u (v k) ≤ b * (a * ↑k)
Module: Mathlib.MeasureTheory.Covering.Vitali

Goal state before:
pp: α : Type u_1
ι : Type u_2
inst✝⁴ : PseudoMetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)
μB : ∀ a ∈ t, μ (closedBall (c a) (3 * r a)) ≤ ↑C * μ (B a)
ht : ∀ a ∈ t, (interior (B a)).Nonempty
h't : ∀ a ∈ t, IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (B a)
hf : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ t, r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), μ (closedBall x (20 * R x)) < ∞
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
A : ∀ a ∈ t', r a ≤ 1
A' : ∀ a ∈ t', (B a).Nonempty
a : ι
ha : a ∈ t'
⊢ 0 ≤ r a

Tactic:
exact nonempty_closedBall.1 ((A' a ha).mono (hB a ha.1))

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

Goal state before:
pp: case inr
α : Type u_1
ι : Type u_2
inst✝⁴ : PseudoMetricSpace α
inst✝³ : MeasurableSpace α
inst✝² : OpensMeasurableSpace α
inst✝¹ : SecondCountableTopology α
μ : Measure α
inst✝ : IsLocallyFiniteMeasure μ
s : Set α
t : Set ι
C : ℝ≥0
r : ι → ℝ
c : ι → α
B : ι → Set α
hB : ∀ a ∈ t, B a ⊆ closedBall (c a) (r a)
μB : ∀ a ∈ t, μ (closedBall (c a) (3 * r a)) ≤ ↑C * μ (B a)
ht : ∀ a ∈ t, (interior (B a)).Nonempty
h't : ∀ a ∈ t, IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (B a)
hf : ∀ x ∈ s, ∀ ε > 0, ∃ a ∈ t, r a ≤ ε ∧ c a = x
R : α → ℝ
hR0 : ∀ (x : α), 0 < R x
hR1 : ∀ (x : α), R x ≤ 1
hRμ : ∀ (x : α), μ (closedBall x (20 * R x)) < ∞
t' : Set ι := {a | a ∈ t ∧ r a ≤ R (c a)}
u : Set ι
ut' : u ⊆ t'
u_disj : u.PairwiseDisjoint B
hu : ∀ a ∈ t', ∃ b ∈ u, (B a ∩ B b).Nonempty ∧ r a ≤ 2 * r b
ut : u ⊆ t
u_count : u.Countable
x : α
x✝ : x ∈ s \ ⋃ a ∈ u, B a
v : Set ι := {a | a ∈ u ∧ (B a ∩ ball x (R x)).Nonempty}
vu : v ⊆ u
Idist_v : ∀ a ∈ v, dist (c a) x ≤ r a + R x
R0 : ℝ := sSup (r '' v)
R0_def : R0 = sSup (r '' v)
R0_bdd : BddAbove (r '' v)
H : R x ≤ R0
⊢ ∃ K, μ (closedBall x K) < ∞ ∧ ∀ a ∈ u, (B a ∩ ball x (R x)).Nonempty → B a ⊆ closedBall x K

Tactic:
have R0pos : 0 < R0 := (hR0 x).trans_le H
      have vnonempty : v.Nonempty := by
        by_contra h
        rw [nonempty_iff_ne_empty, Classical.not_not] at h
        rw [h, image_empty, Real.sSup_empty] at R0_def
        exact lt_irrefl _ (R0pos.trans_le (le_of_eq R0_def))
      obtain ⟨a, hav, R0a⟩ : ∃ a ∈ v, R0 / 2 < r a := by
        obtain ⟨r', r'mem, hr'⟩ : ∃ r' ∈ r '' v, R0 / 2 < r' :=
          exists_lt_of_lt_csSup (vnonempty.image _) (half_lt_self R0pos)
        rcases (mem_image _ _ _).1 r'mem with ⟨a, hav, rfl⟩
        exact ⟨a, hav, hr'⟩
      refine ⟨8 * R0, ?_, ?_⟩
      · apply lt_of_le_of_lt (measure_mono _) (hRμ (c a))
        apply closedBall_subset_closedBall'
        rw [dist_comm]
        linarith [Idist_v a hav, (ut' (vu hav)).2]
      · intro b bu hbx
        refine (hB b (ut bu)).trans ?_
        apply closedBall_subset_closedBall'
        have : r b ≤ R0 := le_csSup R0_bdd (mem_image_of_mem _ ⟨bu, hbx⟩)
        linarith [Idist_v b ⟨bu, hbx⟩]

Goal state after: