mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.AbsoluteValue.Equivalence

Goal state before:
pp: F : Type u_1
inst✝ : Field F
v w : AbsoluteValue F ℝ
h : v.IsEquiv w
hw : ¬w.IsNontrivial
x : F
⊢ v x ^ 1 = w x

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.AbsoluteValue.Equivalence

Goal state before:
pp: F : Type u_1
inst✝ : Field F
v w : AbsoluteValue F ℝ
⊢ v.IsEquiv w ↔ ∃ c, 0 < c ∧ (fun x ↦ v x ^ c) = ⇑w

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.AbsoluteValue.Equivalence

Goal state before:
pp: F : Type u_1
inst✝ : Field F
v w : AbsoluteValue F ℝ
h : v.IsEquiv w
U : Set (WithAbs v)
hU : U ∈ 𝓝 0
⊢ ⇑(WithAbs.congr v w (RingEquiv.refl F)) '' U ∈ 𝓝 0

Tactic:
Metric.mem_nhds_iff

Goal state after:
F : Type u_1
inst✝ : Field F
v w : AbsoluteValue F ℝ
h : v.IsEquiv w
U : Set (WithAbs v)
hU : ∃ ε > 0, Metric.ball 0 ε ⊆ U
⊢ ∃ ε > 0, Metric.ball 0 ε ⊆ ⇑(WithAbs.congr v w (RingEquiv.refl F)) '' 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
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
⊢ ↑f'symm.nnnorm * dist (f b) y * (1 - (↑c * ↑f'symm.nnnorm) ^ n) ≤ ↑f'symm.nnnorm * dist (f b) y * 1

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.AbsoluteValue.Equivalence

Goal state before:
pp: F : Type u_1
inst✝ : Field F
v w : AbsoluteValue F ℝ
h : v.IsEquiv w
U : Set (WithAbs v)
hU : U ∈ 𝓝 0
⊢ ⇑(WithAbs.congr v w (RingEquiv.refl F)) '' U ∈ 𝓝 0

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
hf : ¬AnalyticAt 𝕜 f z₀
⊢ analyticOrderNatAt f z₀ = 0

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.AbsoluteValue.Equivalence

Goal state before:
pp: F : Type u_1
inst✝ : Field F
v w : AbsoluteValue F ℝ
h : v.IsEquiv w
U : Set (WithAbs v)
hU : U ∈ 𝓝 0
⊢ ⇑(WithAbs.congr v w (RingEquiv.refl F)) ⁻¹' ⇑(WithAbs.congr v w (RingEquiv.refl F)) '' U ⊆ U

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
hf : analyticOrderAt f z₀ = ⊤
⊢ ∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = 0

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
hf : ∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = 0
⊢ analyticOrderAt f z₀ = ⊤

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
⊢ EventuallyConst f (𝓝 z₀) ↔ analyticOrderAt (fun x ↦ f x - f z₀) z₀ = ⊤

Tactic:
by

Goal state after:
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
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
⊢ ↑f'symm.nnnorm * dist (f b) y * 1 ≤ ↑f'symm.nnnorm * (((↑f'symm.nnnorm)⁻¹ - ↑c) * ε)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.AbsoluteValue.Equivalence

Goal state before:
pp: F : Type u_1
inst✝ : Field F
v w : AbsoluteValue F ℝ
h : v.IsEquiv w
⊢ IsEmbedding ⇑(WithAbs.congr v w (RingEquiv.refl F))

Tactic:
by

Goal state after:
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
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
⊢ ↑f'symm.nnnorm * (((↑f'symm.nnnorm)⁻¹ - ↑c) * ε) = ε * (1 - ↑c * ↑f'symm.nnnorm)

Tactic:
by

Goal state after:
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
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
⊢ w ∈ closedBall b ε

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
n : ℕ
z₀ : 𝕜
hf : AnalyticAt 𝕜 f z₀
⊢ analyticOrderAt f z₀ = ↑n ↔ ∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = (z - z₀) ^ n • g z

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
hf : AnalyticAt 𝕜 f z₀
hf' : analyticOrderAt f z₀ ≠ ⊤
n : ℕ
⊢ analyticOrderNatAt f z₀ = n ↔ ∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = (z - z₀) ^ n • g z

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
hf : AnalyticAt 𝕜 f z₀
⊢ analyticOrderAt f z₀ ≠ ⊤ ↔
    ∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ f =ᶠ[𝓝 z₀] fun z ↦ (z - z₀) ^ analyticOrderNatAt f z₀ • g z

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.AbsoluteValue.Equivalence

Goal state before:
pp: F : Type u_1
inst✝ : Field F
v w : AbsoluteValue F ℝ
x✝ : IsEmbedding ⇑(WithAbs.congr v w (RingEquiv.refl F)) ∧ Function.Surjective ⇑(WithAbs.congr v w (RingEquiv.refl F))
hi : IsEmbedding ⇑(WithAbs.congr v w (RingEquiv.refl F))
right✝ : Function.Surjective ⇑(WithAbs.congr v w (RingEquiv.refl F))
x : F
h : Filter.Tendsto (fun n ↦ WithAbs.toAbs v x ^ n) Filter.atTop (𝓝 0)
⊢ Filter.Tendsto (fun n ↦ WithAbs.toAbs w x ^ n) Filter.atTop (𝓝 0)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.AbsoluteValue.Equivalence

Goal state before:
pp: F : Type u_1
inst✝ : Field F
v w : AbsoluteValue F ℝ
x✝ : IsEmbedding ⇑(WithAbs.congr v w (RingEquiv.refl F)) ∧ Function.Surjective ⇑(WithAbs.congr v w (RingEquiv.refl F))
hi : IsEmbedding ⇑(WithAbs.congr v w (RingEquiv.refl F))
right✝ : Function.Surjective ⇑(WithAbs.congr v w (RingEquiv.refl F))
x : F
h : Filter.Tendsto (fun n ↦ WithAbs.toAbs w x ^ n) Filter.atTop (𝓝 0)
⊢ Filter.Tendsto (fun n ↦ WithAbs.toAbs v x ^ n) Filter.atTop (𝓝 0)

Tactic:
by

Goal state after:
Module: Mathlib.Analysis.AbsoluteValue.Equivalence

Goal state before:
pp: F : Type u_1
inst✝ : Field F
v w : AbsoluteValue F ℝ
⊢ v.IsEquiv w ↔ IsHomeomorph ⇑(WithAbs.congr v w (RingEquiv.refl F))

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
hf : AnalyticAt 𝕜 f z₀
hz : ¬AnalyticAt 𝕜 f z₀ ∨ f z₀ ≠ 0
⊢ ∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = (z - z₀) ^ 0 • f z

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
⊢ analyticOrderAt f z₀ = 0 ↔ ¬AnalyticAt 𝕜 f z₀ ∨ f z₀ ≠ 0

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
⊢ analyticOrderAt f z₀ ≠ 0 ↔ AnalyticAt 𝕜 f z₀ ∧ f z₀ = 0

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
hf : AnalyticAt 𝕜 f z₀
⊢ analyticOrderAt f z₀ = 0 ↔ f z₀ ≠ 0

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
hf : analyticOrderAt f z₀ ≠ 0
⊢ f z₀ = 0

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
hf : analyticOrderNatAt f z₀ ≠ 0
⊢ f z₀ = 0

Tactic:
by

Goal state after:
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 ↦
      ∑ 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))

Tactic:
refine tendsto_finsetSum _ fun w _ ↦ ?_

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 w : ι
x✝ : w ∈ Finset.univ
⊢ Tendsto
    (fun i ↦
      (WithAbs.equiv (v u)).symm (1 / (1 + (a w)⁻¹ ^ i)) * (WithAbs.equiv (v u)).symm ((WithAbs.equiv (v w)) (z w)))
    atTop (𝓝 (Pi.single w (z w) u))
Module: Mathlib.Analysis.Analytic.Order

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
hf : AnalyticAt 𝕜 f z₀
n : ℕ
h : ∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = 0
⊢ ∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = (z - z₀) ^ n • 0 z

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
hf : AnalyticAt 𝕜 f z₀
n : ℕ
h : ¬∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = 0
m : ℕ := ⋯.choose
g : 𝕜 → E
hg : AnalyticAt 𝕜 g z₀
hg_ne : g z₀ ≠ 0
hm : ∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = (z - z₀) ^ ⋯.choose • g z
hmn : n ≤ ⋯.choose
⊢ AnalyticAt 𝕜 (fun z ↦ (z - z₀) ^ (m - n) • g z) z₀

Tactic:
by

Goal state after:
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 w : ι
x✝ : w ∈ Finset.univ
hw : u = w
⊢ (v u) (a u)⁻¹ < 1

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
hf : AnalyticAt 𝕜 f z₀
n : ℕ
h : ¬∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = 0
m : ℕ := ⋯.choose
g : 𝕜 → E
hg : AnalyticAt 𝕜 g z₀
hg_ne : g z₀ ≠ 0
hm : ∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = (z - z₀) ^ ⋯.choose • g z
hmn : n ≤ ⋯.choose
z : 𝕜
hz : f z = (z - z₀) ^ ⋯.choose • g z
⊢ f z = (z - z₀) ^ n • (z - z₀) ^ (m - n) • g z

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
hf : AnalyticAt 𝕜 f z₀
n : ℕ
h✝ : ¬∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = 0
m : ℕ := ⋯.choose
g : 𝕜 → E
hg : AnalyticAt 𝕜 g z₀
hm : ∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = (z - z₀) ^ ⋯.choose • g z
x✝ : ∃ g, AnalyticAt 𝕜 g z₀ ∧ ∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = (z - z₀) ^ n • g z
h : 𝕜 → E
hh : AnalyticAt 𝕜 h z₀
hfh : ∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = (z - z₀) ^ n • h z
hg_ne : ⋯.choose < n
⊢ ContinuousAt (fun z ↦ (z - z₀) ^ (n - m) • h z) z₀

Tactic:
by

Goal state after:
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 w : ι
x✝ : w ∈ Finset.univ
hw : ¬u = w
⊢ 1 < (v u) (a w)⁻¹

Tactic:
by

Goal state after:
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)
⊢ Tendsto (fun n i ↦ (WithAbs.equiv (v i)).symm (y n)) atTop (𝓝 z)

Tactic:
by

Goal state after:
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
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
C :
  ∀ (n : ℕ) (w : E),
    dist w b ≤ ↑f'symm.nnnorm * (1 - (↑c * ↑f'symm.nnnorm) ^ n) / (1 - ↑c * ↑f'symm.nnnorm) * dist (f b) y →
      w ∈ closedBall b ε
n : ℕ
IH :
  dist (f (u n)) y ≤ (↑c * ↑f'symm.nnnorm) ^ n * dist (f b) y ∧
    dist (u n) b ≤ ↑f'symm.nnnorm * (1 - (↑c * ↑f'symm.nnnorm) ^ n) / (1 - ↑c * ↑f'symm.nnnorm) * dist (f b) y
⊢ ↑f'symm.nnnorm * dist (f (u n)) y + dist (u n) b ≤
    ↑f'symm.nnnorm * ((↑c * ↑f'symm.nnnorm) ^ n * dist (f b) y) +
      ↑f'symm.nnnorm * (1 - (↑c * ↑f'symm.nnnorm) ^ n) / (1 - ↑c * ↑f'symm.nnnorm) * dist (f b) y

Tactic:
by

Goal state after:
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)
⊢ DenseRange ⇑(algebraMap F ((i : ι) → WithAbs (v i)))

Tactic:
by

Goal state after:
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
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
C :
  ∀ (n : ℕ) (w : E),
    dist w b ≤ ↑f'symm.nnnorm * (1 - (↑c * ↑f'symm.nnnorm) ^ n) / (1 - ↑c * ↑f'symm.nnnorm) * dist (f b) y →
      w ∈ closedBall b ε
n : ℕ
IH :
  dist (f (u n)) y ≤ (↑c * ↑f'symm.nnnorm) ^ n * dist (f b) y ∧
    dist (u n) b ≤ ↑f'symm.nnnorm * (1 - (↑c * ↑f'symm.nnnorm) ^ n) / (1 - ↑c * ↑f'symm.nnnorm) * dist (f b) y
⊢ 1 - ↑f'symm.nnnorm * ↑c ≠ 0

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
hf : AnalyticAt 𝕜 f z₀
n : ℕ
h✝ : ¬∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = 0
m : ℕ := ⋯.choose
g : 𝕜 → E
hg : AnalyticAt 𝕜 g z₀
hm : ∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = (z - z₀) ^ ⋯.choose • g z
x✝ : ∃ g, AnalyticAt 𝕜 g z₀ ∧ ∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = (z - z₀) ^ n • g z
h : 𝕜 → E
hh : AnalyticAt 𝕜 h z₀
hfh : ∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = (z - z₀) ^ n • h z
hg_ne : ⋯.choose < n
this : ContinuousAt (fun z ↦ (z - z₀) ^ (n - m) • h z) z₀
z : 𝕜
hz : z ∈ {z₀}ᶜ
hf' : ((z - z₀) ^ n * ((z - z₀) ^ ⋯.choose)⁻¹) • h z = g z
hf'' : f z = (z - z₀) ^ n • h z
⊢ m ≤ n

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
hf : AnalyticAt 𝕜 f z₀
n : ℕ
⊢ ↑n ≤ analyticOrderAt f z₀ ↔ ∃ g, AnalyticAt 𝕜 g z₀ ∧ ∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = (z - z₀) ^ n • g z

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f g : 𝕜 → E
z₀ : 𝕜
hfg : f =ᶠ[𝓝 z₀] g
⊢ analyticOrderAt f z₀ = analyticOrderAt g z₀

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
⊢ AnalyticAt 𝕜 (fun x ↦ 1) 0

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
⊢ (fun x ↦ 1) 0 ≠ 0

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
⊢ ∀ᶠ (z : 𝕜) in 𝓝 0, id z = (z - 0) ^ 1 • (fun x ↦ 1) z

Tactic:
by

Goal state after:
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
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
C :
  ∀ (n : ℕ) (w : E),
    dist w b ≤ ↑f'symm.nnnorm * (1 - (↑c * ↑f'symm.nnnorm) ^ n) / (1 - ↑c * ↑f'symm.nnnorm) * dist (f b) y →
      w ∈ closedBall b ε
n : ℕ
IH :
  dist (f (u n)) y ≤ (↑c * ↑f'symm.nnnorm) ^ n * dist (f b) y ∧
    dist (u n) b ≤ ↑f'symm.nnnorm * (1 - (↑c * ↑f'symm.nnnorm) ^ n) / (1 - ↑c * ↑f'symm.nnnorm) * dist (f b) y
⊢ ↑f'symm.nnnorm * ((↑c * ↑f'symm.nnnorm) ^ n * dist (f b) y) +
      ↑f'symm.nnnorm * (1 - (↑c * ↑f'symm.nnnorm) ^ n) / (1 - ↑c * ↑f'symm.nnnorm) * dist (f b) y =
    ↑f'symm.nnnorm * (1 - (↑c * ↑f'symm.nnnorm) ^ n.succ) / (1 - ↑c * ↑f'symm.nnnorm) * dist (f b) y

Tactic:
by

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

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
f : 𝕜 → E
z₀ : 𝕜
hf : AnalyticAt 𝕜 f z₀
n : ℕ
⊢ ∀ (a : 𝕜 → E),
    (AnalyticAt 𝕜 a z₀ ∧ ∀ᶠ (z : 𝕜) in 𝓝 z₀, (-f) z = (z - z₀) ^ n • a z) ↔
      AnalyticAt 𝕜 ((Equiv.neg (𝕜 → E)) a) z₀ ∧ ∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = (z - z₀) ^ n • (Equiv.neg (𝕜 → E)) a z

Tactic:
by

Goal state after: