mathlib-initiative/mathlib-tactics huggingface dataset:
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 = 1 - ↑c * ↑f'symm.nnnorm

Tactic:
;

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
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 = 1 - ↑c * ↑f'symm.nnnorm
Module: Mathlib.Analysis.Analytic.Order

Goal state before:
pp: case neg
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 = analyticOrderAt f z₀

Tactic:
· rw [analyticOrderAt_of_not_analyticAt hf,
      analyticOrderAt_of_not_analyticAt <| analyticAt_neg.not.2 hf]

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

Goal state before:
pp: case pos
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
hf : AnalyticAt 𝕜 f z₀
hg : AnalyticAt 𝕜 g z₀
n : ℕ
⊢ ((∃ g, AnalyticAt 𝕜 g z₀ ∧ ∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = (z - z₀) ^ n • g z) ∧
      ∃ g_1, AnalyticAt 𝕜 g_1 z₀ ∧ ∀ᶠ (z : 𝕜) in 𝓝 z₀, g z = (z - z₀) ^ n • g_1 z) →
    ∃ g_1, AnalyticAt 𝕜 g_1 z₀ ∧ ∀ᶠ (z : 𝕜) in 𝓝 z₀, (f + g) z = (z - z₀) ^ n • g_1 z

Tactic:
refine fun ⟨⟨F, hF, hF'⟩, ⟨G, hG, hG'⟩⟩ ↦ ⟨F + G, hF.add hG, ?_⟩

Goal state after:
case pos
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
hf : AnalyticAt 𝕜 f z₀
hg : AnalyticAt 𝕜 g z₀
n : ℕ
x✝ :
  (∃ g, AnalyticAt 𝕜 g z₀ ∧ ∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = (z - z₀) ^ n • g z) ∧
    ∃ g_1, AnalyticAt 𝕜 g_1 z₀ ∧ ∀ᶠ (z : 𝕜) in 𝓝 z₀, g z = (z - z₀) ^ n • g_1 z
F : 𝕜 → E
hF : AnalyticAt 𝕜 F z₀
hF' : ∀ᶠ (z : 𝕜) in 𝓝 z₀, f z = (z - z₀) ^ n • F z
G : 𝕜 → E
hG : AnalyticAt 𝕜 G z₀
hG' : ∀ᶠ (z : 𝕜) in 𝓝 z₀, g z = (z - z₀) ^ n • G z
⊢ ∀ᶠ (z : 𝕜) in 𝓝 z₀, (f + g) z = (z - z₀) ^ n • (F + G) z
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
hgf : analyticOrderAt g z₀ < analyticOrderAt f z₀
⊢ analyticOrderAt (g + f) z₀ = analyticOrderAt g z₀

Tactic:
analyticOrderAt_add_eq_left_of_lt hgf

Goal state after:
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
hgf : analyticOrderAt g z₀ < analyticOrderAt f z₀
⊢ analyticOrderAt g z₀ = analyticOrderAt g z₀
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
Ign :
  dist (g (u n)) b ≤ ↑f'symm.nnnorm * (1 - (↑c * ↑f'symm.nnnorm) ^ n.succ) / (1 - ↑c * ↑f'symm.nnnorm) * dist (f b) y
⊢ ↑c * ↑f'symm.nnnorm * ((↑c * ↑f'symm.nnnorm) ^ n * dist (f b) y) =
    ↑c * ↑f'symm.nnnorm * (↑c * ↑f'symm.nnnorm) ^ n * dist (f b) y

Tactic:
;

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
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
Ign :
  dist (g (u n)) b ≤ ↑f'symm.nnnorm * (1 - (↑c * ↑f'symm.nnnorm) ^ n.succ) / (1 - ↑c * ↑f'symm.nnnorm) * dist (f b) y
⊢ ↑c * ↑f'symm.nnnorm * ((↑c * ↑f'symm.nnnorm) ^ n * dist (f b) y) =
    ↑c * ↑f'symm.nnnorm * (↑c * ↑f'symm.nnnorm) ^ n * dist (f b) y
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
g : 𝕜 → E
z₀ : 𝕜
f : 𝕜 → 𝕜
hf : AnalyticAt 𝕜 f z₀
hg : AnalyticAt 𝕜 g z₀
hf' : ¬analyticOrderAt f z₀ = ⊤
hg' : ¬analyticOrderAt g z₀ = ⊤
f' : 𝕜 → 𝕜
h₁f' : AnalyticAt 𝕜 f' z₀
h₂f' : f' z₀ ≠ 0
h₃f' : f =ᶠ[𝓝 z₀] fun z ↦ (z - z₀) ^ analyticOrderNatAt f z₀ • f' z
g' : 𝕜 → E
h₁g' : AnalyticAt 𝕜 g' z₀
h₂g' : g' z₀ ≠ 0
h₃g' : g =ᶠ[𝓝 z₀] fun z ↦ (z - z₀) ^ analyticOrderNatAt g z₀ • g' z
t : Set 𝕜
h₁t : ∀ y ∈ t, f y = (fun z ↦ (z - z₀) ^ analyticOrderNatAt f z₀ • f' z) y
h₂t : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t
h₃t : z₀ ∈ t
s : Set 𝕜
h₁s : ∀ y ∈ s, g y = (fun z ↦ (z - z₀) ^ analyticOrderNatAt g z₀ • g' z) y
h₂s : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
h₃s : z₀ ∈ s
y : 𝕜
hy : y ∈ t ∩ s
⊢ ((y - z₀) ^ analyticOrderNatAt f z₀ * f' y) • (y - z₀) ^ analyticOrderNatAt g z₀ • g' y =
    (y - z₀) ^ (analyticOrderNatAt f z₀ + analyticOrderNatAt g z₀) • f' y • g' y

Tactic:
;

Goal state after:
𝕜 : Type u_1
E : Type u_2
inst✝² : NontriviallyNormedField 𝕜
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace 𝕜 E
g : 𝕜 → E
z₀ : 𝕜
f : 𝕜 → 𝕜
hf : AnalyticAt 𝕜 f z₀
hg : AnalyticAt 𝕜 g z₀
hf' : ¬analyticOrderAt f z₀ = ⊤
hg' : ¬analyticOrderAt g z₀ = ⊤
f' : 𝕜 → 𝕜
h₁f' : AnalyticAt 𝕜 f' z₀
h₂f' : f' z₀ ≠ 0
h₃f' : f =ᶠ[𝓝 z₀] fun z ↦ (z - z₀) ^ analyticOrderNatAt f z₀ • f' z
g' : 𝕜 → E
h₁g' : AnalyticAt 𝕜 g' z₀
h₂g' : g' z₀ ≠ 0
h₃g' : g =ᶠ[𝓝 z₀] fun z ↦ (z - z₀) ^ analyticOrderNatAt g z₀ • g' z
t : Set 𝕜
h₁t : ∀ y ∈ t, f y = (fun z ↦ (z - z₀) ^ analyticOrderNatAt f z₀ • f' z) y
h₂t : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] t
h₃t : z₀ ∈ t
s : Set 𝕜
h₁s : ∀ y ∈ s, g y = (fun z ↦ (z - z₀) ^ analyticOrderNatAt g z₀ • g' z) y
h₂s : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s
h₃s : z₀ ∈ s
y : 𝕜
hy : y ∈ t ∩ s
⊢ ((y - z₀) ^ analyticOrderNatAt f z₀ * f' y) • (y - z₀) ^ analyticOrderNatAt g z₀ • g' y =
    (y - z₀) ^ (analyticOrderNatAt f z₀ + analyticOrderNatAt g z₀) • f' y • g' y
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
⊢ ↑c < ↑‖↑f'.symm‖₊⁻¹

Tactic:
;

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.InverseFunctionTheorem.ApproximatesLinearOn

Goal state before:
pp: case hbc
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
f' : E ≃L[𝕜] F
s : Set E
c : ℝ≥0
hf : ApproximatesLinearOn f (↑f') s c
hc : Subsingleton E ∨ c < ‖↑f'.symm‖₊⁻¹
A : PartialEquiv E F := ⋯
Af : ∀ (z : E), ↑A z = f z
x' : E
x's : x' ∈ s
hx : f x' ∈ f '' s
y' : E
y's : y' ∈ s
hy : f y' ∈ f '' s
⊢ ‖↑A y' - ↑A x' - f' (y' - x')‖ ≤ ↑c * ‖y' - x'‖

Tactic:
;

Goal state after:
case hbc
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
f' : E ≃L[𝕜] F
s : Set E
c : ℝ≥0
hf : ApproximatesLinearOn f (↑f') s c
hc : Subsingleton E ∨ c < ‖↑f'.symm‖₊⁻¹
A : PartialEquiv E F := ⋯
Af : ∀ (z : E), ↑A z = f z
x' : E
x's : x' ∈ s
hx : f x' ∈ f '' s
y' : E
y's : y' ∈ s
hy : f y' ∈ f '' s
⊢ ‖↑A y' - ↑A x' - f' (y' - x')‖ ≤ ↑c * ‖y' - x'‖
Module: Mathlib.Analysis.Analytic.Order

Goal state before:
pp: case pos
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
z₀ : 𝕜
inst✝¹ : CharZero 𝕜
inst✝ : CompleteSpace E
n : ℕ
IH : ∀ {f : 𝕜 → E}, AnalyticAt 𝕜 f z₀ → (↑n ≤ analyticOrderAt f z₀ ↔ ∀ i < n, iteratedDeriv i f z₀ = 0)
f : 𝕜 → E
hf : AnalyticAt 𝕜 f z₀
hfz : f z₀ = 0
⊢ ↑(n + 1) ≤ analyticOrderAt f z₀ ↔ ∀ i < n + 1, iteratedDeriv i f z₀ = 0

pp: case neg
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
z₀ : 𝕜
inst✝¹ : CharZero 𝕜
inst✝ : CompleteSpace E
n : ℕ
IH : ∀ {f : 𝕜 → E}, AnalyticAt 𝕜 f z₀ → (↑n ≤ analyticOrderAt f z₀ ↔ ∀ i < n, iteratedDeriv i f z₀ = 0)
f : 𝕜 → E
hf : AnalyticAt 𝕜 f z₀
hfz : ¬f z₀ = 0
⊢ ↑(n + 1) ≤ analyticOrderAt f z₀ ↔ ∀ i < n + 1, iteratedDeriv i f z₀ = 0

Tactic:
;

Goal state after:
case pos
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
z₀ : 𝕜
inst✝¹ : CharZero 𝕜
inst✝ : CompleteSpace E
n : ℕ
IH : ∀ {f : 𝕜 → E}, AnalyticAt 𝕜 f z₀ → (↑n ≤ analyticOrderAt f z₀ ↔ ∀ i < n, iteratedDeriv i f z₀ = 0)
f : 𝕜 → E
hf : AnalyticAt 𝕜 f z₀
hfz : f z₀ = 0
⊢ ↑(n + 1) ≤ analyticOrderAt f z₀ ↔ ∀ i < n + 1, iteratedDeriv i f z₀ = 0

case neg
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
z₀ : 𝕜
inst✝¹ : CharZero 𝕜
inst✝ : CompleteSpace E
n : ℕ
IH : ∀ {f : 𝕜 → E}, AnalyticAt 𝕜 f z₀ → (↑n ≤ analyticOrderAt f z₀ ↔ ∀ i < n, iteratedDeriv i f z₀ = 0)
f : 𝕜 → E
hf : AnalyticAt 𝕜 f z₀
hfz : ¬f z₀ = 0
⊢ ↑(n + 1) ≤ analyticOrderAt f z₀ ↔ ∀ i < n + 1, iteratedDeriv i f z₀ = 0
Module: Mathlib.Analysis.Calculus.InverseFunctionTheorem.ApproximatesLinearOn

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
f' : E ≃L[𝕜] F
s : Set E
c : ℝ≥0
hf : ApproximatesLinearOn f (↑f') s c
hc : Subsingleton E ∨ c < ‖↑f'.symm‖₊⁻¹
A : PartialEquiv E F := hf.toPartialEquiv hc
Af : ∀ (z : E), ↑A z = f z
x' : E
x's : x' ∈ s
hx : f x' ∈ f '' s
y' : E
y's : y' ∈ s
hy : f y' ∈ f '' s
⊢ ↑‖↑f'.symm‖₊ * (↑c * (↑(‖↑f'.symm‖₊⁻¹ - c)⁻¹ * ‖↑A x' - ↑A y'‖)) =
    ↑‖↑f'.symm‖₊ * ↑(‖↑f'.symm‖₊⁻¹ - c)⁻¹ * ↑c * ‖↑A x' - ↑A y'‖

Tactic:
;

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
f' : E ≃L[𝕜] F
s : Set E
c : ℝ≥0
hf : ApproximatesLinearOn f (↑f') s c
hc : Subsingleton E ∨ c < ‖↑f'.symm‖₊⁻¹
A : PartialEquiv E F := hf.toPartialEquiv hc
Af : ∀ (z : E), ↑A z = f z
x' : E
x's : x' ∈ s
hx : f x' ∈ f '' s
y' : E
y's : y' ∈ s
hy : f y' ∈ f '' s
⊢ ↑‖↑f'.symm‖₊ * (↑c * (↑(‖↑f'.symm‖₊⁻¹ - c)⁻¹ * ‖↑A x' - ↑A y'‖)) =
    ↑‖↑f'.symm‖₊ * ↑(‖↑f'.symm‖₊⁻¹ - c)⁻¹ * ↑c * ‖↑A x' - ↑A y'‖
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv

Goal state before:
pp: f : ℂ → ℂ
f' x : ℂ
h₁ : HasStrictDerivAt f f' x
h₂ : f x ∈ slitPlane
⊢ HasStrictDerivAt (fun t ↦ log (f t)) ((f x)⁻¹ * f') x

Tactic:
;

Goal state after:
f : ℂ → ℂ
f' x : ℂ
h₁ : HasStrictDerivAt f f' x
h₂ : f x ∈ slitPlane
⊢ HasStrictDerivAt (fun t ↦ log (f t)) ((f x)⁻¹ * f') x
Module: Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv

Goal state before:
pp: f : ℂ → ℂ
f' x : ℂ
h₁ : HasDerivAt f f' x
h₂ : f x ∈ slitPlane
⊢ HasDerivAt (fun t ↦ log (f t)) ((f x)⁻¹ * f') x

Tactic:
;

Goal state after:
f : ℂ → ℂ
f' x : ℂ
h₁ : HasDerivAt f f' x
h₂ : f x ∈ slitPlane
⊢ HasDerivAt (fun t ↦ log (f t)) ((f x)⁻¹ * f') x
Module: Mathlib.Analysis.SpecialFunctions.Complex.Analytic

Goal state before:
pp: x✝ : ℝ
hx : 0 < x✝
x : ℝ
⊢ Real.log x = (log ↑x).re

Tactic:
;

Goal state after:
x✝ : ℝ
hx : 0 < x✝
x : ℝ
⊢ Real.log x = (log ↑x).re
Module: Mathlib.AlgebraicTopology.SimplicialSet.NerveAdjunction

Goal state before:
pp: X'✝ X✝ : Truncated 2
Y✝ : Cat
x✝¹ : X'✝ ⟶ X✝
x✝ : X✝ ⟶ nerveFunctor₂.obj Y✝
⊢ (((Cat.Hom.equivFunctor (hoFunctor₂.obj X'✝) Y✝).trans HomotopyCategory.functorEquiv).symm (x✝¹ ≫ x✝)).toFunctor =
    (hoFunctor₂.map x✝¹ ≫
        ((Cat.Hom.equivFunctor (hoFunctor₂.obj X✝) Y✝).trans HomotopyCategory.functorEquiv).symm x✝).toFunctor

Tactic:
;

Goal state after:
X'✝ X✝ : Truncated 2
Y✝ : Cat
x✝¹ : X'✝ ⟶ X✝
x✝ : X✝ ⟶ nerveFunctor₂.obj Y✝
⊢ (((Cat.Hom.equivFunctor (hoFunctor₂.obj X'✝) Y✝).trans HomotopyCategory.functorEquiv).symm (x✝¹ ≫ x✝)).toFunctor =
    (hoFunctor₂.map x✝¹ ≫
        ((Cat.Hom.equivFunctor (hoFunctor₂.obj X✝) Y✝).trans HomotopyCategory.functorEquiv).symm x✝).toFunctor
Module: Mathlib.AlgebraicTopology.SimplicialSet.NerveAdjunction

Goal state before:
pp: C D : Type u
inst✝¹ : SmallCategory C
inst✝ : SmallCategory D
φ : nerveFunctor₂.obj (Cat.of C) ⟶ nerveFunctor₂.obj (Cat.of D)
x y : ↑(Cat.of C)
f : x ⟶ y
⊢ (ConcreteCategory.hom ((nerveFunctor₂.map (functorOfNerveMap φ).toCatHom).app (op { obj := ⦋1⦌, property := ⋯ })))
      (ComposableArrows.mk₁ f) =
    (ConcreteCategory.hom (φ.app (op { obj := ⦋1⦌, property := ⋯ }))) (ComposableArrows.mk₁ f)

Tactic:
exact (nerveMap_app_mk₁ _ _).trans ((nerve.mk₁_homEquiv_apply _).trans
      (ComposableArrows.mk₁_hom _))

Goal state after:
Module: Mathlib.AlgebraicTopology.SimplicialSet.NerveAdjunction

Goal state before:
pp: C D : Type u
inst✝¹ : SmallCategory C
inst✝ : SmallCategory D
X✝ Y✝ : Cat
x✝ : X✝ ⟶ Y✝
⊢ (functorOfNerveMap (nerveFunctor₂.map x✝)).toCatHom.toFunctor = x✝.toFunctor

Tactic:
;

Goal state after:
C D : Type u
inst✝¹ : SmallCategory C
inst✝ : SmallCategory D
X✝ Y✝ : Cat
x✝ : X✝ ⟶ Y✝
⊢ (functorOfNerveMap (nerveFunctor₂.map x✝)).toCatHom.toFunctor = x✝.toFunctor
Module: Mathlib.Analysis.Analytic.Order

Goal state before:
pp: 𝕜 : Type u_1
E : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : CharZero 𝕜
inst✝ : CompleteSpace E
z₀ : 𝕜
f : 𝕜 → E
hf : AnalyticAt 𝕜 f z₀
hzero : f z₀ = 0
n : ℕ
h : ∀ i < n, iteratedDeriv (i + 1) f z₀ = 0
k : ℕ
hk : k < n + 1
⊢ iteratedDeriv k f z₀ = 0

Tactic:
cases k with
  | zero => simpa
  | succ k => exact h k <| by lia

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

Goal state before:
pp: case coe
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : CharZero 𝕜
inst✝ : CompleteSpace E
z₀ : 𝕜
f : 𝕜 → E
hf : AnalyticAt 𝕜 f z₀
hzero : f z₀ = 0
n m a✝ : ℕ
⊢ a✝ = m ↔ m ≤ a✝ ∧ ¬m + 1 ≤ a✝

Tactic:
;

Goal state after:
case coe
𝕜 : Type u_1
E : Type u_2
inst✝⁴ : NontriviallyNormedField 𝕜
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
inst✝¹ : CharZero 𝕜
inst✝ : CompleteSpace E
z₀ : 𝕜
f : 𝕜 → E
hf : AnalyticAt 𝕜 f z₀
hzero : f z₀ = 0
n m a✝ : ℕ
⊢ a✝ = m ↔ m ≤ a✝ ∧ ¬m + 1 ≤ a✝
Module: Mathlib.Analysis.SpecialFunctions.OrdinaryHypergeometric

Goal state before:
pp: case refine_1
𝕂 : Type u_1
𝔸 : Type u_2
inst✝² : RCLike 𝕂
inst✝¹ : NormedDivisionRing 𝔸
inst✝ : NormedAlgebra 𝕂 𝔸
a b c : 𝕂
n : ℕ
h : ordinaryHypergeometricSeries 𝔸 a b c n = 0
⊢ ∃ k < n, ↑k = -a ∨ ↑k = -b ∨ ↑k = -c

pp: case refine_2
𝕂 : Type u_1
𝔸 : Type u_2
inst✝² : RCLike 𝕂
inst✝¹ : NormedDivisionRing 𝔸
inst✝ : NormedAlgebra 𝕂 𝔸
a b c : 𝕂
n : ℕ
zero : ∃ k < n, ↑k = -a ∨ ↑k = -b ∨ ↑k = -c
⊢ ordinaryHypergeometricSeries 𝔸 a b c n = 0

Tactic:
· rw [ordinaryHypergeometricSeries, ofScalars_eq_zero] at h
    simp only [_root_.mul_eq_zero, inv_eq_zero] at h
    rcases h with ((hn | h) | h) | h
    · simp [Nat.factorial_ne_zero] at hn
    all_goals
      obtain ⟨kn, hkn, hn⟩ := (ascPochhammer_eval_eq_zero_iff _ _).1 h
      exact ⟨kn, hkn, by tauto⟩

Goal state after:
case refine_2
𝕂 : Type u_1
𝔸 : Type u_2
inst✝² : RCLike 𝕂
inst✝¹ : NormedDivisionRing 𝔸
inst✝ : NormedAlgebra 𝕂 𝔸
a b c : 𝕂
n : ℕ
zero : ∃ k < n, ↑k = -a ∨ ↑k = -b ∨ ↑k = -c
⊢ ordinaryHypergeometricSeries 𝔸 a b c n = 0
Module: Mathlib.Analysis.Calculus.FDeriv.Extend

Goal state before:
pp: case pos
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]
      (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s)
⊢ HasFDerivWithinAt f f' (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s) x

Tactic:
;

Goal state after:
case pos
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]
      (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s)
⊢ HasFDerivWithinAt f f' (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s) x
Module: Mathlib.Analysis.Analytic.Order

Goal state before:
pp: 𝕜 : Type u_1
inst✝ : NontriviallyNormedField 𝕜
z₀ : 𝕜
n : ℕ
⊢ analyticOrderAt ((fun x ↦ x - z₀) ^ n) z₀ = ↑n

Tactic:
rw [AnalyticAt.analyticOrderAt_eq_natCast (by fun_prop)]
  exact ⟨1, by simp [Pi.one_def, analyticAt_const]⟩

Goal state after:
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
u v : E
u_in : (u, v).1 ∈ B ∩ s
v_in : (u, v).2 ∈ B ∩ s
conv : Convex ℝ (B ∩ s)
⊢ ‖f (u, v).2 - f (u, v).1 - (f' (u, v).2 - f' (u, v).1)‖ ≤ ε * ‖(u, v).2 - (u, v).1‖

Tactic:
have diff : DifferentiableOn ℝ f (B ∩ s) := f_diff.mono inter_subset_right

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
u v : E
u_in : (u, v).1 ∈ B ∩ s
v_in : (u, v).2 ∈ B ∩ s
conv : Convex ℝ (B ∩ s)
diff : DifferentiableOn ℝ f (B ∩ s)
⊢ ‖f (u, v).2 - f (u, v).1 - (f' (u, v).2 - f' (u, v).1)‖ ≤ ε * ‖(u, v).2 - (u, v).1‖