-- Mathlib/Analysis/Calculus/FDeriv/Analytic.lean
theorem HasFPowerSeriesWithinAt.hasStrictFDerivWithinAt (h : HasFPowerSeriesWithinAt f p s x) :
    (fun y ↦ f y.1 - f y.2 - (continuousMultilinearCurryFin1 𝕜 E F (p 1)) (y.1 - y.2))
      =o[𝓝[insert x s ×ˢ insert x s] (x, x)] fun y ↦ y.1 - y.2

Proof state:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
f : E → F
x : E
s : Set E
h : HasFPowerSeriesWithinAt f p s x
⊢ (fun y =>
      f y.1 - f y.2 - ((continuousMultilinearCurryFin1 𝕜 E F) (p 1)) (y.1 - y.2)) =o[𝓝[insert x s ×ˢ insert x s] (x, x)]
    fun y => y.1 - y.2

Tactic:
refine h.isBigO_image_sub_norm_mul_norm_sub.trans_isLittleO (IsLittleO.of_norm_right ?_)

Result:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
f : E → F
x : E
s : Set E
h : HasFPowerSeriesWithinAt f p s x
⊢ (fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖) =o[𝓝[insert x s ×ˢ insert x s] (x, x)] fun x => ‖x.1 - x.2‖

Proof state:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
f : E → F
x : E
s : Set E
h : HasFPowerSeriesWithinAt f p s x
⊢ ‖id (x, x) - (x, x)‖ = 0

Tactic:
rw [_root_.id, sub_self, norm_zero]

Result:
no goals

Proof state:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
f : E → F
x : E
s : Set E
h : HasFPowerSeriesWithinAt f p s x
⊢ Tendsto (fun y => ‖y - (x, x)‖) (𝓝 (x, x)) (𝓝 0)

Tactic:
refine (continuous_id.sub continuous_const).norm.tendsto' _ _ ?_

Result:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
f : E → F
x : E
s : Set E
h : HasFPowerSeriesWithinAt f p s x
⊢ ‖id (x, x) - (x, x)‖ = 0

Proof state:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
f : E → F
x : E
s : Set E
h : HasFPowerSeriesWithinAt f p s x
⊢ Tendsto (fun y => ‖y - (x, x)‖) (𝓝[insert x s ×ˢ insert x s] (x, x)) (𝓝 0)

Tactic:
apply Tendsto.mono_left _ nhdsWithin_le_nhds

Result:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
f : E → F
x : E
s : Set E
h : HasFPowerSeriesWithinAt f p s x
⊢ Tendsto (fun y => ‖y - (x, x)‖) (𝓝 (x, x)) (𝓝 0)

Proof state:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
f : E → F
x : E
s : Set E
h : HasFPowerSeriesWithinAt f p s x
⊢ (fun y => ‖y - (x, x)‖ * ‖y.1 - y.2‖) =o[𝓝[insert x s ×ˢ insert x s] (x, x)] fun x => ‖x.1 - x.2‖

Tactic:
refine isLittleO_iff_exists_eq_mul.2 ⟨fun y => ‖y - (x, x)‖, ?_, EventuallyEq.rfl⟩

Result:
𝕜 : Type u_1
inst✝⁴ : NontriviallyNormedField 𝕜
E : Type u
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace 𝕜 E
F : Type v
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace 𝕜 F
p : FormalMultilinearSeries 𝕜 E F
f : E → F
x : E
s : Set E
h : HasFPowerSeriesWithinAt f p s x
⊢ Tendsto (fun y => ‖y - (x, x)‖) (𝓝[insert x s ×ˢ insert x s] (x, x)) (𝓝 0)