Module: Mathlib.Analysis.Calculus.FDeriv.Symmetric

Goal state before:
pp: E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s : Set E
s_conv : Convex ℝ s
f : E → F
f' : E → E →L[ℝ] F
f'' : E →L[ℝ] E →L[ℝ] F
hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x
x : E
xs : x ∈ s
hx : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x_1 : E) in 𝓝[interior s] x, ‖f' x_1 - f' x - f'' (x_1 - x)‖ ≤ c * ‖x_1 - x‖
v w : E
hv : x + v ∈ interior s
hw : x + v + w ∈ interior s
ε : ℝ
εpos : 0 < ε
δ : ℝ
δpos : δ > 0
sδ : Metric.ball x δ ∩ interior s ⊆ {x_1 | (fun x_2 ↦ ‖f' x_2 - f' x - f'' (x_2 - x)‖ ≤ ε * ‖x_2 - x‖) x_1}
E1 : ∀ᶠ (h : ℝ) in 𝓝[>] 0, h * (‖v‖ + ‖w‖) < δ
E2 : ∀ᶠ (h : ℝ) in 𝓝[>] 0, h < 1
h : ℝ
hδ : h * (‖v‖ + ‖w‖) < δ
h_lt_1 : h < 1
hpos : 0 < h
xt_mem : ∀ t ∈ Icc 0 1, x + h • v + (t * h) • w ∈ interior s
g : ℝ → F :=
  fun t ↦ f (x + h • v + (t * h) • w) - (t * h) • (f' x) w - (t * h ^ 2) • (f'' v) w - ((t * h) ^ 2 / 2) • (f'' w) w
g' : ℝ → F :=
  fun t ↦ (f' (x + h • v + (t * h) • w)) (h • w) - h • (f' x) w - h ^ 2 • (f'' v) w - (t * h ^ 2) • (f'' w) w
hg' : g' = fun t ↦ (f' (x + h • v + (t * h) • w)) (h • w) - h • (f' x) w - h ^ 2 • (f'' v) w - (t * h ^ 2) • (f'' w) w
g_deriv : ∀ t ∈ Icc 0 1, HasDerivWithinAt g (g' t) (Icc 0 1) t
⊢ ‖f (x + h • v + h • w) - f (x + h • v) - h • (f' x) w - h ^ 2 • (f'' v) w - (h ^ 2 / 2) • (f'' w) w‖ ≤
    ε * ‖(‖v‖ + ‖w‖) * ‖w‖ * h ^ 2‖

Tactic:
have g'_bound : ∀ t ∈ Ico (0 : ℝ) 1, ‖g' t‖ ≤ ε * ((‖v‖ + ‖w‖) * ‖w‖) * h ^ 2 := by
    intro t ht
    have I : ‖h • v + (t * h) • w‖ ≤ h * (‖v‖ + ‖w‖) :=
      calc
        ‖h • v + (t * h) • w‖ ≤ ‖h • v‖ + ‖(t * h) • w‖ := norm_add_le _ _
        _ = h * ‖v‖ + t * (h * ‖w‖) := by
          simp only [norm_smul, Real.norm_eq_abs, hpos.le, abs_of_nonneg, abs_mul, ht.left,
            mul_assoc]
        _ ≤ h * ‖v‖ + 1 * (h * ‖w‖) := by gcongr; exact ht.2.le
        _ = h * (‖v‖ + ‖w‖) := by ring
    calc
      ‖g' t‖ = ‖(f' (x + h • v + (t * h) • w) - f' x - f'' (h • v + (t * h) • w)) (h • w)‖ := by
        rw [hg']
        congrm ‖?_‖
        simp only [sub_apply, add_apply, smul_apply, map_add, map_smul]
        module
      _ ≤ ‖f' (x + h • v + (t * h) • w) - f' x - f'' (h • v + (t * h) • w)‖ * ‖h • w‖ :=
        (ContinuousLinearMap.le_opNorm _ _)
      _ ≤ ε * ‖h • v + (t * h) • w‖ * ‖h • w‖ := by
        gcongr
        have H : x + h • v + (t * h) • w ∈ Metric.ball x δ ∩ interior s := by
          refine ⟨?_, xt_mem t ⟨ht.1, ht.2.le⟩⟩
          rw [add_assoc, add_mem_ball_iff_norm]
          exact I.trans_lt hδ
        simpa only [mem_ofPred_eq, add_assoc x, add_sub_cancel_left] using sδ H
      _ ≤ ε * (‖h • v‖ + ‖h • w‖) * ‖h • w‖ := by
        gcongr
        apply (norm_add_le _ _).trans
        gcongr
        simp only [norm_smul, Real.norm_eq_abs, abs_mul, abs_of_nonneg, ht.1, hpos.le, mul_assoc]
        exact mul_le_of_le_one_left (by positivity) ht.2.le
      _ = ε * ((‖v‖ + ‖w‖) * ‖w‖) * h ^ 2 := by
        simp only [norm_smul, Real.norm_eq_abs, abs_of_nonneg, hpos.le]; ring

Goal state after:
E : Type u_1
F : Type u_2
inst✝³ : NormedAddCommGroup E
inst✝² : NormedSpace ℝ E
inst✝¹ : NormedAddCommGroup F
inst✝ : NormedSpace ℝ F
s : Set E
s_conv : Convex ℝ s
f : E → F
f' : E → E →L[ℝ] F
f'' : E →L[ℝ] E →L[ℝ] F
hf : ∀ x ∈ interior s, HasFDerivAt f (f' x) x
x : E
xs : x ∈ s
hx : ∀ ⦃c : ℝ⦄, 0 < c → ∀ᶠ (x_1 : E) in 𝓝[interior s] x, ‖f' x_1 - f' x - f'' (x_1 - x)‖ ≤ c * ‖x_1 - x‖
v w : E
hv : x + v ∈ interior s
hw : x + v + w ∈ interior s
ε : ℝ
εpos : 0 < ε
δ : ℝ
δpos : δ > 0
sδ : Metric.ball x δ ∩ interior s ⊆ {x_1 | (fun x_2 ↦ ‖f' x_2 - f' x - f'' (x_2 - x)‖ ≤ ε * ‖x_2 - x‖) x_1}
E1 : ∀ᶠ (h : ℝ) in 𝓝[>] 0, h * (‖v‖ + ‖w‖) < δ
E2 : ∀ᶠ (h : ℝ) in 𝓝[>] 0, h < 1
h : ℝ
hδ : h * (‖v‖ + ‖w‖) < δ
h_lt_1 : h < 1
hpos : 0 < h
xt_mem : ∀ t ∈ Icc 0 1, x + h • v + (t * h) • w ∈ interior s
g : ℝ → F :=
  fun t ↦ f (x + h • v + (t * h) • w) - (t * h) • (f' x) w - (t * h ^ 2) • (f'' v) w - ((t * h) ^ 2 / 2) • (f'' w) w
g' : ℝ → F :=
  fun t ↦ (f' (x + h • v + (t * h) • w)) (h • w) - h • (f' x) w - h ^ 2 • (f'' v) w - (t * h ^ 2) • (f'' w) w
hg' : g' = fun t ↦ (f' (x + h • v + (t * h) • w)) (h • w) - h • (f' x) w - h ^ 2 • (f'' v) w - (t * h ^ 2) • (f'' w) w
g_deriv : ∀ t ∈ Icc 0 1, HasDerivWithinAt g (g' t) (Icc 0 1) t
g'_bound : ∀ t ∈ Ico 0 1, ‖g' t‖ ≤ ε * ((‖v‖ + ‖w‖) * ‖w‖) * h ^ 2
⊢ ‖f (x + h • v + h • w) - f (x + h • v) - h • (f' x) w - h ^ 2 • (f'' v) w - (h ^ 2 / 2) • (f'' w) w‖ ≤
    ε * ‖(‖v‖ + ‖w‖) * ‖w‖ * h ^ 2‖