Theorem: hasDerivWithinAt_taylor_coeff_within
File: Mathlib/Analysis/Calculus/Taylor.lean

tactic: replace hf :
  HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y := by
  convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
  rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
  exact (derivWithin_of_mem hs ht hf).symm
state_before: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x y : ℝ
k : ℕ
s t : Set ℝ
ht : UniqueDiffWithinAt ℝ t y
hs : s ∈ 𝓝[t] y
hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y
⊢ HasDerivWithinAt (fun z => (((↑k + 1) * ↑k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z)
    ((((↑k + 1) * ↑k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y -
      ((↑k !)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y)
    t y
state_after: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x y : ℝ
k : ℕ
s t : Set ℝ
ht : UniqueDiffWithinAt ℝ t y
hs : s ∈ 𝓝[t] y
hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y
⊢ HasDerivWithinAt (fun z => (((↑k + 1) * ↑k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z)
    ((((↑k + 1) * ↑k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y -
      ((↑k !)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y)
    t y

tactic: have : HasDerivWithinAt (fun t => ((k + 1 : ℝ) * k !)⁻¹ * (x - t) ^ (k + 1))
    (-((k ! : ℝ)⁻¹ * (x - y) ^ k)) t y := by
  have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by
    field_simp; ring
  rw [this]
  exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
state_before: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x y : ℝ
k : ℕ
s t : Set ℝ
ht : UniqueDiffWithinAt ℝ t y
hs : s ∈ 𝓝[t] y
hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y
⊢ HasDerivWithinAt (fun z => (((↑k + 1) * ↑k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z)
    ((((↑k + 1) * ↑k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y -
      ((↑k !)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y)
    t y
state_after: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x y : ℝ
k : ℕ
s t : Set ℝ
ht : UniqueDiffWithinAt ℝ t y
hs : s ∈ 𝓝[t] y
hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y
this : HasDerivWithinAt (fun t => ((↑k + 1) * ↑k !)⁻¹ * (x - t) ^ (k + 1)) (-((↑k !)⁻¹ * (x - y) ^ k)) t y
⊢ HasDerivWithinAt (fun z => (((↑k + 1) * ↑k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z)
    ((((↑k + 1) * ↑k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y -
      ((↑k !)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y)
    t y

tactic: convert this.smul hf using 1
state_before: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x y : ℝ
k : ℕ
s t : Set ℝ
ht : UniqueDiffWithinAt ℝ t y
hs : s ∈ 𝓝[t] y
hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y
this : HasDerivWithinAt (fun t => ((↑k + 1) * ↑k !)⁻¹ * (x - t) ^ (k + 1)) (-((↑k !)⁻¹ * (x - y) ^ k)) t y
⊢ HasDerivWithinAt (fun z => (((↑k + 1) * ↑k !)⁻¹ * (x - z) ^ (k + 1)) • iteratedDerivWithin (k + 1) f s z)
    ((((↑k + 1) * ↑k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y -
      ((↑k !)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y)
    t y
state_after: case h.e'_7
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x y : ℝ
k : ℕ
s t : Set ℝ
ht : UniqueDiffWithinAt ℝ t y
hs : s ∈ 𝓝[t] y
hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y
this : HasDerivWithinAt (fun t => ((↑k + 1) * ↑k !)⁻¹ * (x - t) ^ (k + 1)) (-((↑k !)⁻¹ * (x - y) ^ k)) t y
⊢ (((↑k + 1) * ↑k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y -
      ((↑k !)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y =
    (((↑k + 1) * ↑k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y +
      -((↑k !)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y

tactic: field_simp
state_before: case h.e'_7
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x y : ℝ
k : ℕ
s t : Set ℝ
ht : UniqueDiffWithinAt ℝ t y
hs : s ∈ 𝓝[t] y
hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y
this : HasDerivWithinAt (fun t => ((↑k + 1) * ↑k !)⁻¹ * (x - t) ^ (k + 1)) (-((↑k !)⁻¹ * (x - y) ^ k)) t y
⊢ (((↑k + 1) * ↑k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y -
      ((↑k !)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y =
    (((↑k + 1) * ↑k !)⁻¹ * (x - y) ^ (k + 1)) • iteratedDerivWithin (k + 2) f s y +
      -((↑k !)⁻¹ * (x - y) ^ k) • iteratedDerivWithin (k + 1) f s y
state_after: case h.e'_7
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x y : ℝ
k : ℕ
s t : Set ℝ
ht : UniqueDiffWithinAt ℝ t y
hs : s ∈ 𝓝[t] y
hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y
this : HasDerivWithinAt (fun t => ((↑k + 1) * ↑k !)⁻¹ * (x - t) ^ (k + 1)) (-((↑k !)⁻¹ * (x - y) ^ k)) t y
⊢ ((x - y) ^ (k + 1) / ((↑k + 1) * ↑k !)) • iteratedDerivWithin (k + 2) f s y -
      ((x - y) ^ k / ↑k !) • iteratedDerivWithin (k + 1) f s y =
    ((x - y) ^ (k + 1) / ((↑k + 1) * ↑k !)) • iteratedDerivWithin (k + 2) f s y +
      (-(x - y) ^ k / ↑k !) • iteratedDerivWithin (k + 1) f s y

tactic: rw [neg_div, neg_smul, sub_eq_add_neg]
state_before: case h.e'_7
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x y : ℝ
k : ℕ
s t : Set ℝ
ht : UniqueDiffWithinAt ℝ t y
hs : s ∈ 𝓝[t] y
hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y
this : HasDerivWithinAt (fun t => ((↑k + 1) * ↑k !)⁻¹ * (x - t) ^ (k + 1)) (-((↑k !)⁻¹ * (x - y) ^ k)) t y
⊢ ((x - y) ^ (k + 1) / ((↑k + 1) * ↑k !)) • iteratedDerivWithin (k + 2) f s y -
      ((x - y) ^ k / ↑k !) • iteratedDerivWithin (k + 1) f s y =
    ((x - y) ^ (k + 1) / ((↑k + 1) * ↑k !)) • iteratedDerivWithin (k + 2) f s y +
      (-(x - y) ^ k / ↑k !) • iteratedDerivWithin (k + 1) f s y
state_after: no goals

tactic: convert (hf.mono_of_mem hs).hasDerivWithinAt using 1
state_before: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x y : ℝ
k : ℕ
s t : Set ℝ
ht : UniqueDiffWithinAt ℝ t y
hs : s ∈ 𝓝[t] y
hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y
⊢ HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y
state_after: case h.e'_7
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x y : ℝ
k : ℕ
s t : Set ℝ
ht : UniqueDiffWithinAt ℝ t y
hs : s ∈ 𝓝[t] y
hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y
⊢ iteratedDerivWithin (k + 2) f s y = derivWithin (iteratedDerivWithin (k + 1) f s) t y

tactic: rw [iteratedDerivWithin_succ (ht.mono_nhds (nhdsWithin_le_iff.mpr hs))]
state_before: case h.e'_7
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x y : ℝ
k : ℕ
s t : Set ℝ
ht : UniqueDiffWithinAt ℝ t y
hs : s ∈ 𝓝[t] y
hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y
⊢ iteratedDerivWithin (k + 2) f s y = derivWithin (iteratedDerivWithin (k + 1) f s) t y
state_after: case h.e'_7
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x y : ℝ
k : ℕ
s t : Set ℝ
ht : UniqueDiffWithinAt ℝ t y
hs : s ∈ 𝓝[t] y
hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y
⊢ derivWithin (iteratedDerivWithin (k + 1) f s) s y = derivWithin (iteratedDerivWithin (k + 1) f s) t y

tactic: exact (derivWithin_of_mem hs ht hf).symm
state_before: case h.e'_7
𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x y : ℝ
k : ℕ
s t : Set ℝ
ht : UniqueDiffWithinAt ℝ t y
hs : s ∈ 𝓝[t] y
hf : DifferentiableWithinAt ℝ (iteratedDerivWithin (k + 1) f s) s y
⊢ derivWithin (iteratedDerivWithin (k + 1) f s) s y = derivWithin (iteratedDerivWithin (k + 1) f s) t y
state_after: no goals

tactic: have : -((k ! : ℝ)⁻¹ * (x - y) ^ k) = ((k + 1 : ℝ) * k !)⁻¹ * (-(k + 1) * (x - y) ^ k) := by
  field_simp; ring
state_before: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x y : ℝ
k : ℕ
s t : Set ℝ
ht : UniqueDiffWithinAt ℝ t y
hs : s ∈ 𝓝[t] y
hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y
⊢ HasDerivWithinAt (fun t => ((↑k + 1) * ↑k !)⁻¹ * (x - t) ^ (k + 1)) (-((↑k !)⁻¹ * (x - y) ^ k)) t y
state_after: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x y : ℝ
k : ℕ
s t : Set ℝ
ht : UniqueDiffWithinAt ℝ t y
hs : s ∈ 𝓝[t] y
hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y
this : -((↑k !)⁻¹ * (x - y) ^ k) = ((↑k + 1) * ↑k !)⁻¹ * (-(↑k + 1) * (x - y) ^ k)
⊢ HasDerivWithinAt (fun t => ((↑k + 1) * ↑k !)⁻¹ * (x - t) ^ (k + 1)) (-((↑k !)⁻¹ * (x - y) ^ k)) t y

tactic: rw [this]
state_before: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x y : ℝ
k : ℕ
s t : Set ℝ
ht : UniqueDiffWithinAt ℝ t y
hs : s ∈ 𝓝[t] y
hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y
this : -((↑k !)⁻¹ * (x - y) ^ k) = ((↑k + 1) * ↑k !)⁻¹ * (-(↑k + 1) * (x - y) ^ k)
⊢ HasDerivWithinAt (fun t => ((↑k + 1) * ↑k !)⁻¹ * (x - t) ^ (k + 1)) (-((↑k !)⁻¹ * (x - y) ^ k)) t y
state_after: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x y : ℝ
k : ℕ
s t : Set ℝ
ht : UniqueDiffWithinAt ℝ t y
hs : s ∈ 𝓝[t] y
hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y
this : -((↑k !)⁻¹ * (x - y) ^ k) = ((↑k + 1) * ↑k !)⁻¹ * (-(↑k + 1) * (x - y) ^ k)
⊢ HasDerivWithinAt (fun t => ((↑k + 1) * ↑k !)⁻¹ * (x - t) ^ (k + 1)) (((↑k + 1) * ↑k !)⁻¹ * (-(↑k + 1) * (x - y) ^ k))
    t y

tactic: exact (monomial_has_deriv_aux y x _).hasDerivWithinAt.const_mul _
state_before: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x y : ℝ
k : ℕ
s t : Set ℝ
ht : UniqueDiffWithinAt ℝ t y
hs : s ∈ 𝓝[t] y
hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y
this : -((↑k !)⁻¹ * (x - y) ^ k) = ((↑k + 1) * ↑k !)⁻¹ * (-(↑k + 1) * (x - y) ^ k)
⊢ HasDerivWithinAt (fun t => ((↑k + 1) * ↑k !)⁻¹ * (x - t) ^ (k + 1)) (((↑k + 1) * ↑k !)⁻¹ * (-(↑k + 1) * (x - y) ^ k))
    t y
state_after: no goals

tactic: field_simp
state_before: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x y : ℝ
k : ℕ
s t : Set ℝ
ht : UniqueDiffWithinAt ℝ t y
hs : s ∈ 𝓝[t] y
hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y
⊢ -((↑k !)⁻¹ * (x - y) ^ k) = ((↑k + 1) * ↑k !)⁻¹ * (-(↑k + 1) * (x - y) ^ k)
state_after: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x y : ℝ
k : ℕ
s t : Set ℝ
ht : UniqueDiffWithinAt ℝ t y
hs : s ∈ 𝓝[t] y
hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y
⊢ -((x - y) ^ k * ((↑k + 1) * ↑k !)) = (-1 + -↑k) * (x - y) ^ k * ↑k !

tactic: ring
state_before: 𝕜 : Type u_1
E : Type u_2
F : Type u_3
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℝ E
f : ℝ → E
x y : ℝ
k : ℕ
s t : Set ℝ
ht : UniqueDiffWithinAt ℝ t y
hs : s ∈ 𝓝[t] y
hf : HasDerivWithinAt (iteratedDerivWithin (k + 1) f s) (iteratedDerivWithin (k + 2) f s y) t y
⊢ -((x - y) ^ k * ((↑k + 1) * ↑k !)) = (-1 + -↑k) * (x - y) ^ k * ↑k !
state_after: no goals