-- Mathlib/Analysis/Calculus/ContDiff/FaaDiBruno.lean
theorem HasFTaylorSeriesUpToOn.comp {n : WithTop ℕ∞} {g : F → G} {f : E → F}
    (hg : HasFTaylorSeriesUpToOn n g q t) (hf : HasFTaylorSeriesUpToOn n f p s) (h : MapsTo f s t) :
    HasFTaylorSeriesUpToOn n (g ∘ f) (fun x ↦ (q (f x)).taylorComp (p x)) s

Proof state:
case h.e'_12
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n : WithTop ℕ∞
g : F → G
f : E → F
hg : HasFTaylorSeriesUpToOn n g q t
hf : HasFTaylorSeriesUpToOn n f p s
h : MapsTo f s t
m : ℕ
hm : ↑m < n
x : E
hx : x ∈ s
A :
  ∀ (c : OrderedFinpartition m),
    HasFDerivWithinAt (fun x => (q (f x)).compAlongOrderedFinpartition (p x) c)
      (∑ i : Option (Fin c.length), ((q (f x)).compAlongOrderedFinpartition (p x) (c.extend i)).curryLeft) s x
B :
  HasFDerivWithinAt (fun x => (q (f x)).taylorComp (p x) m)
    (∑ c : OrderedFinpartition m,
      ∑ i : Option (Fin c.length), ((q (f x)).compAlongOrderedFinpartition (p x) (c.extend i)).curryLeft)
    s x
this :
  ∑ c : OrderedFinpartition m, ∑ i : Option (Fin c.length), (q (f x)).compAlongOrderedFinpartition (p x) (c.extend i) =
    (q (f x)).taylorComp (p x) (m + 1)
⊢ (∑ c : OrderedFinpartition m,
        ∑ i : Option (Fin c.length), (q (f x)).compAlongOrderedFinpartition (p x) (c.extend i)).curryLeft =
    ∑ c : OrderedFinpartition m,
      ∑ i : Option (Fin c.length), ((q (f x)).compAlongOrderedFinpartition (p x) (c.extend i)).curryLeft

Tactic:
ext v

Result:
case h.e'_12.h.H
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n : WithTop ℕ∞
g : F → G
f : E → F
hg : HasFTaylorSeriesUpToOn n g q t
hf : HasFTaylorSeriesUpToOn n f p s
h : MapsTo f s t
m : ℕ
hm : ↑m < n
x : E
hx : x ∈ s
A :
  ∀ (c : OrderedFinpartition m),
    HasFDerivWithinAt (fun x => (q (f x)).compAlongOrderedFinpartition (p x) c)
      (∑ i : Option (Fin c.length), ((q (f x)).compAlongOrderedFinpartition (p x) (c.extend i)).curryLeft) s x
B :
  HasFDerivWithinAt (fun x => (q (f x)).taylorComp (p x) m)
    (∑ c : OrderedFinpartition m,
      ∑ i : Option (Fin c.length), ((q (f x)).compAlongOrderedFinpartition (p x) (c.extend i)).curryLeft)
    s x
this :
  ∑ c : OrderedFinpartition m, ∑ i : Option (Fin c.length), (q (f x)).compAlongOrderedFinpartition (p x) (c.extend i) =
    (q (f x)).taylorComp (p x) (m + 1)
v : E
x✝ : Fin m → E
⊢ ((∑ c : OrderedFinpartition m,
            ∑ i : Option (Fin c.length), (q (f x)).compAlongOrderedFinpartition (p x) (c.extend i)).curryLeft
        v)
      x✝ =
    ((∑ c : OrderedFinpartition m,
          ∑ i : Option (Fin c.length), ((q (f x)).compAlongOrderedFinpartition (p x) (c.extend i)).curryLeft)
        v)
      x✝

Proof state:
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n : WithTop ℕ∞
g : F → G
f : E → F
hg : HasFTaylorSeriesUpToOn n g q t
hf : HasFTaylorSeriesUpToOn n f p s
h : MapsTo f s t
m : ℕ
hm : ↑m < n
x : E
hx : x ∈ s
c : OrderedFinpartition m
B : ContinuousMultilinearMap 𝕜 (fun i => F) G →L[𝕜]
  ContinuousMultilinearMap 𝕜 (fun i => ContinuousMultilinearMap 𝕜 (fun i => E) F)
    (ContinuousMultilinearMap 𝕜 (fun i => E) G) :=
  compAlongOrderedFinpartitionL 𝕜 E F G c
⊢ HasFDerivWithinAt (fun y => (B (q (f y) c.length)) fun i => p y (c.partSize i))
    (∑ i : Option (Fin c.length), ((q (f x)).compAlongOrderedFinpartition (p x) (c.extend i)).curryLeft) s x

Tactic:
have cm : (c.length : WithTop ℕ∞) ≤ m := mod_cast OrderedFinpartition.length_le c

Result:
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type u_2
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type u_3
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type u_4
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n : WithTop ℕ∞
g : F → G
f : E → F
hg : HasFTaylorSeriesUpToOn n g q t
hf : HasFTaylorSeriesUpToOn n f p s
h : MapsTo f s t
m : ℕ
hm : ↑m < n
x : E
hx : x ∈ s
c : OrderedFinpartition m
B : ContinuousMultilinearMap 𝕜 (fun i => F) G →L[𝕜]
  ContinuousMultilinearMap 𝕜 (fun i => ContinuousMultilinearMap 𝕜 (fun i => E) F)
    (ContinuousMultilinearMap 𝕜 (fun i => E) G) :=
  compAlongOrderedFinpartitionL 𝕜 E F G c
cm : ↑c.length ≤ ↑m
⊢ HasFDerivWithinAt (fun y => (B (q (f y) c.length)) fun i => p y (c.partSize i))
    (∑ i : Option (Fin c.length), ((q (f x)).compAlongOrderedFinpartition (p x) (c.extend i)).curryLeft) s x