-- 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 fderivWithin
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
⊢ HasFDerivWithinAt (fun x => (q (f x)).taylorComp (p x) m) ((q (f x)).taylorComp (p x) m.succ).curryLeft s x

Tactic:
have 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 :=
  HasFDerivWithinAt.sum (fun c _ ↦ A c)

Result:
case fderivWithin
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
⊢ HasFDerivWithinAt (fun x => (q (f x)).taylorComp (p x) m) ((q (f x)).taylorComp (p x) m.succ).curryLeft s 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
cm : ↑c.length ≤ ↑m
cp : ∀ (i : Fin c.length), ↑(c.partSize i) ≤ ↑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

Tactic:
have I i : HasFDerivWithinAt (fun x ↦ p x (c.partSize i))
    (p x (c.partSize i).succ).curryLeft s x :=
  hf.fderivWithin (c.partSize i) ((cp i).trans_lt hm) x hx

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
cp : ∀ (i : Fin c.length), ↑(c.partSize i) ≤ ↑m
I : ∀ (i : Fin c.length), HasFDerivWithinAt (fun x => p x (c.partSize i)) (p x (c.partSize i).succ).curryLeft s x
⊢ 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

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
c : OrderedFinpartition m
x✝ : c ∈ Finset.univ
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
i : Fin c.length
⊢ ↑(c.partSize i) ≤ ↑m

Tactic:
exact_mod_cast OrderedFinpartition.partSize_le c i

Result:
no goals

Proof state:
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
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
cp : ∀ (i : Fin c.length), ↑(c.partSize i) ≤ ↑m
I : ∀ (i : Fin c.length), HasFDerivWithinAt (fun x => p x (c.partSize i)) (p x (c.partSize i).succ).curryLeft s x
J : HasFDerivWithinAt (fun x => q x c.length) (q (f x) c.length.succ).curryLeft t (f x)
K : HasFDerivWithinAt f ((continuousMultilinearCurryFin1 𝕜 E F) (p x 1)) s x
e_8✝ : ContinuousMultilinearMap.instAddCommGroup = SeminormedAddCommGroup.toAddCommGroup
he✝ : ContinuousMultilinearMap.instModule = NormedSpace.toModule
e_10✝ : ContinuousMultilinearMap.instTopologicalSpace = UniformSpace.toTopologicalSpace
⊢ ∑ i : Option (Fin c.length), ((q (f x)).compAlongOrderedFinpartition (p x) (c.extend i)).curryLeft =
    (B.flipMultilinear fun i => p x (c.partSize i)).comp
        ((q (f x) c.length.succ).curryLeft.comp ((continuousMultilinearCurryFin1 𝕜 E F) (p x 1))) +
      ∑ i : Fin c.length,
        ((B (((fun x => q x c.length) ∘ f) x)).toContinuousLinearMap (fun j => p x (c.partSize j)) i).comp
          (p x (c.partSize i).succ).curryLeft

Tactic:
simp only [B, Nat.succ_eq_add_one, Fintype.sum_option, comp_apply, faaDiBruno_aux1,
  faaDiBruno_aux2]

Result:
no goals