l3lab/ntp-mathlib huggingface dataset:
theorem hasFTaylorSeriesUpToOn_pi :
    HasFTaylorSeriesUpToOn n (fun x i => φ i x)
        (fun x m => ContinuousMultilinearMap.pi fun i => p' i x m) s ↔
      ∀ i, HasFTaylorSeriesUpToOn n (φ i) (p' i) s := by
  set pr := @ContinuousLinearMap.proj 𝕜 _ ι F' _ _ _
  letI : ∀ (m : ℕ) (i : ι), NormedSpace 𝕜 (E[×m]→L[𝕜] F' i) := fun m i => inferInstance
  

Proof state:
𝕜 : Type u_1
inst✝¹⁴ : NontriviallyNormedField 𝕜
D : Type uD
inst✝¹³ : NormedAddCommGroup D
inst✝¹² : NormedSpace 𝕜 D
E : Type uE
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
F : Type uF
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
G : Type uG
inst✝⁷ : NormedAddCommGroup G
inst✝⁶ : NormedSpace 𝕜 G
X : Type u_2
inst✝⁵ : NormedAddCommGroup X
inst✝⁴ : NormedSpace 𝕜 X
s s₁ t u : Set E
f f₁ : E → F
g : F → G
x x₀ : E
c : F
b : E × F → G
m n : ℕ∞
p : E → FormalMultilinearSeries 𝕜 E F
ι : Type u_3
ι' : Type u_4
inst✝³ : Fintype ι
inst✝² : Fintype ι'
F' : ι → Type u_5
inst✝¹ : (i : ι) → NormedAddCommGroup (F' i)
inst✝ : (i : ι) → NormedSpace 𝕜 (F' i)
φ : (i : ι) → E → F' i
p' : (i : ι) → E → FormalMultilinearSeries 𝕜 E (F' i)
Φ : E → (i : ι) → F' i
P' : E → FormalMultilinearSeries 𝕜 E ((i : ι) → F' i)
pr : (i : ι) → ((i : ι) → F' i) →L[𝕜] F' i := ContinuousLinearMap.proj
this : (m : ℕ) → (i : ι) → NormedSpace 𝕜 (ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) := fun m i => inferInstance
⊢ HasFTaylorSeriesUpToOn n (fun x i => φ i x) (fun x m => ContinuousMultilinearMap.pi fun i => p' i x m) s ↔
    ∀ (i : ι), HasFTaylorSeriesUpToOn n (φ i) (p' i) s

Next tactic:
set L : ∀ m : ℕ, (∀ i, E[×m]→L[𝕜] F' i) ≃ₗᵢ[𝕜] E[×m]→L[𝕜] ∀ i, F' i := fun m =>
    ContinuousMultilinearMap.piₗᵢ _ _
theorem hasFTaylorSeriesUpToOn_pi :
    HasFTaylorSeriesUpToOn n (fun x i => φ i x)
        (fun x m => ContinuousMultilinearMap.pi fun i => p' i x m) s ↔
      ∀ i, HasFTaylorSeriesUpToOn n (φ i) (p' i) s := by
  set pr := @ContinuousLinearMap.proj 𝕜 _ ι F' _ _ _
  letI : ∀ (m : ℕ) (i : ι), NormedSpace 𝕜 (E[×m]→L[𝕜] F' i) := fun m i => inferInstance
  set L : ∀ m : ℕ, (∀ i, E[×m]→L[𝕜] F' i) ≃ₗᵢ[𝕜] E[×m]→L[𝕜] ∀ i, F' i := fun m =>
    ContinuousMultilinearMap.piₗᵢ _ _
  

Proof state:
𝕜 : Type u_1
inst✝¹⁴ : NontriviallyNormedField 𝕜
D : Type uD
inst✝¹³ : NormedAddCommGroup D
inst✝¹² : NormedSpace 𝕜 D
E : Type uE
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
F : Type uF
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
G : Type uG
inst✝⁷ : NormedAddCommGroup G
inst✝⁶ : NormedSpace 𝕜 G
X : Type u_2
inst✝⁵ : NormedAddCommGroup X
inst✝⁴ : NormedSpace 𝕜 X
s s₁ t u : Set E
f f₁ : E → F
g : F → G
x x₀ : E
c : F
b : E × F → G
m n : ℕ∞
p : E → FormalMultilinearSeries 𝕜 E F
ι : Type u_3
ι' : Type u_4
inst✝³ : Fintype ι
inst✝² : Fintype ι'
F' : ι → Type u_5
inst✝¹ : (i : ι) → NormedAddCommGroup (F' i)
inst✝ : (i : ι) → NormedSpace 𝕜 (F' i)
φ : (i : ι) → E → F' i
p' : (i : ι) → E → FormalMultilinearSeries 𝕜 E (F' i)
Φ : E → (i : ι) → F' i
P' : E → FormalMultilinearSeries 𝕜 E ((i : ι) → F' i)
pr : (i : ι) → ((i : ι) → F' i) →L[𝕜] F' i := ContinuousLinearMap.proj
this : (m : ℕ) → (i : ι) → NormedSpace 𝕜 (ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) := fun m i => inferInstance
L : (m : ℕ) →
  ((i : ι) → ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) ≃ₗᵢ[𝕜]
    ContinuousMultilinearMap 𝕜 (fun i => E) ((i : ι) → F' i) :=
  fun m => ContinuousMultilinearMap.piₗᵢ 𝕜 fun i => E
⊢ HasFTaylorSeriesUpToOn n (fun x i => φ i x) (fun x m => ContinuousMultilinearMap.pi fun i => p' i x m) s ↔
    ∀ (i : ι), HasFTaylorSeriesUpToOn n (φ i) (p' i) s

Next tactic:
refine' ⟨fun h i => _, fun h => ⟨fun x hx => _, _, _⟩⟩
theorem hasFTaylorSeriesUpToOn_pi :
    HasFTaylorSeriesUpToOn n (fun x i => φ i x)
        (fun x m => ContinuousMultilinearMap.pi fun i => p' i x m) s ↔
      ∀ i, HasFTaylorSeriesUpToOn n (φ i) (p' i) s := by
  set pr := @ContinuousLinearMap.proj 𝕜 _ ι F' _ _ _
  letI : ∀ (m : ℕ) (i : ι), NormedSpace 𝕜 (E[×m]→L[𝕜] F' i) := fun m i => inferInstance
  set L : ∀ m : ℕ, (∀ i, E[×m]→L[𝕜] F' i) ≃ₗᵢ[𝕜] E[×m]→L[𝕜] ∀ i, F' i := fun m =>
    ContinuousMultilinearMap.piₗᵢ _ _
  refine' ⟨fun h i => _, fun h => ⟨fun x hx => _, _, _⟩⟩
  · 

Proof state:
case refine'_1
𝕜 : Type u_1
inst✝¹⁴ : NontriviallyNormedField 𝕜
D : Type uD
inst✝¹³ : NormedAddCommGroup D
inst✝¹² : NormedSpace 𝕜 D
E : Type uE
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
F : Type uF
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
G : Type uG
inst✝⁷ : NormedAddCommGroup G
inst✝⁶ : NormedSpace 𝕜 G
X : Type u_2
inst✝⁵ : NormedAddCommGroup X
inst✝⁴ : NormedSpace 𝕜 X
s s₁ t u : Set E
f f₁ : E → F
g : F → G
x x₀ : E
c : F
b : E × F → G
m n : ℕ∞
p : E → FormalMultilinearSeries 𝕜 E F
ι : Type u_3
ι' : Type u_4
inst✝³ : Fintype ι
inst✝² : Fintype ι'
F' : ι → Type u_5
inst✝¹ : (i : ι) → NormedAddCommGroup (F' i)
inst✝ : (i : ι) → NormedSpace 𝕜 (F' i)
φ : (i : ι) → E → F' i
p' : (i : ι) → E → FormalMultilinearSeries 𝕜 E (F' i)
Φ : E → (i : ι) → F' i
P' : E → FormalMultilinearSeries 𝕜 E ((i : ι) → F' i)
pr : (i : ι) → ((i : ι) → F' i) →L[𝕜] F' i := ContinuousLinearMap.proj
this : (m : ℕ) → (i : ι) → NormedSpace 𝕜 (ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) := fun m i => inferInstance
L : (m : ℕ) →
  ((i : ι) → ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) ≃ₗᵢ[𝕜]
    ContinuousMultilinearMap 𝕜 (fun i => E) ((i : ι) → F' i) :=
  fun m => ContinuousMultilinearMap.piₗᵢ 𝕜 fun i => E
h : HasFTaylorSeriesUpToOn n (fun x i => φ i x) (fun x m => ContinuousMultilinearMap.pi fun i => p' i x m) s
i : ι
⊢ HasFTaylorSeriesUpToOn n (φ i) (p' i) s

Next tactic:
convert h.continuousLinearMap_comp (pr i)
theorem hasFTaylorSeriesUpToOn_pi :
    HasFTaylorSeriesUpToOn n (fun x i => φ i x)
        (fun x m => ContinuousMultilinearMap.pi fun i => p' i x m) s ↔
      ∀ i, HasFTaylorSeriesUpToOn n (φ i) (p' i) s := by
  set pr := @ContinuousLinearMap.proj 𝕜 _ ι F' _ _ _
  letI : ∀ (m : ℕ) (i : ι), NormedSpace 𝕜 (E[×m]→L[𝕜] F' i) := fun m i => inferInstance
  set L : ∀ m : ℕ, (∀ i, E[×m]→L[𝕜] F' i) ≃ₗᵢ[𝕜] E[×m]→L[𝕜] ∀ i, F' i := fun m =>
    ContinuousMultilinearMap.piₗᵢ _ _
  refine' ⟨fun h i => _, fun h => ⟨fun x hx => _, _, _⟩⟩
  · convert h.continuousLinearMap_comp (pr i)
  · 

Proof state:
case refine'_2
𝕜 : Type u_1
inst✝¹⁴ : NontriviallyNormedField 𝕜
D : Type uD
inst✝¹³ : NormedAddCommGroup D
inst✝¹² : NormedSpace 𝕜 D
E : Type uE
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
F : Type uF
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
G : Type uG
inst✝⁷ : NormedAddCommGroup G
inst✝⁶ : NormedSpace 𝕜 G
X : Type u_2
inst✝⁵ : NormedAddCommGroup X
inst✝⁴ : NormedSpace 𝕜 X
s s₁ t u : Set E
f f₁ : E → F
g : F → G
x✝ x₀ : E
c : F
b : E × F → G
m n : ℕ∞
p : E → FormalMultilinearSeries 𝕜 E F
ι : Type u_3
ι' : Type u_4
inst✝³ : Fintype ι
inst✝² : Fintype ι'
F' : ι → Type u_5
inst✝¹ : (i : ι) → NormedAddCommGroup (F' i)
inst✝ : (i : ι) → NormedSpace 𝕜 (F' i)
φ : (i : ι) → E → F' i
p' : (i : ι) → E → FormalMultilinearSeries 𝕜 E (F' i)
Φ : E → (i : ι) → F' i
P' : E → FormalMultilinearSeries 𝕜 E ((i : ι) → F' i)
pr : (i : ι) → ((i : ι) → F' i) →L[𝕜] F' i := ContinuousLinearMap.proj
this : (m : ℕ) → (i : ι) → NormedSpace 𝕜 (ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) := fun m i => inferInstance
L : (m : ℕ) →
  ((i : ι) → ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) ≃ₗᵢ[𝕜]
    ContinuousMultilinearMap 𝕜 (fun i => E) ((i : ι) → F' i) :=
  fun m => ContinuousMultilinearMap.piₗᵢ 𝕜 fun i => E
h : ∀ (i : ι), HasFTaylorSeriesUpToOn n (φ i) (p' i) s
x : E
hx : x ∈ s
⊢ ContinuousMultilinearMap.uncurry0 (ContinuousMultilinearMap.pi fun i => p' i x 0) = fun i => φ i x

Next tactic:
ext1 i
theorem hasFTaylorSeriesUpToOn_pi :
    HasFTaylorSeriesUpToOn n (fun x i => φ i x)
        (fun x m => ContinuousMultilinearMap.pi fun i => p' i x m) s ↔
      ∀ i, HasFTaylorSeriesUpToOn n (φ i) (p' i) s := by
  set pr := @ContinuousLinearMap.proj 𝕜 _ ι F' _ _ _
  letI : ∀ (m : ℕ) (i : ι), NormedSpace 𝕜 (E[×m]→L[𝕜] F' i) := fun m i => inferInstance
  set L : ∀ m : ℕ, (∀ i, E[×m]→L[𝕜] F' i) ≃ₗᵢ[𝕜] E[×m]→L[𝕜] ∀ i, F' i := fun m =>
    ContinuousMultilinearMap.piₗᵢ _ _
  refine' ⟨fun h i => _, fun h => ⟨fun x hx => _, _, _⟩⟩
  · convert h.continuousLinearMap_comp (pr i)
  · ext1 i
    

Proof state:
case refine'_2.h
𝕜 : Type u_1
inst✝¹⁴ : NontriviallyNormedField 𝕜
D : Type uD
inst✝¹³ : NormedAddCommGroup D
inst✝¹² : NormedSpace 𝕜 D
E : Type uE
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
F : Type uF
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
G : Type uG
inst✝⁷ : NormedAddCommGroup G
inst✝⁶ : NormedSpace 𝕜 G
X : Type u_2
inst✝⁵ : NormedAddCommGroup X
inst✝⁴ : NormedSpace 𝕜 X
s s₁ t u : Set E
f f₁ : E → F
g : F → G
x✝ x₀ : E
c : F
b : E × F → G
m n : ℕ∞
p : E → FormalMultilinearSeries 𝕜 E F
ι : Type u_3
ι' : Type u_4
inst✝³ : Fintype ι
inst✝² : Fintype ι'
F' : ι → Type u_5
inst✝¹ : (i : ι) → NormedAddCommGroup (F' i)
inst✝ : (i : ι) → NormedSpace 𝕜 (F' i)
φ : (i : ι) → E → F' i
p' : (i : ι) → E → FormalMultilinearSeries 𝕜 E (F' i)
Φ : E → (i : ι) → F' i
P' : E → FormalMultilinearSeries 𝕜 E ((i : ι) → F' i)
pr : (i : ι) → ((i : ι) → F' i) →L[𝕜] F' i := ContinuousLinearMap.proj
this : (m : ℕ) → (i : ι) → NormedSpace 𝕜 (ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) := fun m i => inferInstance
L : (m : ℕ) →
  ((i : ι) → ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) ≃ₗᵢ[𝕜]
    ContinuousMultilinearMap 𝕜 (fun i => E) ((i : ι) → F' i) :=
  fun m => ContinuousMultilinearMap.piₗᵢ 𝕜 fun i => E
h : ∀ (i : ι), HasFTaylorSeriesUpToOn n (φ i) (p' i) s
x : E
hx : x ∈ s
i : ι
⊢ ContinuousMultilinearMap.uncurry0 (ContinuousMultilinearMap.pi fun i => p' i x 0) i = φ i x

Next tactic:
exact (h i).zero_eq x hx
theorem hasFTaylorSeriesUpToOn_pi :
    HasFTaylorSeriesUpToOn n (fun x i => φ i x)
        (fun x m => ContinuousMultilinearMap.pi fun i => p' i x m) s ↔
      ∀ i, HasFTaylorSeriesUpToOn n (φ i) (p' i) s := by
  set pr := @ContinuousLinearMap.proj 𝕜 _ ι F' _ _ _
  letI : ∀ (m : ℕ) (i : ι), NormedSpace 𝕜 (E[×m]→L[𝕜] F' i) := fun m i => inferInstance
  set L : ∀ m : ℕ, (∀ i, E[×m]→L[𝕜] F' i) ≃ₗᵢ[𝕜] E[×m]→L[𝕜] ∀ i, F' i := fun m =>
    ContinuousMultilinearMap.piₗᵢ _ _
  refine' ⟨fun h i => _, fun h => ⟨fun x hx => _, _, _⟩⟩
  · convert h.continuousLinearMap_comp (pr i)
  · ext1 i
    exact (h i).zero_eq x hx
  · 

Proof state:
case refine'_3
𝕜 : Type u_1
inst✝¹⁴ : NontriviallyNormedField 𝕜
D : Type uD
inst✝¹³ : NormedAddCommGroup D
inst✝¹² : NormedSpace 𝕜 D
E : Type uE
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
F : Type uF
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
G : Type uG
inst✝⁷ : NormedAddCommGroup G
inst✝⁶ : NormedSpace 𝕜 G
X : Type u_2
inst✝⁵ : NormedAddCommGroup X
inst✝⁴ : NormedSpace 𝕜 X
s s₁ t u : Set E
f f₁ : E → F
g : F → G
x x₀ : E
c : F
b : E × F → G
m n : ℕ∞
p : E → FormalMultilinearSeries 𝕜 E F
ι : Type u_3
ι' : Type u_4
inst✝³ : Fintype ι
inst✝² : Fintype ι'
F' : ι → Type u_5
inst✝¹ : (i : ι) → NormedAddCommGroup (F' i)
inst✝ : (i : ι) → NormedSpace 𝕜 (F' i)
φ : (i : ι) → E → F' i
p' : (i : ι) → E → FormalMultilinearSeries 𝕜 E (F' i)
Φ : E → (i : ι) → F' i
P' : E → FormalMultilinearSeries 𝕜 E ((i : ι) → F' i)
pr : (i : ι) → ((i : ι) → F' i) →L[𝕜] F' i := ContinuousLinearMap.proj
this : (m : ℕ) → (i : ι) → NormedSpace 𝕜 (ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) := fun m i => inferInstance
L : (m : ℕ) →
  ((i : ι) → ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) ≃ₗᵢ[𝕜]
    ContinuousMultilinearMap 𝕜 (fun i => E) ((i : ι) → F' i) :=
  fun m => ContinuousMultilinearMap.piₗᵢ 𝕜 fun i => E
h : ∀ (i : ι), HasFTaylorSeriesUpToOn n (φ i) (p' i) s
⊢ ∀ (m : ℕ),
    ↑m < n →
      ∀ x ∈ s,
        HasFDerivWithinAt (fun x => ContinuousMultilinearMap.pi fun i => p' i x m)
          (ContinuousMultilinearMap.curryLeft (ContinuousMultilinearMap.pi fun i => p' i x (Nat.succ m))) s x

Next tactic:
intro m hm x hx
theorem hasFTaylorSeriesUpToOn_pi :
    HasFTaylorSeriesUpToOn n (fun x i => φ i x)
        (fun x m => ContinuousMultilinearMap.pi fun i => p' i x m) s ↔
      ∀ i, HasFTaylorSeriesUpToOn n (φ i) (p' i) s := by
  set pr := @ContinuousLinearMap.proj 𝕜 _ ι F' _ _ _
  letI : ∀ (m : ℕ) (i : ι), NormedSpace 𝕜 (E[×m]→L[𝕜] F' i) := fun m i => inferInstance
  set L : ∀ m : ℕ, (∀ i, E[×m]→L[𝕜] F' i) ≃ₗᵢ[𝕜] E[×m]→L[𝕜] ∀ i, F' i := fun m =>
    ContinuousMultilinearMap.piₗᵢ _ _
  refine' ⟨fun h i => _, fun h => ⟨fun x hx => _, _, _⟩⟩
  · convert h.continuousLinearMap_comp (pr i)
  · ext1 i
    exact (h i).zero_eq x hx
  · intro m hm x hx
    

Proof state:
case refine'_3
𝕜 : Type u_1
inst✝¹⁴ : NontriviallyNormedField 𝕜
D : Type uD
inst✝¹³ : NormedAddCommGroup D
inst✝¹² : NormedSpace 𝕜 D
E : Type uE
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
F : Type uF
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
G : Type uG
inst✝⁷ : NormedAddCommGroup G
inst✝⁶ : NormedSpace 𝕜 G
X : Type u_2
inst✝⁵ : NormedAddCommGroup X
inst✝⁴ : NormedSpace 𝕜 X
s s₁ t u : Set E
f f₁ : E → F
g : F → G
x✝ x₀ : E
c : F
b : E × F → G
m✝ n : ℕ∞
p : E → FormalMultilinearSeries 𝕜 E F
ι : Type u_3
ι' : Type u_4
inst✝³ : Fintype ι
inst✝² : Fintype ι'
F' : ι → Type u_5
inst✝¹ : (i : ι) → NormedAddCommGroup (F' i)
inst✝ : (i : ι) → NormedSpace 𝕜 (F' i)
φ : (i : ι) → E → F' i
p' : (i : ι) → E → FormalMultilinearSeries 𝕜 E (F' i)
Φ : E → (i : ι) → F' i
P' : E → FormalMultilinearSeries 𝕜 E ((i : ι) → F' i)
pr : (i : ι) → ((i : ι) → F' i) →L[𝕜] F' i := ContinuousLinearMap.proj
this : (m : ℕ) → (i : ι) → NormedSpace 𝕜 (ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) := fun m i => inferInstance
L : (m : ℕ) →
  ((i : ι) → ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) ≃ₗᵢ[𝕜]
    ContinuousMultilinearMap 𝕜 (fun i => E) ((i : ι) → F' i) :=
  fun m => ContinuousMultilinearMap.piₗᵢ 𝕜 fun i => E
h : ∀ (i : ι), HasFTaylorSeriesUpToOn n (φ i) (p' i) s
m : ℕ
hm : ↑m < n
x : E
hx : x ∈ s
⊢ HasFDerivWithinAt (fun x => ContinuousMultilinearMap.pi fun i => p' i x m)
    (ContinuousMultilinearMap.curryLeft (ContinuousMultilinearMap.pi fun i => p' i x (Nat.succ m))) s x

Next tactic:
have := hasFDerivWithinAt_pi.2 fun i => (h i).fderivWithin m hm x hx
theorem hasFTaylorSeriesUpToOn_pi :
    HasFTaylorSeriesUpToOn n (fun x i => φ i x)
        (fun x m => ContinuousMultilinearMap.pi fun i => p' i x m) s ↔
      ∀ i, HasFTaylorSeriesUpToOn n (φ i) (p' i) s := by
  set pr := @ContinuousLinearMap.proj 𝕜 _ ι F' _ _ _
  letI : ∀ (m : ℕ) (i : ι), NormedSpace 𝕜 (E[×m]→L[𝕜] F' i) := fun m i => inferInstance
  set L : ∀ m : ℕ, (∀ i, E[×m]→L[𝕜] F' i) ≃ₗᵢ[𝕜] E[×m]→L[𝕜] ∀ i, F' i := fun m =>
    ContinuousMultilinearMap.piₗᵢ _ _
  refine' ⟨fun h i => _, fun h => ⟨fun x hx => _, _, _⟩⟩
  · convert h.continuousLinearMap_comp (pr i)
  · ext1 i
    exact (h i).zero_eq x hx
  · intro m hm x hx
    have := hasFDerivWithinAt_pi.2 fun i => (h i).fderivWithin m hm x hx
    

Proof state:
case refine'_3
𝕜 : Type u_1
inst✝¹⁴ : NontriviallyNormedField 𝕜
D : Type uD
inst✝¹³ : NormedAddCommGroup D
inst✝¹² : NormedSpace 𝕜 D
E : Type uE
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
F : Type uF
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
G : Type uG
inst✝⁷ : NormedAddCommGroup G
inst✝⁶ : NormedSpace 𝕜 G
X : Type u_2
inst✝⁵ : NormedAddCommGroup X
inst✝⁴ : NormedSpace 𝕜 X
s s₁ t u : Set E
f f₁ : E → F
g : F → G
x✝ x₀ : E
c : F
b : E × F → G
m✝ n : ℕ∞
p : E → FormalMultilinearSeries 𝕜 E F
ι : Type u_3
ι' : Type u_4
inst✝³ : Fintype ι
inst✝² : Fintype ι'
F' : ι → Type u_5
inst✝¹ : (i : ι) → NormedAddCommGroup (F' i)
inst✝ : (i : ι) → NormedSpace 𝕜 (F' i)
φ : (i : ι) → E → F' i
p' : (i : ι) → E → FormalMultilinearSeries 𝕜 E (F' i)
Φ : E → (i : ι) → F' i
P' : E → FormalMultilinearSeries 𝕜 E ((i : ι) → F' i)
pr : (i : ι) → ((i : ι) → F' i) →L[𝕜] F' i := ContinuousLinearMap.proj
this✝ : (m : ℕ) → (i : ι) → NormedSpace 𝕜 (ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) := fun m i => inferInstance
L : (m : ℕ) →
  ((i : ι) → ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) ≃ₗᵢ[𝕜]
    ContinuousMultilinearMap 𝕜 (fun i => E) ((i : ι) → F' i) :=
  fun m => ContinuousMultilinearMap.piₗᵢ 𝕜 fun i => E
h : ∀ (i : ι), HasFTaylorSeriesUpToOn n (φ i) (p' i) s
m : ℕ
hm : ↑m < n
x : E
hx : x ∈ s
this :
  HasFDerivWithinAt (fun x i => p' i x m)
    (ContinuousLinearMap.pi fun i => ContinuousMultilinearMap.curryLeft (p' i x (Nat.succ m))) s x
⊢ HasFDerivWithinAt (fun x => ContinuousMultilinearMap.pi fun i => p' i x m)
    (ContinuousMultilinearMap.curryLeft (ContinuousMultilinearMap.pi fun i => p' i x (Nat.succ m))) s x

Next tactic:
convert (L m).hasFDerivAt.comp_hasFDerivWithinAt x this
theorem hasFTaylorSeriesUpToOn_pi :
    HasFTaylorSeriesUpToOn n (fun x i => φ i x)
        (fun x m => ContinuousMultilinearMap.pi fun i => p' i x m) s ↔
      ∀ i, HasFTaylorSeriesUpToOn n (φ i) (p' i) s := by
  set pr := @ContinuousLinearMap.proj 𝕜 _ ι F' _ _ _
  letI : ∀ (m : ℕ) (i : ι), NormedSpace 𝕜 (E[×m]→L[𝕜] F' i) := fun m i => inferInstance
  set L : ∀ m : ℕ, (∀ i, E[×m]→L[𝕜] F' i) ≃ₗᵢ[𝕜] E[×m]→L[𝕜] ∀ i, F' i := fun m =>
    ContinuousMultilinearMap.piₗᵢ _ _
  refine' ⟨fun h i => _, fun h => ⟨fun x hx => _, _, _⟩⟩
  · convert h.continuousLinearMap_comp (pr i)
  · ext1 i
    exact (h i).zero_eq x hx
  · intro m hm x hx
    have := hasFDerivWithinAt_pi.2 fun i => (h i).fderivWithin m hm x hx
    convert (L m).hasFDerivAt.comp_hasFDerivWithinAt x this
  · 

Proof state:
case refine'_4
𝕜 : Type u_1
inst✝¹⁴ : NontriviallyNormedField 𝕜
D : Type uD
inst✝¹³ : NormedAddCommGroup D
inst✝¹² : NormedSpace 𝕜 D
E : Type uE
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
F : Type uF
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
G : Type uG
inst✝⁷ : NormedAddCommGroup G
inst✝⁶ : NormedSpace 𝕜 G
X : Type u_2
inst✝⁵ : NormedAddCommGroup X
inst✝⁴ : NormedSpace 𝕜 X
s s₁ t u : Set E
f f₁ : E → F
g : F → G
x x₀ : E
c : F
b : E × F → G
m n : ℕ∞
p : E → FormalMultilinearSeries 𝕜 E F
ι : Type u_3
ι' : Type u_4
inst✝³ : Fintype ι
inst✝² : Fintype ι'
F' : ι → Type u_5
inst✝¹ : (i : ι) → NormedAddCommGroup (F' i)
inst✝ : (i : ι) → NormedSpace 𝕜 (F' i)
φ : (i : ι) → E → F' i
p' : (i : ι) → E → FormalMultilinearSeries 𝕜 E (F' i)
Φ : E → (i : ι) → F' i
P' : E → FormalMultilinearSeries 𝕜 E ((i : ι) → F' i)
pr : (i : ι) → ((i : ι) → F' i) →L[𝕜] F' i := ContinuousLinearMap.proj
this : (m : ℕ) → (i : ι) → NormedSpace 𝕜 (ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) := fun m i => inferInstance
L : (m : ℕ) →
  ((i : ι) → ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) ≃ₗᵢ[𝕜]
    ContinuousMultilinearMap 𝕜 (fun i => E) ((i : ι) → F' i) :=
  fun m => ContinuousMultilinearMap.piₗᵢ 𝕜 fun i => E
h : ∀ (i : ι), HasFTaylorSeriesUpToOn n (φ i) (p' i) s
⊢ ∀ (m : ℕ), ↑m ≤ n → ContinuousOn (fun x => ContinuousMultilinearMap.pi fun i => p' i x m) s

Next tactic:
intro m hm
theorem hasFTaylorSeriesUpToOn_pi :
    HasFTaylorSeriesUpToOn n (fun x i => φ i x)
        (fun x m => ContinuousMultilinearMap.pi fun i => p' i x m) s ↔
      ∀ i, HasFTaylorSeriesUpToOn n (φ i) (p' i) s := by
  set pr := @ContinuousLinearMap.proj 𝕜 _ ι F' _ _ _
  letI : ∀ (m : ℕ) (i : ι), NormedSpace 𝕜 (E[×m]→L[𝕜] F' i) := fun m i => inferInstance
  set L : ∀ m : ℕ, (∀ i, E[×m]→L[𝕜] F' i) ≃ₗᵢ[𝕜] E[×m]→L[𝕜] ∀ i, F' i := fun m =>
    ContinuousMultilinearMap.piₗᵢ _ _
  refine' ⟨fun h i => _, fun h => ⟨fun x hx => _, _, _⟩⟩
  · convert h.continuousLinearMap_comp (pr i)
  · ext1 i
    exact (h i).zero_eq x hx
  · intro m hm x hx
    have := hasFDerivWithinAt_pi.2 fun i => (h i).fderivWithin m hm x hx
    convert (L m).hasFDerivAt.comp_hasFDerivWithinAt x this
  · intro m hm
    

Proof state:
case refine'_4
𝕜 : Type u_1
inst✝¹⁴ : NontriviallyNormedField 𝕜
D : Type uD
inst✝¹³ : NormedAddCommGroup D
inst✝¹² : NormedSpace 𝕜 D
E : Type uE
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
F : Type uF
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
G : Type uG
inst✝⁷ : NormedAddCommGroup G
inst✝⁶ : NormedSpace 𝕜 G
X : Type u_2
inst✝⁵ : NormedAddCommGroup X
inst✝⁴ : NormedSpace 𝕜 X
s s₁ t u : Set E
f f₁ : E → F
g : F → G
x x₀ : E
c : F
b : E × F → G
m✝ n : ℕ∞
p : E → FormalMultilinearSeries 𝕜 E F
ι : Type u_3
ι' : Type u_4
inst✝³ : Fintype ι
inst✝² : Fintype ι'
F' : ι → Type u_5
inst✝¹ : (i : ι) → NormedAddCommGroup (F' i)
inst✝ : (i : ι) → NormedSpace 𝕜 (F' i)
φ : (i : ι) → E → F' i
p' : (i : ι) → E → FormalMultilinearSeries 𝕜 E (F' i)
Φ : E → (i : ι) → F' i
P' : E → FormalMultilinearSeries 𝕜 E ((i : ι) → F' i)
pr : (i : ι) → ((i : ι) → F' i) →L[𝕜] F' i := ContinuousLinearMap.proj
this : (m : ℕ) → (i : ι) → NormedSpace 𝕜 (ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) := fun m i => inferInstance
L : (m : ℕ) →
  ((i : ι) → ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) ≃ₗᵢ[𝕜]
    ContinuousMultilinearMap 𝕜 (fun i => E) ((i : ι) → F' i) :=
  fun m => ContinuousMultilinearMap.piₗᵢ 𝕜 fun i => E
h : ∀ (i : ι), HasFTaylorSeriesUpToOn n (φ i) (p' i) s
m : ℕ
hm : ↑m ≤ n
⊢ ContinuousOn (fun x => ContinuousMultilinearMap.pi fun i => p' i x m) s

Next tactic:
have := continuousOn_pi.2 fun i => (h i).cont m hm
theorem hasFTaylorSeriesUpToOn_pi :
    HasFTaylorSeriesUpToOn n (fun x i => φ i x)
        (fun x m => ContinuousMultilinearMap.pi fun i => p' i x m) s ↔
      ∀ i, HasFTaylorSeriesUpToOn n (φ i) (p' i) s := by
  set pr := @ContinuousLinearMap.proj 𝕜 _ ι F' _ _ _
  letI : ∀ (m : ℕ) (i : ι), NormedSpace 𝕜 (E[×m]→L[𝕜] F' i) := fun m i => inferInstance
  set L : ∀ m : ℕ, (∀ i, E[×m]→L[𝕜] F' i) ≃ₗᵢ[𝕜] E[×m]→L[𝕜] ∀ i, F' i := fun m =>
    ContinuousMultilinearMap.piₗᵢ _ _
  refine' ⟨fun h i => _, fun h => ⟨fun x hx => _, _, _⟩⟩
  · convert h.continuousLinearMap_comp (pr i)
  · ext1 i
    exact (h i).zero_eq x hx
  · intro m hm x hx
    have := hasFDerivWithinAt_pi.2 fun i => (h i).fderivWithin m hm x hx
    convert (L m).hasFDerivAt.comp_hasFDerivWithinAt x this
  · intro m hm
    have := continuousOn_pi.2 fun i => (h i).cont m hm
    

Proof state:
case refine'_4
𝕜 : Type u_1
inst✝¹⁴ : NontriviallyNormedField 𝕜
D : Type uD
inst✝¹³ : NormedAddCommGroup D
inst✝¹² : NormedSpace 𝕜 D
E : Type uE
inst✝¹¹ : NormedAddCommGroup E
inst✝¹⁰ : NormedSpace 𝕜 E
F : Type uF
inst✝⁹ : NormedAddCommGroup F
inst✝⁸ : NormedSpace 𝕜 F
G : Type uG
inst✝⁷ : NormedAddCommGroup G
inst✝⁶ : NormedSpace 𝕜 G
X : Type u_2
inst✝⁵ : NormedAddCommGroup X
inst✝⁴ : NormedSpace 𝕜 X
s s₁ t u : Set E
f f₁ : E → F
g : F → G
x x₀ : E
c : F
b : E × F → G
m✝ n : ℕ∞
p : E → FormalMultilinearSeries 𝕜 E F
ι : Type u_3
ι' : Type u_4
inst✝³ : Fintype ι
inst✝² : Fintype ι'
F' : ι → Type u_5
inst✝¹ : (i : ι) → NormedAddCommGroup (F' i)
inst✝ : (i : ι) → NormedSpace 𝕜 (F' i)
φ : (i : ι) → E → F' i
p' : (i : ι) → E → FormalMultilinearSeries 𝕜 E (F' i)
Φ : E → (i : ι) → F' i
P' : E → FormalMultilinearSeries 𝕜 E ((i : ι) → F' i)
pr : (i : ι) → ((i : ι) → F' i) →L[𝕜] F' i := ContinuousLinearMap.proj
this✝ : (m : ℕ) → (i : ι) → NormedSpace 𝕜 (ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) := fun m i => inferInstance
L : (m : ℕ) →
  ((i : ι) → ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) ≃ₗᵢ[𝕜]
    ContinuousMultilinearMap 𝕜 (fun i => E) ((i : ι) → F' i) :=
  fun m => ContinuousMultilinearMap.piₗᵢ 𝕜 fun i => E
h : ∀ (i : ι), HasFTaylorSeriesUpToOn n (φ i) (p' i) s
m : ℕ
hm : ↑m ≤ n
this : ContinuousOn (fun y i => p' i y m) s
⊢ ContinuousOn (fun x => ContinuousMultilinearMap.pi fun i => p' i x m) s

Next tactic:
convert (L m).continuous.comp_continuousOn this