l3lab/ntp-mathlib huggingface dataset:
/-- Bounding the norm of the iterated derivative of `B (f x) (g x)` within a set in terms of the
iterated derivatives of `f` and `g` when `B` is bilinear. This lemma is an auxiliary version
assuming all spaces live in the same universe, to enable an induction. Use instead
`ContinuousLinearMap.norm_iteratedFDerivWithin_le_of_bilinear` that removes this assumption. -/
theorem ContinuousLinearMap.norm_iteratedFDerivWithin_le_of_bilinear_aux {Du Eu Fu Gu : Type u}
    [NormedAddCommGroup Du] [NormedSpace 𝕜 Du] [NormedAddCommGroup Eu] [NormedSpace 𝕜 Eu]
    [NormedAddCommGroup Fu] [NormedSpace 𝕜 Fu] [NormedAddCommGroup Gu] [NormedSpace 𝕜 Gu]
    (B : Eu →L[𝕜] Fu →L[𝕜] Gu) {f : Du → Eu} {g : Du → Fu} {n : ℕ} {s : Set Du} {x : Du}
    (hf : ContDiffOn 𝕜 n f s) (hg : ContDiffOn 𝕜 n g s) (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) :
    ‖iteratedFDerivWithin 𝕜 n (fun y => B (f y) (g y)) s x‖ ≤
      ‖B‖ * ∑ i in Finset.range (n + 1), (n.choose i : ℝ) * ‖iteratedFDerivWithin 𝕜 i f s x‖ *
        ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖ := by
  /- We argue by induction on `n`. The bound is trivial for `n = 0`. For `n + 1`, we write
    the `(n+1)`-th derivative as the `n`-th derivative of the derivative `B f g' + B f' g`,
    and apply the inductive assumption to each of those two terms. For this induction to make sense,
    the spaces of linear maps that appear in the induction should be in the same universe as the
    original spaces, which explains why we assume in the lemma that all spaces live in the same
    universe. -/
  induction' n with n IH generalizing Eu Fu Gu
  · simp only [Nat.zero_eq, norm_iteratedFDerivWithin_zero, zero_add, Finset.range_one,
      Finset.sum_singleton, Nat.choose_self, Nat.cast_one, one_mul, Nat.sub_zero, ← mul_assoc]
    

Proof state:
case zero
𝕜 : Type u_1
inst✝¹⁶ : NontriviallyNormedField 𝕜
D : Type uD
inst✝¹⁵ : NormedAddCommGroup D
inst✝¹⁴ : NormedSpace 𝕜 D
E : Type uE
inst✝¹³ : NormedAddCommGroup E
inst✝¹² : NormedSpace 𝕜 E
F : Type uF
inst✝¹¹ : NormedAddCommGroup F
inst✝¹⁰ : NormedSpace 𝕜 F
G : Type uG
inst✝⁹ : NormedAddCommGroup G
inst✝⁸ : NormedSpace 𝕜 G
s✝ s₁ t u : Set E
Du : Type u
inst✝⁷ : NormedAddCommGroup Du
inst✝⁶ : NormedSpace 𝕜 Du
s : Set Du
x : Du
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
Eu Fu Gu : Type u
inst✝⁵ : NormedAddCommGroup Eu
inst✝⁴ : NormedSpace 𝕜 Eu
inst✝³ : NormedAddCommGroup Fu
inst✝² : NormedSpace 𝕜 Fu
inst✝¹ : NormedAddCommGroup Gu
inst✝ : NormedSpace 𝕜 Gu
B : Eu →L[𝕜] Fu →L[𝕜] Gu
f : Du → Eu
g : Du → Fu
hf : ContDiffOn 𝕜 (↑Nat.zero) f s
hg : ContDiffOn 𝕜 (↑Nat.zero) g s
⊢ ‖(B (f x)) (g x)‖ ≤ ‖B‖ * ‖f x‖ * ‖g x‖

Next tactic:
apply B.le_op_norm₂
set_option maxHeartbeats 700000 in -- 3.5× the default limit
/-- Bounding the norm of the iterated derivative of `B (f x) (g x)` within a set in terms of the
iterated derivatives of `f` and `g` when `B` is bilinear:
`‖D^n (x ↦ B (f x) (g x))‖ ≤ ‖B‖ ∑_{k ≤ n} n.choose k ‖D^k f‖ ‖D^{n-k} g‖` -/
theorem ContinuousLinearMap.norm_iteratedFDerivWithin_le_of_bilinear (B : E →L[𝕜] F →L[𝕜] G)
    {f : D → E} {g : D → F} {N : ℕ∞} {s : Set D} {x : D} (hf : ContDiffOn 𝕜 N f s)
    (hg : ContDiffOn 𝕜 N g s) (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) {n : ℕ} (hn : (n : ℕ∞) ≤ N) :
    ‖iteratedFDerivWithin 𝕜 n (fun y => B (f y) (g y)) s x‖ ≤
      ‖B‖ * ∑ i in Finset.range (n + 1), (n.choose i : ℝ) * ‖iteratedFDerivWithin 𝕜 i f s x‖ *
        ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖ := by
  /- We reduce the bound to the case where all spaces live in the same universe (in which we
    already have proved the result), by using linear isometries between the spaces and their `ULift`
    to a common universe. These linear isometries preserve the norm of the iterated derivative. -/
  

Proof state:
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
D : Type uD
inst✝⁷ : NormedAddCommGroup D
inst✝⁶ : NormedSpace 𝕜 D
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s✝ s₁ t u : Set E
B : E →L[𝕜] F →L[𝕜] G
f : D → E
g : D → F
N : ℕ∞
s : Set D
x : D
hf : ContDiffOn 𝕜 N f s
hg : ContDiffOn 𝕜 N g s
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
hn : ↑n ≤ N
⊢ ‖iteratedFDerivWithin 𝕜 n (fun y => (B (f y)) (g y)) s x‖ ≤
    ‖B‖ *
      ∑ i in Finset.range (n + 1),
        ↑(Nat.choose n i) * ‖iteratedFDerivWithin 𝕜 i f s x‖ * ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖

Next tactic:
let Du : Type max uD uE uF uG := ULift.{max uE uF uG, uD} D
set_option maxHeartbeats 700000 in -- 3.5× the default limit
/-- Bounding the norm of the iterated derivative of `B (f x) (g x)` within a set in terms of the
iterated derivatives of `f` and `g` when `B` is bilinear:
`‖D^n (x ↦ B (f x) (g x))‖ ≤ ‖B‖ ∑_{k ≤ n} n.choose k ‖D^k f‖ ‖D^{n-k} g‖` -/
theorem ContinuousLinearMap.norm_iteratedFDerivWithin_le_of_bilinear (B : E →L[𝕜] F →L[𝕜] G)
    {f : D → E} {g : D → F} {N : ℕ∞} {s : Set D} {x : D} (hf : ContDiffOn 𝕜 N f s)
    (hg : ContDiffOn 𝕜 N g s) (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) {n : ℕ} (hn : (n : ℕ∞) ≤ N) :
    ‖iteratedFDerivWithin 𝕜 n (fun y => B (f y) (g y)) s x‖ ≤
      ‖B‖ * ∑ i in Finset.range (n + 1), (n.choose i : ℝ) * ‖iteratedFDerivWithin 𝕜 i f s x‖ *
        ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖ := by
  /- We reduce the bound to the case where all spaces live in the same universe (in which we
    already have proved the result), by using linear isometries between the spaces and their `ULift`
    to a common universe. These linear isometries preserve the norm of the iterated derivative. -/
  let Du : Type max uD uE uF uG := ULift.{max uE uF uG, uD} D
  

Proof state:
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
D : Type uD
inst✝⁷ : NormedAddCommGroup D
inst✝⁶ : NormedSpace 𝕜 D
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s✝ s₁ t u : Set E
B : E →L[𝕜] F →L[𝕜] G
f : D → E
g : D → F
N : ℕ∞
s : Set D
x : D
hf : ContDiffOn 𝕜 N f s
hg : ContDiffOn 𝕜 N g s
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
hn : ↑n ≤ N
Du : Type (max uD uE uF uG) := ULift.{max uE uF uG, uD} D
⊢ ‖iteratedFDerivWithin 𝕜 n (fun y => (B (f y)) (g y)) s x‖ ≤
    ‖B‖ *
      ∑ i in Finset.range (n + 1),
        ↑(Nat.choose n i) * ‖iteratedFDerivWithin 𝕜 i f s x‖ * ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖

Next tactic:
let Eu : Type max uD uE uF uG := ULift.{max uD uF uG, uE} E
set_option maxHeartbeats 700000 in -- 3.5× the default limit
/-- Bounding the norm of the iterated derivative of `B (f x) (g x)` within a set in terms of the
iterated derivatives of `f` and `g` when `B` is bilinear:
`‖D^n (x ↦ B (f x) (g x))‖ ≤ ‖B‖ ∑_{k ≤ n} n.choose k ‖D^k f‖ ‖D^{n-k} g‖` -/
theorem ContinuousLinearMap.norm_iteratedFDerivWithin_le_of_bilinear (B : E →L[𝕜] F →L[𝕜] G)
    {f : D → E} {g : D → F} {N : ℕ∞} {s : Set D} {x : D} (hf : ContDiffOn 𝕜 N f s)
    (hg : ContDiffOn 𝕜 N g s) (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) {n : ℕ} (hn : (n : ℕ∞) ≤ N) :
    ‖iteratedFDerivWithin 𝕜 n (fun y => B (f y) (g y)) s x‖ ≤
      ‖B‖ * ∑ i in Finset.range (n + 1), (n.choose i : ℝ) * ‖iteratedFDerivWithin 𝕜 i f s x‖ *
        ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖ := by
  /- We reduce the bound to the case where all spaces live in the same universe (in which we
    already have proved the result), by using linear isometries between the spaces and their `ULift`
    to a common universe. These linear isometries preserve the norm of the iterated derivative. -/
  let Du : Type max uD uE uF uG := ULift.{max uE uF uG, uD} D
  let Eu : Type max uD uE uF uG := ULift.{max uD uF uG, uE} E
  

Proof state:
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
D : Type uD
inst✝⁷ : NormedAddCommGroup D
inst✝⁶ : NormedSpace 𝕜 D
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s✝ s₁ t u : Set E
B : E →L[𝕜] F →L[𝕜] G
f : D → E
g : D → F
N : ℕ∞
s : Set D
x : D
hf : ContDiffOn 𝕜 N f s
hg : ContDiffOn 𝕜 N g s
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
hn : ↑n ≤ N
Du : Type (max uD uE uF uG) := ULift.{max uE uF uG, uD} D
Eu : Type (max uD uE uF uG) := ULift.{max uD uF uG, uE} E
⊢ ‖iteratedFDerivWithin 𝕜 n (fun y => (B (f y)) (g y)) s x‖ ≤
    ‖B‖ *
      ∑ i in Finset.range (n + 1),
        ↑(Nat.choose n i) * ‖iteratedFDerivWithin 𝕜 i f s x‖ * ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖

Next tactic:
let Fu : Type max uD uE uF uG := ULift.{max uD uE uG, uF} F
set_option maxHeartbeats 700000 in -- 3.5× the default limit
/-- Bounding the norm of the iterated derivative of `B (f x) (g x)` within a set in terms of the
iterated derivatives of `f` and `g` when `B` is bilinear:
`‖D^n (x ↦ B (f x) (g x))‖ ≤ ‖B‖ ∑_{k ≤ n} n.choose k ‖D^k f‖ ‖D^{n-k} g‖` -/
theorem ContinuousLinearMap.norm_iteratedFDerivWithin_le_of_bilinear (B : E →L[𝕜] F →L[𝕜] G)
    {f : D → E} {g : D → F} {N : ℕ∞} {s : Set D} {x : D} (hf : ContDiffOn 𝕜 N f s)
    (hg : ContDiffOn 𝕜 N g s) (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) {n : ℕ} (hn : (n : ℕ∞) ≤ N) :
    ‖iteratedFDerivWithin 𝕜 n (fun y => B (f y) (g y)) s x‖ ≤
      ‖B‖ * ∑ i in Finset.range (n + 1), (n.choose i : ℝ) * ‖iteratedFDerivWithin 𝕜 i f s x‖ *
        ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖ := by
  /- We reduce the bound to the case where all spaces live in the same universe (in which we
    already have proved the result), by using linear isometries between the spaces and their `ULift`
    to a common universe. These linear isometries preserve the norm of the iterated derivative. -/
  let Du : Type max uD uE uF uG := ULift.{max uE uF uG, uD} D
  let Eu : Type max uD uE uF uG := ULift.{max uD uF uG, uE} E
  let Fu : Type max uD uE uF uG := ULift.{max uD uE uG, uF} F
  

Proof state:
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
D : Type uD
inst✝⁷ : NormedAddCommGroup D
inst✝⁶ : NormedSpace 𝕜 D
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s✝ s₁ t u : Set E
B : E →L[𝕜] F →L[𝕜] G
f : D → E
g : D → F
N : ℕ∞
s : Set D
x : D
hf : ContDiffOn 𝕜 N f s
hg : ContDiffOn 𝕜 N g s
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
hn : ↑n ≤ N
Du : Type (max uD uE uF uG) := ULift.{max uE uF uG, uD} D
Eu : Type (max uD uE uF uG) := ULift.{max uD uF uG, uE} E
Fu : Type (max uD uE uF uG) := ULift.{max uD uE uG, uF} F
⊢ ‖iteratedFDerivWithin 𝕜 n (fun y => (B (f y)) (g y)) s x‖ ≤
    ‖B‖ *
      ∑ i in Finset.range (n + 1),
        ↑(Nat.choose n i) * ‖iteratedFDerivWithin 𝕜 i f s x‖ * ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖

Next tactic:
let Gu : Type max uD uE uF uG := ULift.{max uD uE uF, uG} G
set_option maxHeartbeats 700000 in -- 3.5× the default limit
/-- Bounding the norm of the iterated derivative of `B (f x) (g x)` within a set in terms of the
iterated derivatives of `f` and `g` when `B` is bilinear:
`‖D^n (x ↦ B (f x) (g x))‖ ≤ ‖B‖ ∑_{k ≤ n} n.choose k ‖D^k f‖ ‖D^{n-k} g‖` -/
theorem ContinuousLinearMap.norm_iteratedFDerivWithin_le_of_bilinear (B : E →L[𝕜] F →L[𝕜] G)
    {f : D → E} {g : D → F} {N : ℕ∞} {s : Set D} {x : D} (hf : ContDiffOn 𝕜 N f s)
    (hg : ContDiffOn 𝕜 N g s) (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) {n : ℕ} (hn : (n : ℕ∞) ≤ N) :
    ‖iteratedFDerivWithin 𝕜 n (fun y => B (f y) (g y)) s x‖ ≤
      ‖B‖ * ∑ i in Finset.range (n + 1), (n.choose i : ℝ) * ‖iteratedFDerivWithin 𝕜 i f s x‖ *
        ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖ := by
  /- We reduce the bound to the case where all spaces live in the same universe (in which we
    already have proved the result), by using linear isometries between the spaces and their `ULift`
    to a common universe. These linear isometries preserve the norm of the iterated derivative. -/
  let Du : Type max uD uE uF uG := ULift.{max uE uF uG, uD} D
  let Eu : Type max uD uE uF uG := ULift.{max uD uF uG, uE} E
  let Fu : Type max uD uE uF uG := ULift.{max uD uE uG, uF} F
  let Gu : Type max uD uE uF uG := ULift.{max uD uE uF, uG} G
  

Proof state:
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
D : Type uD
inst✝⁷ : NormedAddCommGroup D
inst✝⁶ : NormedSpace 𝕜 D
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s✝ s₁ t u : Set E
B : E →L[𝕜] F →L[𝕜] G
f : D → E
g : D → F
N : ℕ∞
s : Set D
x : D
hf : ContDiffOn 𝕜 N f s
hg : ContDiffOn 𝕜 N g s
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
hn : ↑n ≤ N
Du : Type (max uD uE uF uG) := ULift.{max uE uF uG, uD} D
Eu : Type (max uD uE uF uG) := ULift.{max uD uF uG, uE} E
Fu : Type (max uD uE uF uG) := ULift.{max uD uE uG, uF} F
Gu : Type (max uD uE uF uG) := ULift.{max uD uE uF, uG} G
⊢ ‖iteratedFDerivWithin 𝕜 n (fun y => (B (f y)) (g y)) s x‖ ≤
    ‖B‖ *
      ∑ i in Finset.range (n + 1),
        ↑(Nat.choose n i) * ‖iteratedFDerivWithin 𝕜 i f s x‖ * ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖

Next tactic:
have isoD : Du ≃ₗᵢ[𝕜] D := LinearIsometryEquiv.ulift 𝕜 D
set_option maxHeartbeats 700000 in -- 3.5× the default limit
/-- Bounding the norm of the iterated derivative of `B (f x) (g x)` within a set in terms of the
iterated derivatives of `f` and `g` when `B` is bilinear:
`‖D^n (x ↦ B (f x) (g x))‖ ≤ ‖B‖ ∑_{k ≤ n} n.choose k ‖D^k f‖ ‖D^{n-k} g‖` -/
theorem ContinuousLinearMap.norm_iteratedFDerivWithin_le_of_bilinear (B : E →L[𝕜] F →L[𝕜] G)
    {f : D → E} {g : D → F} {N : ℕ∞} {s : Set D} {x : D} (hf : ContDiffOn 𝕜 N f s)
    (hg : ContDiffOn 𝕜 N g s) (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) {n : ℕ} (hn : (n : ℕ∞) ≤ N) :
    ‖iteratedFDerivWithin 𝕜 n (fun y => B (f y) (g y)) s x‖ ≤
      ‖B‖ * ∑ i in Finset.range (n + 1), (n.choose i : ℝ) * ‖iteratedFDerivWithin 𝕜 i f s x‖ *
        ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖ := by
  /- We reduce the bound to the case where all spaces live in the same universe (in which we
    already have proved the result), by using linear isometries between the spaces and their `ULift`
    to a common universe. These linear isometries preserve the norm of the iterated derivative. -/
  let Du : Type max uD uE uF uG := ULift.{max uE uF uG, uD} D
  let Eu : Type max uD uE uF uG := ULift.{max uD uF uG, uE} E
  let Fu : Type max uD uE uF uG := ULift.{max uD uE uG, uF} F
  let Gu : Type max uD uE uF uG := ULift.{max uD uE uF, uG} G
  have isoD : Du ≃ₗᵢ[𝕜] D := LinearIsometryEquiv.ulift 𝕜 D
  

Proof state:
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
D : Type uD
inst✝⁷ : NormedAddCommGroup D
inst✝⁶ : NormedSpace 𝕜 D
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s✝ s₁ t u : Set E
B : E →L[𝕜] F →L[𝕜] G
f : D → E
g : D → F
N : ℕ∞
s : Set D
x : D
hf : ContDiffOn 𝕜 N f s
hg : ContDiffOn 𝕜 N g s
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
hn : ↑n ≤ N
Du : Type (max uD uE uF uG) := ULift.{max uE uF uG, uD} D
Eu : Type (max uD uE uF uG) := ULift.{max uD uF uG, uE} E
Fu : Type (max uD uE uF uG) := ULift.{max uD uE uG, uF} F
Gu : Type (max uD uE uF uG) := ULift.{max uD uE uF, uG} G
isoD : Du ≃ₗᵢ[𝕜] D
⊢ ‖iteratedFDerivWithin 𝕜 n (fun y => (B (f y)) (g y)) s x‖ ≤
    ‖B‖ *
      ∑ i in Finset.range (n + 1),
        ↑(Nat.choose n i) * ‖iteratedFDerivWithin 𝕜 i f s x‖ * ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖

Next tactic:
have isoE : Eu ≃ₗᵢ[𝕜] E := LinearIsometryEquiv.ulift 𝕜 E
set_option maxHeartbeats 700000 in -- 3.5× the default limit
/-- Bounding the norm of the iterated derivative of `B (f x) (g x)` within a set in terms of the
iterated derivatives of `f` and `g` when `B` is bilinear:
`‖D^n (x ↦ B (f x) (g x))‖ ≤ ‖B‖ ∑_{k ≤ n} n.choose k ‖D^k f‖ ‖D^{n-k} g‖` -/
theorem ContinuousLinearMap.norm_iteratedFDerivWithin_le_of_bilinear (B : E →L[𝕜] F →L[𝕜] G)
    {f : D → E} {g : D → F} {N : ℕ∞} {s : Set D} {x : D} (hf : ContDiffOn 𝕜 N f s)
    (hg : ContDiffOn 𝕜 N g s) (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) {n : ℕ} (hn : (n : ℕ∞) ≤ N) :
    ‖iteratedFDerivWithin 𝕜 n (fun y => B (f y) (g y)) s x‖ ≤
      ‖B‖ * ∑ i in Finset.range (n + 1), (n.choose i : ℝ) * ‖iteratedFDerivWithin 𝕜 i f s x‖ *
        ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖ := by
  /- We reduce the bound to the case where all spaces live in the same universe (in which we
    already have proved the result), by using linear isometries between the spaces and their `ULift`
    to a common universe. These linear isometries preserve the norm of the iterated derivative. -/
  let Du : Type max uD uE uF uG := ULift.{max uE uF uG, uD} D
  let Eu : Type max uD uE uF uG := ULift.{max uD uF uG, uE} E
  let Fu : Type max uD uE uF uG := ULift.{max uD uE uG, uF} F
  let Gu : Type max uD uE uF uG := ULift.{max uD uE uF, uG} G
  have isoD : Du ≃ₗᵢ[𝕜] D := LinearIsometryEquiv.ulift 𝕜 D
  have isoE : Eu ≃ₗᵢ[𝕜] E := LinearIsometryEquiv.ulift 𝕜 E
  

Proof state:
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
D : Type uD
inst✝⁷ : NormedAddCommGroup D
inst✝⁶ : NormedSpace 𝕜 D
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s✝ s₁ t u : Set E
B : E →L[𝕜] F →L[𝕜] G
f : D → E
g : D → F
N : ℕ∞
s : Set D
x : D
hf : ContDiffOn 𝕜 N f s
hg : ContDiffOn 𝕜 N g s
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
hn : ↑n ≤ N
Du : Type (max uD uE uF uG) := ULift.{max uE uF uG, uD} D
Eu : Type (max uD uE uF uG) := ULift.{max uD uF uG, uE} E
Fu : Type (max uD uE uF uG) := ULift.{max uD uE uG, uF} F
Gu : Type (max uD uE uF uG) := ULift.{max uD uE uF, uG} G
isoD : Du ≃ₗᵢ[𝕜] D
isoE : Eu ≃ₗᵢ[𝕜] E
⊢ ‖iteratedFDerivWithin 𝕜 n (fun y => (B (f y)) (g y)) s x‖ ≤
    ‖B‖ *
      ∑ i in Finset.range (n + 1),
        ↑(Nat.choose n i) * ‖iteratedFDerivWithin 𝕜 i f s x‖ * ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖

Next tactic:
have isoF : Fu ≃ₗᵢ[𝕜] F := LinearIsometryEquiv.ulift 𝕜 F
set_option maxHeartbeats 700000 in -- 3.5× the default limit
/-- Bounding the norm of the iterated derivative of `B (f x) (g x)` within a set in terms of the
iterated derivatives of `f` and `g` when `B` is bilinear:
`‖D^n (x ↦ B (f x) (g x))‖ ≤ ‖B‖ ∑_{k ≤ n} n.choose k ‖D^k f‖ ‖D^{n-k} g‖` -/
theorem ContinuousLinearMap.norm_iteratedFDerivWithin_le_of_bilinear (B : E →L[𝕜] F →L[𝕜] G)
    {f : D → E} {g : D → F} {N : ℕ∞} {s : Set D} {x : D} (hf : ContDiffOn 𝕜 N f s)
    (hg : ContDiffOn 𝕜 N g s) (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) {n : ℕ} (hn : (n : ℕ∞) ≤ N) :
    ‖iteratedFDerivWithin 𝕜 n (fun y => B (f y) (g y)) s x‖ ≤
      ‖B‖ * ∑ i in Finset.range (n + 1), (n.choose i : ℝ) * ‖iteratedFDerivWithin 𝕜 i f s x‖ *
        ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖ := by
  /- We reduce the bound to the case where all spaces live in the same universe (in which we
    already have proved the result), by using linear isometries between the spaces and their `ULift`
    to a common universe. These linear isometries preserve the norm of the iterated derivative. -/
  let Du : Type max uD uE uF uG := ULift.{max uE uF uG, uD} D
  let Eu : Type max uD uE uF uG := ULift.{max uD uF uG, uE} E
  let Fu : Type max uD uE uF uG := ULift.{max uD uE uG, uF} F
  let Gu : Type max uD uE uF uG := ULift.{max uD uE uF, uG} G
  have isoD : Du ≃ₗᵢ[𝕜] D := LinearIsometryEquiv.ulift 𝕜 D
  have isoE : Eu ≃ₗᵢ[𝕜] E := LinearIsometryEquiv.ulift 𝕜 E
  have isoF : Fu ≃ₗᵢ[𝕜] F := LinearIsometryEquiv.ulift 𝕜 F
  

Proof state:
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
D : Type uD
inst✝⁷ : NormedAddCommGroup D
inst✝⁶ : NormedSpace 𝕜 D
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s✝ s₁ t u : Set E
B : E →L[𝕜] F →L[𝕜] G
f : D → E
g : D → F
N : ℕ∞
s : Set D
x : D
hf : ContDiffOn 𝕜 N f s
hg : ContDiffOn 𝕜 N g s
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
hn : ↑n ≤ N
Du : Type (max uD uE uF uG) := ULift.{max uE uF uG, uD} D
Eu : Type (max uD uE uF uG) := ULift.{max uD uF uG, uE} E
Fu : Type (max uD uE uF uG) := ULift.{max uD uE uG, uF} F
Gu : Type (max uD uE uF uG) := ULift.{max uD uE uF, uG} G
isoD : Du ≃ₗᵢ[𝕜] D
isoE : Eu ≃ₗᵢ[𝕜] E
isoF : Fu ≃ₗᵢ[𝕜] F
⊢ ‖iteratedFDerivWithin 𝕜 n (fun y => (B (f y)) (g y)) s x‖ ≤
    ‖B‖ *
      ∑ i in Finset.range (n + 1),
        ↑(Nat.choose n i) * ‖iteratedFDerivWithin 𝕜 i f s x‖ * ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖

Next tactic:
have isoG : Gu ≃ₗᵢ[𝕜] G := LinearIsometryEquiv.ulift 𝕜 G
set_option maxHeartbeats 700000 in -- 3.5× the default limit
/-- Bounding the norm of the iterated derivative of `B (f x) (g x)` within a set in terms of the
iterated derivatives of `f` and `g` when `B` is bilinear:
`‖D^n (x ↦ B (f x) (g x))‖ ≤ ‖B‖ ∑_{k ≤ n} n.choose k ‖D^k f‖ ‖D^{n-k} g‖` -/
theorem ContinuousLinearMap.norm_iteratedFDerivWithin_le_of_bilinear (B : E →L[𝕜] F →L[𝕜] G)
    {f : D → E} {g : D → F} {N : ℕ∞} {s : Set D} {x : D} (hf : ContDiffOn 𝕜 N f s)
    (hg : ContDiffOn 𝕜 N g s) (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) {n : ℕ} (hn : (n : ℕ∞) ≤ N) :
    ‖iteratedFDerivWithin 𝕜 n (fun y => B (f y) (g y)) s x‖ ≤
      ‖B‖ * ∑ i in Finset.range (n + 1), (n.choose i : ℝ) * ‖iteratedFDerivWithin 𝕜 i f s x‖ *
        ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖ := by
  /- We reduce the bound to the case where all spaces live in the same universe (in which we
    already have proved the result), by using linear isometries between the spaces and their `ULift`
    to a common universe. These linear isometries preserve the norm of the iterated derivative. -/
  let Du : Type max uD uE uF uG := ULift.{max uE uF uG, uD} D
  let Eu : Type max uD uE uF uG := ULift.{max uD uF uG, uE} E
  let Fu : Type max uD uE uF uG := ULift.{max uD uE uG, uF} F
  let Gu : Type max uD uE uF uG := ULift.{max uD uE uF, uG} G
  have isoD : Du ≃ₗᵢ[𝕜] D := LinearIsometryEquiv.ulift 𝕜 D
  have isoE : Eu ≃ₗᵢ[𝕜] E := LinearIsometryEquiv.ulift 𝕜 E
  have isoF : Fu ≃ₗᵢ[𝕜] F := LinearIsometryEquiv.ulift 𝕜 F
  have isoG : Gu ≃ₗᵢ[𝕜] G := LinearIsometryEquiv.ulift 𝕜 G
  -- lift `f` and `g` to versions `fu` and `gu` on the lifted spaces.
  

Proof state:
𝕜 : Type u_1
inst✝⁸ : NontriviallyNormedField 𝕜
D : Type uD
inst✝⁷ : NormedAddCommGroup D
inst✝⁶ : NormedSpace 𝕜 D
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
F : Type uF
inst✝³ : NormedAddCommGroup F
inst✝² : NormedSpace 𝕜 F
G : Type uG
inst✝¹ : NormedAddCommGroup G
inst✝ : NormedSpace 𝕜 G
s✝ s₁ t u : Set E
B : E →L[𝕜] F →L[𝕜] G
f : D → E
g : D → F
N : ℕ∞
s : Set D
x : D
hf : ContDiffOn 𝕜 N f s
hg : ContDiffOn 𝕜 N g s
hs : UniqueDiffOn 𝕜 s
hx : x ∈ s
n : ℕ
hn : ↑n ≤ N
Du : Type (max uD uE uF uG) := ULift.{max uE uF uG, uD} D
Eu : Type (max uD uE uF uG) := ULift.{max uD uF uG, uE} E
Fu : Type (max uD uE uF uG) := ULift.{max uD uE uG, uF} F
Gu : Type (max uD uE uF uG) := ULift.{max uD uE uF, uG} G
isoD : Du ≃ₗᵢ[𝕜] D
isoE : Eu ≃ₗᵢ[𝕜] E
isoF : Fu ≃ₗᵢ[𝕜] F
isoG : Gu ≃ₗᵢ[𝕜] G
⊢ ‖iteratedFDerivWithin 𝕜 n (fun y => (B (f y)) (g y)) s x‖ ≤
    ‖B‖ *
      ∑ i in Finset.range (n + 1),
        ↑(Nat.choose n i) * ‖iteratedFDerivWithin 𝕜 i f s x‖ * ‖iteratedFDerivWithin 𝕜 (n - i) g s x‖

Next tactic:
set fu : Du → Eu := isoE.symm ∘ f ∘ isoD with hfu