-- Mathlib/Analysis/Calculus/ContDiff/Bounds.lean
theorem norm_iteratedFDerivWithin_prod_le [DecidableEq ι] [NormOneClass A'] {u : Finset ι}
    {f : ι → E → A'} {N : WithTop ℕ∞} (hf : ∀ i ∈ u, ContDiffOn 𝕜 N (f i) s)
    (hs : UniqueDiffOn 𝕜 s) {x : E} (hx : x ∈ s) {n : ℕ} (hn : n ≤ N) :
    ‖iteratedFDerivWithin 𝕜 n (∏ j ∈ u, f j ·) s x‖ ≤
      ∑ p ∈ u.sym n, (p : Multiset ι).multinomial *
        ∏ j ∈ u, ‖iteratedFDerivWithin 𝕜 (Multiset.count j p) (f j) s x‖

Proof state:
case insert
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
s : Set E
ι : Type u_2
A' : Type u_4
inst✝³ : NormedCommRing A'
inst✝² : NormedAlgebra 𝕜 A'
inst✝¹ : DecidableEq ι
inst✝ : NormOneClass A'
f : ι → E → A'
N : WithTop ℕ∞
hs : UniqueDiffOn 𝕜 s
x : E
hx : x ∈ s
i : ι
u : Finset ι
hi : i ∉ u
n : ℕ
hn : ↑n ≤ N
hf : ContDiffOn 𝕜 N (f i) s ∧ ∀ a ∈ u, ContDiffOn 𝕜 N (f a) s
m : Fin (n + 1)
a✝ : m ∈ Finset.univ
IH :
  ‖iteratedFDerivWithin 𝕜 (n - ↑m) (fun x => ∏ j ∈ u, f j x) s x‖ ≤
    ∑ p ∈ u.sym (n - ↑m), ↑(↑p).multinomial * ∏ j ∈ u, ‖iteratedFDerivWithin 𝕜 (Multiset.count j ↑p) (f j) s x‖
p : Sym ι (n - ↑m)
hp : ∀ a ∈ p, a ∈ u
hip : i ∉ p
⊢ ↑(n.choose ↑m) * ‖iteratedFDerivWithin 𝕜 (↑m) (f i) s x‖ *
      (↑(↑p).multinomial * ∏ x_1 ∈ u, ‖iteratedFDerivWithin 𝕜 (Multiset.count x_1 ↑p) (f x_1) s x‖) =
    ↑(↑(Sym.fill i m p)).multinomial *
      (‖iteratedFDerivWithin 𝕜 (Multiset.count i ↑(Sym.fill i m p)) (f i) s x‖ *
        ∏ x_1 ∈ u, ‖iteratedFDerivWithin 𝕜 (Multiset.count x_1 ↑(Sym.fill i m p)) (f x_1) s x‖)

Tactic:
rw [Sym.count_coe_fill_self_of_not_mem hip, Sym.multinomial_coe_fill_of_not_mem hip]

Result:
case insert
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
s : Set E
ι : Type u_2
A' : Type u_4
inst✝³ : NormedCommRing A'
inst✝² : NormedAlgebra 𝕜 A'
inst✝¹ : DecidableEq ι
inst✝ : NormOneClass A'
f : ι → E → A'
N : WithTop ℕ∞
hs : UniqueDiffOn 𝕜 s
x : E
hx : x ∈ s
i : ι
u : Finset ι
hi : i ∉ u
n : ℕ
hn : ↑n ≤ N
hf : ContDiffOn 𝕜 N (f i) s ∧ ∀ a ∈ u, ContDiffOn 𝕜 N (f a) s
m : Fin (n + 1)
a✝ : m ∈ Finset.univ
IH :
  ‖iteratedFDerivWithin 𝕜 (n - ↑m) (fun x => ∏ j ∈ u, f j x) s x‖ ≤
    ∑ p ∈ u.sym (n - ↑m), ↑(↑p).multinomial * ∏ j ∈ u, ‖iteratedFDerivWithin 𝕜 (Multiset.count j ↑p) (f j) s x‖
p : Sym ι (n - ↑m)
hp : ∀ a ∈ p, a ∈ u
hip : i ∉ p
⊢ ↑(n.choose ↑m) * ‖iteratedFDerivWithin 𝕜 (↑m) (f i) s x‖ *
      (↑(↑p).multinomial * ∏ x_1 ∈ u, ‖iteratedFDerivWithin 𝕜 (Multiset.count x_1 ↑p) (f x_1) s x‖) =
    ↑(n.choose ↑m * (↑p).multinomial) *
      (‖iteratedFDerivWithin 𝕜 (↑m) (f i) s x‖ *
        ∏ x_1 ∈ u, ‖iteratedFDerivWithin 𝕜 (Multiset.count x_1 ↑(Sym.fill i m p)) (f x_1) s x‖)

Proof state:
case insert
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
s : Set E
ι : Type u_2
A' : Type u_4
inst✝³ : NormedCommRing A'
inst✝² : NormedAlgebra 𝕜 A'
inst✝¹ : DecidableEq ι
inst✝ : NormOneClass A'
f : ι → E → A'
N : WithTop ℕ∞
hs : UniqueDiffOn 𝕜 s
x : E
hx : x ∈ s
i : ι
u : Finset ι
hi : i ∉ u
IH :
  (∀ i ∈ u, ContDiffOn 𝕜 N (f i) s) →
    ∀ {n : ℕ},
      ↑n ≤ N →
        ‖iteratedFDerivWithin 𝕜 n (fun x => ∏ j ∈ u, f j x) s x‖ ≤
          ∑ p ∈ u.sym n, ↑(↑p).multinomial * ∏ j ∈ u, ‖iteratedFDerivWithin 𝕜 (Multiset.count j ↑p) (f j) s x‖
n : ℕ
hn : ↑n ≤ N
hf : ContDiffOn 𝕜 N (f i) s ∧ ∀ a ∈ u, ContDiffOn 𝕜 N (f a) s
⊢ ∑ i_1 : Fin (n + 1),
      ↑(n.choose ↑i_1) * ‖iteratedFDerivWithin 𝕜 (↑i_1) (f i) s x‖ *
        ‖iteratedFDerivWithin 𝕜 (n - ↑i_1) (fun y => ∏ i ∈ u, f i y) s x‖ ≤
    ∑ x_1 : Fin (n + 1),
      ∑ x_2 : { x // x ∈ u.sym (n - ↑x_1) },
        ↑(↑(Sym.fill i x_1 ↑x_2)).multinomial *
          ∏ x_3 ∈ insert i u, ‖iteratedFDerivWithin 𝕜 (Multiset.count x_3 ↑(Sym.fill i x_1 ↑x_2)) (f x_3) s x‖

Tactic:
refine Finset.sum_le_sum ?_

Result:
case insert
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
s : Set E
ι : Type u_2
A' : Type u_4
inst✝³ : NormedCommRing A'
inst✝² : NormedAlgebra 𝕜 A'
inst✝¹ : DecidableEq ι
inst✝ : NormOneClass A'
f : ι → E → A'
N : WithTop ℕ∞
hs : UniqueDiffOn 𝕜 s
x : E
hx : x ∈ s
i : ι
u : Finset ι
hi : i ∉ u
IH :
  (∀ i ∈ u, ContDiffOn 𝕜 N (f i) s) →
    ∀ {n : ℕ},
      ↑n ≤ N →
        ‖iteratedFDerivWithin 𝕜 n (fun x => ∏ j ∈ u, f j x) s x‖ ≤
          ∑ p ∈ u.sym n, ↑(↑p).multinomial * ∏ j ∈ u, ‖iteratedFDerivWithin 𝕜 (Multiset.count j ↑p) (f j) s x‖
n : ℕ
hn : ↑n ≤ N
hf : ContDiffOn 𝕜 N (f i) s ∧ ∀ a ∈ u, ContDiffOn 𝕜 N (f a) s
⊢ ∀ i_1 ∈ Finset.univ,
    ↑(n.choose ↑i_1) * ‖iteratedFDerivWithin 𝕜 (↑i_1) (f i) s x‖ *
        ‖iteratedFDerivWithin 𝕜 (n - ↑i_1) (fun y => ∏ i ∈ u, f i y) s x‖ ≤
      ∑ x_1 : { x // x ∈ u.sym (n - ↑i_1) },
        ↑(↑(Sym.fill i i_1 ↑x_1)).multinomial *
          ∏ x_2 ∈ insert i u, ‖iteratedFDerivWithin 𝕜 (Multiset.count x_2 ↑(Sym.fill i i_1 ↑x_1)) (f x_2) s x‖

Proof state:
case insert
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
s : Set E
ι : Type u_2
A' : Type u_4
inst✝³ : NormedCommRing A'
inst✝² : NormedAlgebra 𝕜 A'
inst✝¹ : DecidableEq ι
inst✝ : NormOneClass A'
f : ι → E → A'
N : WithTop ℕ∞
hs : UniqueDiffOn 𝕜 s
x : E
hx : x ∈ s
i : ι
u : Finset ι
hi : i ∉ u
n : ℕ
hn : ↑n ≤ N
hf : ContDiffOn 𝕜 N (f i) s ∧ ∀ a ∈ u, ContDiffOn 𝕜 N (f a) s
m : Fin (n + 1)
a✝ : m ∈ Finset.univ
IH :
  ‖iteratedFDerivWithin 𝕜 (n - ↑m) (fun x => ∏ j ∈ u, f j x) s x‖ ≤
    ∑ p ∈ u.sym (n - ↑m), ↑(↑p).multinomial * ∏ j ∈ u, ‖iteratedFDerivWithin 𝕜 (Multiset.count j ↑p) (f j) s x‖
p : Sym ι (n - ↑m)
hp : ∀ a ∈ p, a ∈ u
hip : i ∉ p
⊢ ∀ x_1 ∈ u,
    ‖iteratedFDerivWithin 𝕜 (Multiset.count x_1 ↑p) (f x_1) s x‖ =
      ‖iteratedFDerivWithin 𝕜 (Multiset.count x_1 ↑(Sym.fill i m p)) (f x_1) s x‖

Tactic:
intro j hj

Result:
case insert
𝕜 : Type u_1
inst✝⁶ : NontriviallyNormedField 𝕜
E : Type uE
inst✝⁵ : NormedAddCommGroup E
inst✝⁴ : NormedSpace 𝕜 E
s : Set E
ι : Type u_2
A' : Type u_4
inst✝³ : NormedCommRing A'
inst✝² : NormedAlgebra 𝕜 A'
inst✝¹ : DecidableEq ι
inst✝ : NormOneClass A'
f : ι → E → A'
N : WithTop ℕ∞
hs : UniqueDiffOn 𝕜 s
x : E
hx : x ∈ s
i : ι
u : Finset ι
hi : i ∉ u
n : ℕ
hn : ↑n ≤ N
hf : ContDiffOn 𝕜 N (f i) s ∧ ∀ a ∈ u, ContDiffOn 𝕜 N (f a) s
m : Fin (n + 1)
a✝ : m ∈ Finset.univ
IH :
  ‖iteratedFDerivWithin 𝕜 (n - ↑m) (fun x => ∏ j ∈ u, f j x) s x‖ ≤
    ∑ p ∈ u.sym (n - ↑m), ↑(↑p).multinomial * ∏ j ∈ u, ‖iteratedFDerivWithin 𝕜 (Multiset.count j ↑p) (f j) s x‖
p : Sym ι (n - ↑m)
hp : ∀ a ∈ p, a ∈ u
hip : i ∉ p
j : ι
hj : j ∈ u
⊢ ‖iteratedFDerivWithin 𝕜 (Multiset.count j ↑p) (f j) s x‖ =
    ‖iteratedFDerivWithin 𝕜 (Multiset.count j ↑(Sym.fill i m p)) (f j) s x‖