-- Module: Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno
Goal:
𝕜 : Type u_1
inst : NontriviallyNormedField 𝕜
E : Type u_2
inst_1 : NormedAddCommGroup E
inst_2 : NormedSpace 𝕜 E
F : Type u_3
inst_3 : NormedAddCommGroup F
inst_4 : NormedSpace 𝕜 F
G : Type u_4
inst_5 : NormedAddCommGroup G
inst_6 : NormedSpace 𝕜 G
s : Set E
t : Set F
q : F → FormalMultilinearSeries 𝕜 F G
p : E → FormalMultilinearSeries 𝕜 E F
n_1 : ℕ
c_1 : OrderedFinpartition n_1
n : ℕ
c : OrderedFinpartition (n + 1)
h : ¬range (ι := Fin (c.partSize 0)) (c.emb 0) = {0}
B : c.partSize (c.index 0) - 1 + 1 = c.partSize (c.index 0)
i : Fin c.length
j : Fin (c.partSize i)
neg :
  ∀ (j : Fin (c.partSize (c.index 0))),
    c.emb (c.index 0) j =
      (if h_1 : c.index 0 = c.index 0 then
          (fun
                (i :
                  Fin
                    (update (β := fun (a : Fin c.length) ↦ ℕ) c.partSize (c.index 0) (c.partSize (c.index 0) - 1)
                        (c.index 0) +
                      1)) ↦
              cases (motive :=
                fun
                  (x :
                    Fin
                      (update (β := fun (a : Fin c.length) ↦ ℕ) c.partSize (c.index 0) (c.partSize (c.index 0) - 1)
                          (c.index 0) +
                        1)) ↦
                Fin (n + 1)) 0
                (succ ∘
                  fun
                    (j :
                      Fin
                        (update (β := fun (a : Fin c.length) ↦ ℕ) c.partSize (c.index 0) (c.partSize (c.index 0) - 1)
                          (c.index 0))) ↦
                  (c.emb (c.index 0) (Fin.cast ⋯ j.succ)).pred ⋯)
                i) ∘
            Fin.cast ⋯
        else
          succ ∘
            (fun
                  (j :
                    Fin
                      (update (β := fun (a : Fin c.length) ↦ ℕ) c.partSize (c.index 0) (c.partSize (c.index 0) - 1)
                        (c.index 0))) ↦
                if h : c.index 0 = c.index 0 then (c.emb (c.index 0) (Fin.cast ⋯ j.succ)).pred ⋯
                else (c.emb (c.index 0) (Fin.cast ⋯ j)).pred ⋯) ∘
              Fin.cast ⋯)
        ⟨(↑j : ℕ), ⋯⟩
neg₁ :
  ∀ (hi : i ≠ c.index 0),
    c.emb i j =
      (if h_1 : i = c.index 0 then
          (fun
                (i :
                  Fin
                    (update (β := fun (a : Fin c.length) ↦ ℕ) c.partSize (c.index 0) (c.partSize (c.index 0) - 1)
                        (c.index 0) +
                      1)) ↦
              cases (motive :=
                fun
                  (x :
                    Fin
                      (update (β := fun (a : Fin c.length) ↦ ℕ) c.partSize (c.index 0) (c.partSize (c.index 0) - 1)
                          (c.index 0) +
                        1)) ↦
                Fin (n + 1)) 0
                (succ ∘
                  fun
                    (j :
                      Fin
                        (update (β := fun (a : Fin c.length) ↦ ℕ) c.partSize (c.index 0) (c.partSize (c.index 0) - 1)
                          (c.index 0))) ↦
                  (c.emb (c.index 0) (Fin.cast ⋯ j.succ)).pred ⋯)
                i) ∘
            Fin.cast ⋯
        else
          succ ∘
            (fun
                  (j :
                    Fin
                      (update (β := fun (a : Fin c.length) ↦ ℕ) c.partSize (c.index 0) (c.partSize (c.index 0) - 1)
                        i)) ↦
                if h : i = c.index 0 then (c.emb i (Fin.cast ⋯ j.succ)).pred ⋯ else (c.emb i (Fin.cast ⋯ j)).pred ⋯) ∘
              Fin.cast ⋯)
        ⟨(↑j : ℕ), ⋯⟩
⊢ c.emb i j =
    (if h_1 : i = c.index 0 then
        (fun
              (i :
                Fin
                  (update (β := fun (a : Fin c.length) ↦ ℕ) c.partSize (c.index 0) (c.partSize (c.index 0) - 1)
                      (c.index 0) +
                    1)) ↦
            cases (motive :=
              fun
                (x :
                  Fin
                    (update (β := fun (a : Fin c.length) ↦ ℕ) c.partSize (c.index 0) (c.partSize (c.index 0) - 1)
                        (c.index 0) +
                      1)) ↦
              Fin (n + 1)) 0
              (succ ∘
                fun
                  (j :
                    Fin
                      (update (β := fun (a : Fin c.length) ↦ ℕ) c.partSize (c.index 0) (c.partSize (c.index 0) - 1)
                        (c.index 0))) ↦
                (c.emb (c.index 0) (Fin.cast ⋯ j.succ)).pred ⋯)
              i) ∘
          Fin.cast ⋯
      else
        succ ∘
          (fun
                (j :
                  Fin
                    (update (β := fun (a : Fin c.length) ↦ ℕ) c.partSize (c.index 0) (c.partSize (c.index 0) - 1) i)) ↦
              if h : i = c.index 0 then (c.emb i (Fin.cast ⋯ j.succ)).pred ⋯ else (c.emb i (Fin.cast ⋯ j)).pred ⋯) ∘
            Fin.cast ⋯)
      ⟨(↑j : ℕ), ⋯⟩

Tactic:
rcases eq_or_ne i (c.index 0) with rfl | hi

Premises used: OrderedFinpartition.index, eq_or_ne, OfNat.ofNat, rfl