-- 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}
A : c.length - 1 + 1 = c.length
pos :
  ∀ (i : Fin (c.length - 1 + 1)),
    partSize (n := n + 1)
        { length := c.length - 1 + 1,
          partSize :=
            cons (α := fun (a : Fin (c.length - 1 + 1)) ↦ ℕ) 1 fun (i : Fin (c.length - 1)) ↦
              c.partSize (Fin.cast ⋯ i.succ),
          partSize_pos := ⋯,
          emb := fun (i : Fin (c.length - 1 + 1)) ↦
            cases (motive := fun (x : Fin (c.length - 1 + 1)) ↦
              Fin
                  (cons (α := fun (a : Fin (c.length - 1 + 1)) ↦ ℕ) 1
                    (fun (i : Fin (c.length - 1)) ↦ c.partSize (Fin.cast ⋯ i.succ)) x) →
                Fin (n + 1))
              (fun
                  (x :
                    Fin
                      (cons (α := fun (a : Fin (c.length - 1 + 1)) ↦ ℕ) 1
                        (fun (i : Fin (c.length - 1)) ↦ c.partSize (Fin.cast ⋯ i.succ)) 0)) ↦
                0)
              (fun (m : Fin (c.length - 1)) ↦
                succ ∘ fun (j : Fin (c.partSize (Fin.cast ⋯ m.succ))) ↦ (c.emb (Fin.cast ⋯ m.succ) j).pred ⋯)
              i,
          emb_strictMono := ⋯, parts_strictMono := ⋯, disjoint := ⋯, cover := ⋯ : OrderedFinpartition (n + 1) }
        i =
      c.partSize ⟨(↑i : ℕ), ⋯⟩
⊢ HEq (α :=
    Fin
        (length (n := n + 1)
          { length := c.length - 1 + 1,
            partSize :=
              cons (α := fun (a : Fin (c.length - 1 + 1)) ↦ ℕ) 1 fun (i : Fin (c.length - 1)) ↦
                c.partSize (Fin.cast ⋯ i.succ),
            partSize_pos := ⋯,
            emb := fun (i : Fin (c.length - 1 + 1)) ↦
              cases (motive := fun (x : Fin (c.length - 1 + 1)) ↦
                Fin
                    (cons (α := fun (a : Fin (c.length - 1 + 1)) ↦ ℕ) 1
                      (fun (i : Fin (c.length - 1)) ↦ c.partSize (Fin.cast ⋯ i.succ)) x) →
                  Fin (n + 1))
                (fun
                    (x :
                      Fin
                        (cons (α := fun (a : Fin (c.length - 1 + 1)) ↦ ℕ) 1
                          (fun (i : Fin (c.length - 1)) ↦ c.partSize (Fin.cast ⋯ i.succ)) 0)) ↦
                  0)
                (fun (m : Fin (c.length - 1)) ↦
                  succ ∘ fun (j : Fin (c.partSize (Fin.cast ⋯ m.succ))) ↦ (c.emb (Fin.cast ⋯ m.succ) j).pred ⋯)
                i,
            emb_strictMono := ⋯, parts_strictMono := ⋯, disjoint := ⋯, cover := ⋯ }) →
      ℕ)
    (partSize
      { length := c.length - 1 + 1,
        partSize :=
          cons (α := fun (a : Fin (c.length - 1 + 1)) ↦ ℕ) 1 fun (i : Fin (c.length - 1)) ↦
            c.partSize (Fin.cast ⋯ i.succ),
        partSize_pos := ⋯,
        emb := fun (i : Fin (c.length - 1 + 1)) ↦
          cases (motive := fun (x : Fin (c.length - 1 + 1)) ↦
            Fin
                (cons (α := fun (a : Fin (c.length - 1 + 1)) ↦ ℕ) 1
                  (fun (i : Fin (c.length - 1)) ↦ c.partSize (Fin.cast ⋯ i.succ)) x) →
              Fin (n + 1))
            (fun
                (x :
                  Fin
                    (cons (α := fun (a : Fin (c.length - 1 + 1)) ↦ ℕ) 1
                      (fun (i : Fin (c.length - 1)) ↦ c.partSize (Fin.cast ⋯ i.succ)) 0)) ↦
              0)
            (fun (m : Fin (c.length - 1)) ↦
              succ ∘ fun (j : Fin (c.partSize (Fin.cast ⋯ m.succ))) ↦ (c.emb (Fin.cast ⋯ m.succ) j).pred ⋯)
            i,
        emb_strictMono := ⋯, parts_strictMono := ⋯, disjoint := ⋯, cover := ⋯ })
    c.partSize

Tactic:
refine (Fin.heq_fun_iff A).mpr (fun i ↦ ?_)

Premises used: Iff.mpr, Fin.heq_fun_iff