-- 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 : ℕ
c_1 c : OrderedFinpartition n
i : Fin (c.length + 1)
hi : Membership.mem (γ := Set (Fin (c.length + 1))) univ i
j : Fin (c.length + 1)
hj : Membership.mem (γ := Set (Fin (c.length + 1))) univ j
hij : i ≠ j
h : j < i
zero :
  ∀ (hi : Membership.mem (γ := Set (Fin (c.length + 1))) univ 0) (hij : 0 ≠ j) (h : j < 0),
    onFun (β := Set (Fin (n + 1))) Disjoint
      (fun (m : Fin (c.length + 1)) ↦
        range (ι := Fin (cons (α := fun (a : Fin (c.length + 1)) ↦ ℕ) 1 c.partSize m))
          (cases (motive := fun (x : Fin (c.length + 1)) ↦
            Fin (cons (α := fun (a : Fin (c.length + 1)) ↦ ℕ) 1 c.partSize x) → Fin (n + 1))
            (fun (x : Fin (cons (α := fun (a : Fin (c.length + 1)) ↦ ℕ) 1 c.partSize 0)) ↦ 0)
            (fun (m : Fin c.length) ↦ Fin.succ ∘ c.emb m) m))
      0 j
succ :
  ∀ (i : Fin c.length),
    (∀ (hi : Membership.mem (γ := Set (Fin (c.length + 1))) univ i.castSucc) (hij : i.castSucc ≠ j)
        (h : j < i.castSucc),
        onFun (β := Set (Fin (n + 1))) Disjoint
          (fun (m : Fin (c.length + 1)) ↦
            range (ι := Fin (cons (α := fun (a : Fin (c.length + 1)) ↦ ℕ) 1 c.partSize m))
              (cases (motive := fun (x : Fin (c.length + 1)) ↦
                Fin (cons (α := fun (a : Fin (c.length + 1)) ↦ ℕ) 1 c.partSize x) → Fin (n + 1))
                (fun (x : Fin (cons (α := fun (a : Fin (c.length + 1)) ↦ ℕ) 1 c.partSize 0)) ↦ 0)
                (fun (m : Fin c.length) ↦ Fin.succ ∘ c.emb m) m))
          i.castSucc j) →
      ∀ (hi : Membership.mem (γ := Set (Fin (c.length + 1))) univ i.succ) (hij : i.succ ≠ j) (h : j < i.succ),
        onFun (β := Set (Fin (n + 1))) Disjoint
          (fun (m : Fin (c.length + 1)) ↦
            range (ι := Fin (cons (α := fun (a : Fin (c.length + 1)) ↦ ℕ) 1 c.partSize m))
              (cases (motive := fun (x : Fin (c.length + 1)) ↦
                Fin (cons (α := fun (a : Fin (c.length + 1)) ↦ ℕ) 1 c.partSize x) → Fin (n + 1))
                (fun (x : Fin (cons (α := fun (a : Fin (c.length + 1)) ↦ ℕ) 1 c.partSize 0)) ↦ 0)
                (fun (m : Fin c.length) ↦ Fin.succ ∘ c.emb m) m))
          i.succ j
⊢ onFun (β := Set (Fin (n + 1))) Disjoint
    (fun (m : Fin (c.length + 1)) ↦
      range (ι := Fin (cons (α := fun (a : Fin (c.length + 1)) ↦ ℕ) 1 c.partSize m))
        (cases (motive := fun (x : Fin (c.length + 1)) ↦
          Fin (cons (α := fun (a : Fin (c.length + 1)) ↦ ℕ) 1 c.partSize x) → Fin (n + 1))
          (fun (x : Fin (cons (α := fun (a : Fin (c.length + 1)) ↦ ℕ) 1 c.partSize 0)) ↦ 0)
          (fun (m : Fin c.length) ↦ Fin.succ ∘ c.emb m) m))
    i j

Tactic:
induction i using Fin.induction with
    | zero => simp at h
    | succ i =>
      induction j using Fin.induction with
      | zero =>
        simp only [onFun, cases_succ, cases_zero]
        apply Set.disjoint_iff_forall_ne.2
        simp only [mem_range, comp_apply, exists_prop', cons_zero, ne_eq, and_imp,
          Nonempty.forall, forall_const, forall_eq', forall_exists_index, forall_apply_eq_imp_iff]
        exact fun _ ↦ succ_ne_zero _
      | succ j =>
        simp only [onFun, cases_succ]
        apply Set.disjoint_iff_forall_ne.2
        simp only [mem_range, comp_apply, ne_eq, forall_exists_index, forall_apply_eq_imp_iff,
          succ_inj]
        intro a b
        apply c.emb_ne_emb_of_ne (by simpa using hij)

Premises used: Iff.mpr, Set.mem_range, Function.onFun, Fin.succ, Nonempty.forall, and_imp, Fin.cases_zero, forall_exists_index, forall_eq', Fin.succ_inj, ne_eq, Set.disjoint_iff_forall_ne, Fin.succ_ne_zero, OrderedFinpartition.emb_ne_emb_of_ne, forall_const, Function.comp_apply, Fin.cases_succ, forall_apply_eq_imp_iff, Fin.cons_zero, exists_prop'