@[simp]
theorem listDecode_encode_list (l : List (Σn, L.BoundedFormula α n)) :
    (listDecode (l.bind fun φ => φ.2.listEncode)).1 = l.headI := by
  suffices h : ∀ (φ : Σn, L.BoundedFormula α n) (l),
      (listDecode (listEncode φ.2 ++ l)).1 = φ ∧ (listDecode (listEncode φ.2 ++ l)).2.1 = l
  · induction' l with φ l _
    · rw [List.nil_bind]
      simp [listDecode]
    · rw [cons_bind, (h φ _).1, headI_cons]
  · rintro ⟨n, φ⟩
    induction' φ with _ _ _ _ φ_n φ_l φ_R ts _ _ _ ih1 ih2 _ _ ih <;> intro l
    · rw [listEncode, singleton_append, listDecode]
      simp only [eq_self_iff_true, heq_iff_eq, and_self_iff]
    · rw [listEncode, cons_append, cons_append, listDecode, dif_pos]
      · simp only [eq_mp_eq_cast, cast_eq, eq_self_iff_true, heq_iff_eq, and_self_iff, nil_append]
      · simp only [eq_self_iff_true, heq_iff_eq, and_self_iff]
    · rw [listEncode, cons_append, cons_append, singleton_append, cons_append, listDecode]
      · have h : ∀ i : Fin φ_l, ((List.map Sum.getLeft? (List.map (fun i : Fin φ_l =>
          Sum.inl (⟨(⟨φ_n, rel φ_R ts⟩ : Σn, L.BoundedFormula α n).fst, ts i⟩ :
            Σn, L.Term (Sum α (Fin n)))) (finRange φ_l) ++ l)).get? ↑i).join = some ⟨_, ts i⟩ := by
          intro i
          simp only [Option.join, map_append, map_map, Option.bind_eq_some, id.def, exists_eq_right,
            get?_eq_some, length_append, length_map, length_finRange]
          refine' ⟨lt_of_lt_of_le i.2 le_self_add, _⟩
          rw [get_append, get_map]
          · simp only [Sum.getLeft?, get_finRange, Fin.eta, Function.comp_apply, eq_self_iff_true,
              heq_iff_eq, and_self_iff]
          · simp only [length_map, length_finRange, is_lt]
        rw [dif_pos]
        swap
        · exact fun i => Option.isSome_iff_exists.2 ⟨⟨_, ts i⟩, h i⟩
        rw [dif_pos]
        swap
        · intro i
          obtain ⟨h1, h2⟩ := Option.eq_some_iff_get_eq.1 (h i)
          rw [h2]
        simp only [Sigma.mk.inj_iff, heq_eq_eq, rel.injEq, true_and]
        refine' ⟨funext fun i => _, _⟩
        · obtain ⟨h1, h2⟩ := Option.eq_some_iff_get_eq.1 (h i)
          rw [eq_mp_eq_cast, cast_eq_iff_heq]
          exact (Sigma.ext_iff.1 ((Sigma.eta (Option.get _ h1)).trans h2)).2
        rw [List.drop_append_eq_append_drop, length_map, length_finRange, Nat.sub_self, drop,
          drop_eq_nil_of_le, nil_append]
        rw [length_map, length_finRange]
    · rw [listEncode, List.append_assoc, cons_append, listDecode]
      simp only [] at *
      

Proof state:
case h.mk.imp
L : Language
M : Type w
N : Type u_1
P : Type u_2
inst✝² : Structure L M
inst✝¹ : Structure L N
inst✝ : Structure L P
α : Type u'
β : Type v'
l✝ : List ((n : ℕ) × BoundedFormula L α n)
n n✝ : ℕ
f₁✝ f₂✝ : BoundedFormula L α n✝
ih1 :
  ∀ (l : List ((k : ℕ) × Term L (α ⊕ Fin k) ⊕ (n : ℕ) × Relations L n ⊕ ℕ)),
    (listDecode (listEncode f₁✝ ++ l)).1 = { fst := n✝, snd := f₁✝ } ∧ ↑(listDecode (listEncode f₁✝ ++ l)).2 = l
ih2 :
  ∀ (l : List ((k : ℕ) × Term L (α ⊕ Fin k) ⊕ (n : ℕ) × Relations L n ⊕ ℕ)),
    (listDecode (listEncode f₂✝ ++ l)).1 = { fst := n✝, snd := f₂✝ } ∧ ↑(listDecode (listEncode f₂✝ ++ l)).2 = l
l : List ((k : ℕ) × Term L (α ⊕ Fin k) ⊕ (n : ℕ) × Relations L n ⊕ ℕ)
⊢ sigmaImp (listDecode (listEncode f₁✝ ++ (listEncode f₂✝ ++ l))).1
        (listDecode ↑(listDecode (listEncode f₁✝ ++ (listEncode f₂✝ ++ l))).2).1 =
      { fst := n✝, snd := f₁✝ ⟹ f₂✝ } ∧
    ↑(listDecode ↑(listDecode (listEncode f₁✝ ++ (listEncode f₂✝ ++ l))).2).2 = l

Next tactic:
rw [(ih1 _).1, (ih1 _).2, (ih2 _).1, (ih2 _).2, sigmaImp]