next hc =>
  simp at h
  have mem_next_iff := exploreεClosure_spec.mem_next_iff h inv
  intro j
  apply Iff.intro
  . intro hj
    cases (mem_next_iff j).mp hj with
    | inl hj => exact .inl hj
    | inr hj =>
      refine .inr ⟨target', ?_, hj⟩
      simp [charStep, Node.charStep, hn, hc]
  . intro hj
    cases hj with
    | inl hj => exact (mem_next_iff j).mpr (.inl hj)
    | inr hj =>
      simp [charStep, Node.charStep, hn, hc] at hj
      exact (mem_next_iff j).mpr (.inr hj)
case inl
nfa : NFA
c : Char
pos : String.Pos
next : SparseSet nfa.nodes.size
saveSlots : Data.Vec (Array (Option String.Pos)) nfa.nodes.size
target : Fin nfa.nodes.size
matched' : Option (Array (Option String.Pos))
next' : SparseSet nfa.nodes.size
saveSlots' : Data.Vec (Array (Option String.Pos)) nfa.nodes.size
inv : ∀ (i j : Fin nfa.nodes.size), i ∈ next → ↑j ∈ nfa.εStep ↑i → j ∈ next
cs : Data.Classes
target' : ℕ
hn : nfa[↑target] = Node.sparse cs target'
h✝ : c ∈ cs
h :
  (let_fun isLt := ⋯;
    let currentSave := saveSlots.get ↑target ⋯;
    exploreεClosure nfa pos next currentSave none saveSlots ⟨target', isLt⟩ #[]) =
    (matched', next', saveSlots')
⊢ ∀ (j : Fin nfa.nodes.size), j ∈ next' ↔ j ∈ next ∨ ∃ i ∈ nfa.charStep (↑target) c, ↑j ∈ nfa.εClosure i

case inr
nfa : NFA
c : Char
pos : String.Pos
next : SparseSet nfa.nodes.size
saveSlots : Data.Vec (Array (Option String.Pos)) nfa.nodes.size
target : Fin nfa.nodes.size
matched' : Option (Array (Option String.Pos))
next' : SparseSet nfa.nodes.size
saveSlots' : Data.Vec (Array (Option String.Pos)) nfa.nodes.size
inv : ∀ (i j : Fin nfa.nodes.size), i ∈ next → ↑j ∈ nfa.εStep ↑i → j ∈ next
cs : Data.Classes
target' : ℕ
hn : nfa[↑target] = Node.sparse cs target'
h✝ : c ∉ cs
h : (none, next, saveSlots) = (matched', next', saveSlots')
⊢ ∀ (j : Fin nfa.nodes.size), j ∈ next' ↔ j ∈ next ∨ ∃ i ∈ nfa.charStep (↑target) c, ↑j ∈ nfa.εClosure i
case inr
nfa : NFA
c : Char
pos : String.Pos
next : SparseSet nfa.nodes.size
saveSlots : Data.Vec (Array (Option String.Pos)) nfa.nodes.size
target : Fin nfa.nodes.size
matched' : Option (Array (Option String.Pos))
next' : SparseSet nfa.nodes.size
saveSlots' : Data.Vec (Array (Option String.Pos)) nfa.nodes.size
inv : ∀ (i j : Fin nfa.nodes.size), i ∈ next → ↑j ∈ nfa.εStep ↑i → j ∈ next
cs : Data.Classes
target' : ℕ
hn : nfa[↑target] = Node.sparse cs target'
h✝ : c ∉ cs
h : (none, next, saveSlots) = (matched', next', saveSlots')
⊢ ∀ (j : Fin nfa.nodes.size), j ∈ next' ↔ j ∈ next ∨ ∃ i ∈ nfa.charStep (↑target) c, ↑j ∈ nfa.εClosure i