Theorem: Regex.VM.exploreεClosure_mem_done_iff
Proof state before:
case h_4
nfa : NFA
pos : String.Pos
next : SparseSet nfa.nodes.size
currentSave : Array (Option String.Pos)
matched : Option (Array (Option String.Pos))
saveSlots : Data.Vec (Array (Option String.Pos)) nfa.nodes.size
target : Fin nfa.nodes.size
stack : Array (StackEntry 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 ∈ next, nfa[i] = Node.done) ↔ matched.isSome = true
hmem : target ∉ next
inv' : nfa[target] ≠ Node.done → ((∃ i ∈ next.insert target, nfa[i] = Node.done) ↔ matched.isSome = true)
heq✝ : nfa[↑target] = Node.done
h :
  εClosure nfa pos (next.insert target) currentSave (HOrElse.hOrElse matched fun x => some currentSave)
      (saveSlots.set ↑target ⋯ currentSave) stack =
    (matched', next', saveSlots')
⊢ (∃ i ∈ next', nfa[i] = Node.done) ↔ matched'.isSome = true

case h_5
nfa : NFA
pos : String.Pos
next : SparseSet nfa.nodes.size
currentSave : Array (Option String.Pos)
matched : Option (Array (Option String.Pos))
saveSlots : Data.Vec (Array (Option String.Pos)) nfa.nodes.size
target : Fin nfa.nodes.size
stack : Array (StackEntry 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 ∈ next, nfa[i] = Node.done) ↔ matched.isSome = true
hmem : target ∉ next
inv' : nfa[target] ≠ Node.done → ((∃ i ∈ next.insert target, nfa[i] = Node.done) ↔ matched.isSome = true)
c✝ : Char
next✝ : ℕ
heq✝ : nfa[↑target] = Node.char c✝ next✝
h :
  εClosure nfa pos (next.insert target) currentSave matched (saveSlots.set ↑target ⋯ currentSave) stack =
    (matched', next', saveSlots')
⊢ (∃ i ∈ next', nfa[i] = Node.done) ↔ matched'.isSome = true

case h_6
nfa : NFA
pos : String.Pos
next : SparseSet nfa.nodes.size
currentSave : Array (Option String.Pos)
matched : Option (Array (Option String.Pos))
saveSlots : Data.Vec (Array (Option String.Pos)) nfa.nodes.size
target : Fin nfa.nodes.size
stack : Array (StackEntry 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 ∈ next, nfa[i] = Node.done) ↔ matched.isSome = true
hmem : target ∉ next
inv' : nfa[target] ≠ Node.done → ((∃ i ∈ next.insert target, nfa[i] = Node.done) ↔ matched.isSome = true)
cs✝ : Data.Classes
next✝ : ℕ
heq✝ : nfa[↑target] = Node.sparse cs✝ next✝
h :
  εClosure nfa pos (next.insert target) currentSave matched (saveSlots.set ↑target ⋯ currentSave) stack =
    (matched', next', saveSlots')
⊢ (∃ i ∈ next', nfa[i] = Node.done) ↔ matched'.isSome = true

case h_7
nfa : NFA
pos : String.Pos
next : SparseSet nfa.nodes.size
currentSave : Array (Option String.Pos)
matched : Option (Array (Option String.Pos))
saveSlots : Data.Vec (Array (Option String.Pos)) nfa.nodes.size
target : Fin nfa.nodes.size
stack : Array (StackEntry 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 ∈ next, nfa[i] = Node.done) ↔ matched.isSome = true
hmem : target ∉ next
inv' : nfa[target] ≠ Node.done → ((∃ i ∈ next.insert target, nfa[i] = Node.done) ↔ matched.isSome = true)
heq✝ : nfa[↑target] = Node.fail
h : εClosure nfa pos (next.insert target) currentSave matched saveSlots stack = (matched', next', saveSlots')
⊢ (∃ i ∈ next', nfa[i] = Node.done) ↔ matched'.isSome = true
Tactic:
next hn =>
  have lhs : ∃ i, i ∈ next.insert target ∧ nfa[i] = .done := ⟨target, SparseSet.mem_insert, hn⟩
  have rhs : (matched <|> currentSave).isSome := by
    cases matched <;> simp
  have inv'' : (∃ i, i ∈ next.insert target ∧ nfa[i] = .done) ↔ (matched <|> currentSave).isSome := by
    simp only [lhs, rhs]
  exact εClosure_mem_done_iff h inv''
Proof state after:
case h_5
nfa : NFA
pos : String.Pos
next : SparseSet nfa.nodes.size
currentSave : Array (Option String.Pos)
matched : Option (Array (Option String.Pos))
saveSlots : Data.Vec (Array (Option String.Pos)) nfa.nodes.size
target : Fin nfa.nodes.size
stack : Array (StackEntry 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 ∈ next, nfa[i] = Node.done) ↔ matched.isSome = true
hmem : target ∉ next
inv' : nfa[target] ≠ Node.done → ((∃ i ∈ next.insert target, nfa[i] = Node.done) ↔ matched.isSome = true)
c✝ : Char
next✝ : ℕ
heq✝ : nfa[↑target] = Node.char c✝ next✝
h :
  εClosure nfa pos (next.insert target) currentSave matched (saveSlots.set ↑target ⋯ currentSave) stack =
    (matched', next', saveSlots')
⊢ (∃ i ∈ next', nfa[i] = Node.done) ↔ matched'.isSome = true

case h_6
nfa : NFA
pos : String.Pos
next : SparseSet nfa.nodes.size
currentSave : Array (Option String.Pos)
matched : Option (Array (Option String.Pos))
saveSlots : Data.Vec (Array (Option String.Pos)) nfa.nodes.size
target : Fin nfa.nodes.size
stack : Array (StackEntry 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 ∈ next, nfa[i] = Node.done) ↔ matched.isSome = true
hmem : target ∉ next
inv' : nfa[target] ≠ Node.done → ((∃ i ∈ next.insert target, nfa[i] = Node.done) ↔ matched.isSome = true)
cs✝ : Data.Classes
next✝ : ℕ
heq✝ : nfa[↑target] = Node.sparse cs✝ next✝
h :
  εClosure nfa pos (next.insert target) currentSave matched (saveSlots.set ↑target ⋯ currentSave) stack =
    (matched', next', saveSlots')
⊢ (∃ i ∈ next', nfa[i] = Node.done) ↔ matched'.isSome = true

case h_7
nfa : NFA
pos : String.Pos
next : SparseSet nfa.nodes.size
currentSave : Array (Option String.Pos)
matched : Option (Array (Option String.Pos))
saveSlots : Data.Vec (Array (Option String.Pos)) nfa.nodes.size
target : Fin nfa.nodes.size
stack : Array (StackEntry 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 ∈ next, nfa[i] = Node.done) ↔ matched.isSome = true
hmem : target ∉ next
inv' : nfa[target] ≠ Node.done → ((∃ i ∈ next.insert target, nfa[i] = Node.done) ↔ matched.isSome = true)
heq✝ : nfa[↑target] = Node.fail
h : εClosure nfa pos (next.insert target) currentSave matched saveSlots stack = (matched', next', saveSlots')
⊢ (∃ i ∈ next', nfa[i] = Node.done) ↔ matched'.isSome = true