Theorem: Regex.VM.lower_inv_exploreεClosure
Proof state before:
case inl
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 : LowerInvExploreεClosure nfa next target stack
h✝¹ : target ∉ next
offset✝ target' : ℕ
hn : nfa[↑target] = Node.save offset✝ target'
isLt : (Node.save offset✝ target').inBounds nfa.nodes.size
h✝ : offset✝ < currentSave.size
h :
  exploreεClosure nfa pos (next.insert target) (currentSave.set ⟨offset✝, ⋯⟩ (some pos)) matched saveSlots ⟨target', ⋯⟩
      (stack.push (StackEntry.restore currentSave)) =
    (matched', next', saveSlots')
⊢ LowerInvεClosure nfa next' #[]

case inr
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 : LowerInvExploreεClosure nfa next target stack
h✝¹ : target ∉ next
offset✝ target' : ℕ
hn : nfa[↑target] = Node.save offset✝ target'
isLt : (Node.save offset✝ target').inBounds nfa.nodes.size
h✝ : ¬offset✝ < currentSave.size
h :
  exploreεClosure nfa pos (next.insert target) currentSave matched saveSlots ⟨target', ⋯⟩ stack =
    (matched', next', saveSlots')
⊢ LowerInvεClosure nfa next' #[]
Tactic:
next =>
  have inv' : LowerInvExploreεClosure nfa (next.insert target) ⟨target', isLt⟩ (stack.push (.restore currentSave)) := by
    intro i j hi hj
    cases SparseSet.eq_or_mem_of_mem_insert hi with
    | inl htarget =>
      subst htarget
      simp [εStep, Node.εStep, hn] at hj
      exact .inr (.inl (Fin.eq_of_val_eq hj))
    | inr hnext =>
      match inv i j hnext hj with
      | .inl hnext => exact .inl (SparseSet.mem_insert_of_mem hnext)
      | .inr (.inl htarget) => exact .inl (htarget ▸ SparseSet.mem_insert)
      | .inr (.inr hstack) => exact .inr (.inr ((Array.mem_push ..).mpr (.inl hstack)))
  exact lower_inv_exploreεClosure h inv'
Proof state after:
case inr
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 : LowerInvExploreεClosure nfa next target stack
h✝¹ : target ∉ next
offset✝ target' : ℕ
hn : nfa[↑target] = Node.save offset✝ target'
isLt : (Node.save offset✝ target').inBounds nfa.nodes.size
h✝ : ¬offset✝ < currentSave.size
h :
  exploreεClosure nfa pos (next.insert target) currentSave matched saveSlots ⟨target', ⋯⟩ stack =
    (matched', next', saveSlots')
⊢ LowerInvεClosure nfa next' #[]