Theorem: Regex.VM.lower_inv_exploreεClosure
Proof state before:
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✝ : target ∉ next
h :
  (match hn : nfa[↑target] with
    | Node.epsilon target' =>
      exploreεClosure nfa pos (next.insert target) currentSave matched saveSlots ⟨target', ⋯⟩ stack
    | Node.split target₁ target₂ =>
      exploreεClosure nfa pos (next.insert target) currentSave matched saveSlots ⟨target₁, ⋯⟩
        (stack.push (StackEntry.explore ⟨target₂, ⋯⟩))
    | Node.save offset target' =>
      if h : offset < currentSave.size then
        exploreεClosure nfa pos (next.insert target) (currentSave.set ⟨offset, ⋯⟩ (some pos)) matched saveSlots
          ⟨target', ⋯⟩ (stack.push (StackEntry.restore currentSave))
      else exploreεClosure nfa pos (next.insert target) currentSave matched saveSlots ⟨target', ⋯⟩ stack
    | Node.done =>
      εClosure nfa pos (next.insert target) currentSave (HOrElse.hOrElse matched fun x => some currentSave)
        (saveSlots.set ↑target ⋯ currentSave) stack
    | Node.char c next_1 =>
      εClosure nfa pos (next.insert target) currentSave matched (saveSlots.set ↑target ⋯ currentSave) stack
    | Node.sparse cs next_1 =>
      εClosure nfa pos (next.insert target) currentSave matched (saveSlots.set ↑target ⋯ currentSave) stack
    | Node.fail => εClosure nfa pos (next.insert target) currentSave matched saveSlots stack) =
    (matched', next', saveSlots')
⊢ LowerInvεClosure nfa next' #[]
Tactic:
split at h
Proof state after:
case h_1
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✝ : target ∉ next
target'✝ : ℕ
heq✝ : nfa[↑target] = Node.epsilon target'✝
h :
  exploreεClosure nfa pos (next.insert target) currentSave matched saveSlots ⟨target'✝, ⋯⟩ stack =
    (matched', next', saveSlots')
⊢ LowerInvεClosure nfa next' #[]

case h_2
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✝ : target ∉ next
target₁✝ target₂✝ : ℕ
heq✝ : nfa[↑target] = Node.split target₁✝ target₂✝
h :
  exploreεClosure nfa pos (next.insert target) currentSave matched saveSlots ⟨target₁✝, ⋯⟩
      (stack.push (StackEntry.explore ⟨target₂✝, ⋯⟩)) =
    (matched', next', saveSlots')
⊢ LowerInvεClosure nfa next' #[]

case h_3
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✝ : target ∉ next
offset✝ target'✝ : ℕ
heq✝ : nfa[↑target] = Node.save offset✝ target'✝
h :
  (if h : offset✝ < currentSave.size then
      exploreεClosure nfa pos (next.insert target) (currentSave.set ⟨offset✝, ⋯⟩ (some pos)) matched saveSlots
        ⟨target'✝, ⋯⟩ (stack.push (StackEntry.restore currentSave))
    else exploreεClosure nfa pos (next.insert target) currentSave matched saveSlots ⟨target'✝, ⋯⟩ stack) =
    (matched', next', saveSlots')
⊢ LowerInvεClosure nfa next' #[]

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 : LowerInvExploreεClosure nfa next target stack
h✝ : target ∉ next
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')
⊢ LowerInvεClosure nfa next' #[]

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 : LowerInvExploreεClosure nfa next target stack
h✝ : target ∉ next
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')
⊢ LowerInvεClosure nfa next' #[]

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 : LowerInvExploreεClosure nfa next target stack
h✝ : target ∉ next
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')
⊢ LowerInvεClosure nfa next' #[]

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 : LowerInvExploreεClosure nfa next target stack
h✝ : target ∉ next
heq✝ : nfa[↑target] = Node.fail
h : εClosure nfa pos (next.insert target) currentSave matched saveSlots stack = (matched', next', saveSlots')
⊢ LowerInvεClosure nfa next' #[]