Theorem: Regex.VM.upper_bound_exploreεClosure
Proof state before:
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
i : Fin nfa.nodes.size
inv : UpperInvExploreεClosure nfa i 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')
⊢ UpperBoundεClosure nfa i (next.insert target) 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
i : Fin nfa.nodes.size
inv : UpperInvExploreεClosure nfa i 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')
⊢ UpperBoundεClosure nfa i (next.insert target) 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
i : Fin nfa.nodes.size
inv : UpperInvExploreεClosure nfa i 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')
⊢ UpperBoundεClosure nfa i (next.insert target) 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
i : Fin nfa.nodes.size
inv : UpperInvExploreεClosure nfa i 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')
⊢ UpperBoundεClosure nfa i (next.insert target) 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
i : Fin nfa.nodes.size
inv : UpperInvExploreεClosure nfa i target stack
h✝ : target ∉ next
heq✝ : nfa[↑target] = Node.fail
h : εClosure nfa pos (next.insert target) currentSave matched saveSlots stack = (matched', next', saveSlots')
⊢ UpperBoundεClosure nfa i (next.insert target) next'
Tactic:
next _ target' hn =>
  have isLt := nfa.inBounds' target hn
  split at h
  next =>
    have inv' : UpperInvExploreεClosure nfa i ⟨target', isLt⟩ (stack.push (.restore currentSave)) := by
      refine ⟨εClosure_snoc inv.1 (by simp [εStep, Node.εStep, hn]), ?_⟩
      intro j hj
      cases (Array.mem_push ..).mp hj with
      | inl hj => exact inv.2 j hj
      | inr hj => contradiction
    exact upper_bound_exploreεClosure h inv'
  next =>
    have inv' : UpperInvExploreεClosure nfa i ⟨target', isLt⟩ stack :=
      ⟨εClosure_snoc inv.1 (by simp [εStep, Node.εStep, hn]), inv.2⟩
    exact upper_bound_exploreεClosure h inv'
Proof state after:
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
i : Fin nfa.nodes.size
inv : UpperInvExploreεClosure nfa i 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')
⊢ UpperBoundεClosure nfa i (next.insert target) 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
i : Fin nfa.nodes.size
inv : UpperInvExploreεClosure nfa i 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')
⊢ UpperBoundεClosure nfa i (next.insert target) 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
i : Fin nfa.nodes.size
inv : UpperInvExploreεClosure nfa i 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')
⊢ UpperBoundεClosure nfa i (next.insert target) 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
i : Fin nfa.nodes.size
inv : UpperInvExploreεClosure nfa i target stack
h✝ : target ∉ next
heq✝ : nfa[↑target] = Node.fail
h : εClosure nfa pos (next.insert target) currentSave matched saveSlots stack = (matched', next', saveSlots')
⊢ UpperBoundεClosure nfa i (next.insert target) next'