Theorem: Regex.VM.upper_bound_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
i : Fin nfa.nodes.size
inv : UpperInvExploreεClosure nfa i target stack
h✝ : target ∉ next
h :
  (let next' := next.insert target;
    match hn : nfa[target] with
    | Node.epsilon target' =>
      let_fun isLt := ⋯;
      exploreεClosure nfa pos next' currentSave matched saveSlots ⟨target', isLt⟩ stack
    | Node.split target₁ target₂ =>
      let_fun isLt := ⋯;
      exploreεClosure nfa pos next' currentSave matched saveSlots ⟨target₁, ⋯⟩
        (stack.push (StackEntry.explore ⟨target₂, ⋯⟩))
    | Node.save offset target' =>
      let_fun isLt := ⋯;
      if h : offset < currentSave.size then
        let nextSave := currentSave.set ⟨offset, h⟩ (some pos);
        let stack' := stack.push (StackEntry.restore currentSave);
        exploreεClosure nfa pos next' nextSave matched saveSlots ⟨target', isLt⟩ stack'
      else exploreεClosure nfa pos next' currentSave matched saveSlots ⟨target', isLt⟩ stack
    | Node.done =>
      let matched' := HOrElse.hOrElse matched fun x => some currentSave;
      let saveSlots' := saveSlots.set ↑target ⋯ currentSave;
      εClosure nfa pos next' currentSave matched' saveSlots' stack
    | Node.char c next =>
      let saveSlots' := saveSlots.set ↑target ⋯ currentSave;
      εClosure nfa pos next' currentSave matched saveSlots' stack
    | Node.sparse cs next =>
      let saveSlots' := saveSlots.set ↑target ⋯ currentSave;
      εClosure nfa pos next' currentSave matched saveSlots' stack
    | Node.fail => εClosure nfa pos next' currentSave matched saveSlots stack) =
    (matched', next', saveSlots')
this : UpperBoundεClosure nfa i (next.insert target) next'
⊢ UpperBoundεClosure nfa i next next'
Tactic:
intro j hj
Proof state after:
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
h :
  (let next' := next.insert target;
    match hn : nfa[target] with
    | Node.epsilon target' =>
      let_fun isLt := ⋯;
      exploreεClosure nfa pos next' currentSave matched saveSlots ⟨target', isLt⟩ stack
    | Node.split target₁ target₂ =>
      let_fun isLt := ⋯;
      exploreεClosure nfa pos next' currentSave matched saveSlots ⟨target₁, ⋯⟩
        (stack.push (StackEntry.explore ⟨target₂, ⋯⟩))
    | Node.save offset target' =>
      let_fun isLt := ⋯;
      if h : offset < currentSave.size then
        let nextSave := currentSave.set ⟨offset, h⟩ (some pos);
        let stack' := stack.push (StackEntry.restore currentSave);
        exploreεClosure nfa pos next' nextSave matched saveSlots ⟨target', isLt⟩ stack'
      else exploreεClosure nfa pos next' currentSave matched saveSlots ⟨target', isLt⟩ stack
    | Node.done =>
      let matched' := HOrElse.hOrElse matched fun x => some currentSave;
      let saveSlots' := saveSlots.set ↑target ⋯ currentSave;
      εClosure nfa pos next' currentSave matched' saveSlots' stack
    | Node.char c next =>
      let saveSlots' := saveSlots.set ↑target ⋯ currentSave;
      εClosure nfa pos next' currentSave matched saveSlots' stack
    | Node.sparse cs next =>
      let saveSlots' := saveSlots.set ↑target ⋯ currentSave;
      εClosure nfa pos next' currentSave matched saveSlots' stack
    | Node.fail => εClosure nfa pos next' currentSave matched saveSlots stack) =
    (matched', next', saveSlots')
this : UpperBoundεClosure nfa i (next.insert target) next'
j : Fin nfa.nodes.size
hj : j ∈ next'
⊢ j ∈ next ∨ ↑j ∈ nfa.εClosure ↑i