have currentPath'  (k : Fin nfa.nodes.size) (hk : k ∈ next') :
  ∃ (s : Substring) (l m r : List Char), s.ValidFor l m r ∧ it.next.ValidFor (m.reverse ++ l.reverse) r ∧ nfa.reaches k m := by
  have ⟨_, mem_next_iff⟩ := eachStepChar_spec.mem_next_iff hs hemp
  have ⟨i, isLt, _, ⟨j, step, cls⟩⟩ := (mem_next_iff k).mp hk
  cases (mem_current'_iff current'[i]).mp (SparseSet.mem_get isLt) with
  | inl mem_current =>
    have ⟨s, l, m, r, sv, iv, prev⟩ := inv.currentPath current'[i] mem_current
    have ⟨r', hr⟩ := iv.exists_cons_of_not_atEnd atEnd
    subst hr
    exact ⟨s.expand, l, m ++ [it.curr], r', sv.expand, by simp [iv.next], prev.snoc step cls⟩
  | inr cls' =>
    have ⟨l, r, iv⟩ := v.validFor
    have ⟨r', hr⟩ := iv.exists_cons_of_not_atEnd atEnd
    subst hr
    let startPos : String.Pos := ⟨String.utf8Len l⟩
    refine ⟨⟨it.toString, startPos, startPos + it.curr⟩, l.reverse, [it.curr], r', ?_, ?_, ?_⟩
    . apply Substring.ValidFor.of_eq <;> simp [iv.toString, List.reverseAux_eq]
    . apply String.Iterator.ValidFor.of_eq <;> simp [iv.next.toString, iv.next.pos]
    . have prev : nfa.reaches current'[i] [] := .nil cls'
      exact prev.snoc step cls
nfa : NFA
saveSize : ℕ
it : String.Iterator
current next : SparseSet nfa.nodes.size
saveSlots : Vec (Array (Option String.Pos)) nfa.nodes.size
lastMatch matched : Option (Array (Option String.Pos))
haystack : String
inv : captureNext.go.Inv nfa it haystack current lastMatch
v : it.Valid
eqs : haystack = it.toString
hemp : next.isEmpty = true
atEnd : ¬it.atEnd = true
h✝ : ¬(current.isEmpty && lastMatch.isSome) = true
hm : lastMatch.isNone = true
explored :
  Option (Array (Option String.Pos)) × SparseSet nfa.nodes.size × Vec (Array (Option String.Pos)) nfa.nodes.size
fst✝ : Option (Array (Option String.Pos))
current' : SparseSet nfa.nodes.size
saveSlots' : Vec (Array (Option String.Pos)) nfa.nodes.size
he :
  exploreεClosure nfa it.pos current (captureNext.initSave saveSize) none saveSlots nfa.start #[] =
    (fst✝, current', saveSlots')
mem_current'_iff : ∀ (i : Fin nfa.nodes.size), i ∈ current' ↔ i ∈ current ∨ ↑i ∈ nfa.εClosure ↑nfa.start
stepped : Option (Array (Option String.Pos)) × SparseSet nfa.nodes.size × Vec (Array (Option String.Pos)) nfa.nodes.size
matched' : Option (Array (Option String.Pos))
next' : SparseSet nfa.nodes.size
saveSlots'' : Vec (Array (Option String.Pos)) nfa.nodes.size
hs : eachStepChar nfa it.curr it.next.i current' next saveSlots' = (matched', next', saveSlots'')
h :
  captureNext.go nfa saveSize it.next next' current.clear saveSlots'' (HOrElse.hOrElse matched' fun x => lastMatch) =
    matched
⊢ matched.isSome = true →
    ∃ s l m r,
      Substring.ValidFor l m r s ∧ haystack = { data := l ++ m ++ r } ∧ ∃ i, nfa.reaches i m ∧ nfa[i] = Node.done
nfa : NFA
saveSize : ℕ
it : String.Iterator
current next : SparseSet nfa.nodes.size
saveSlots : Vec (Array (Option String.Pos)) nfa.nodes.size
lastMatch matched : Option (Array (Option String.Pos))
haystack : String
inv : captureNext.go.Inv nfa it haystack current lastMatch
v : it.Valid
eqs : haystack = it.toString
hemp : next.isEmpty = true
atEnd : ¬it.atEnd = true
h✝ : ¬(current.isEmpty && lastMatch.isSome) = true
hm : lastMatch.isNone = true
explored :
  Option (Array (Option String.Pos)) × SparseSet nfa.nodes.size × Vec (Array (Option String.Pos)) nfa.nodes.size
fst✝ : Option (Array (Option String.Pos))
current' : SparseSet nfa.nodes.size
saveSlots' : Vec (Array (Option String.Pos)) nfa.nodes.size
he :
  exploreεClosure nfa it.pos current (captureNext.initSave saveSize) none saveSlots nfa.start #[] =
    (fst✝, current', saveSlots')
mem_current'_iff : ∀ (i : Fin nfa.nodes.size), i ∈ current' ↔ i ∈ current ∨ ↑i ∈ nfa.εClosure ↑nfa.start
stepped : Option (Array (Option String.Pos)) × SparseSet nfa.nodes.size × Vec (Array (Option String.Pos)) nfa.nodes.size
matched' : Option (Array (Option String.Pos))
next' : SparseSet nfa.nodes.size
saveSlots'' : Vec (Array (Option String.Pos)) nfa.nodes.size
hs : eachStepChar nfa it.curr it.next.i current' next saveSlots' = (matched', next', saveSlots'')
h :
  captureNext.go nfa saveSize it.next next' current.clear saveSlots'' (HOrElse.hOrElse matched' fun x => lastMatch) =
    matched
currentPath' :
  ∀ k ∈ next',
    ∃ s l m r,
      Substring.ValidFor l m r s ∧ String.Iterator.ValidFor (m.reverse ++ l.reverse) r it.next ∧ nfa.reaches k m
⊢ matched.isSome = true →
    ∃ s l m r,
      Substring.ValidFor l m r s ∧ haystack = { data := l ++ m ++ r } ∧ ∃ i, nfa.reaches i m ∧ nfa[i] = Node.done