Theorem: Regex.VM.εClosure_subset (RegexCorrectness/VM/Auxiliary.lean)

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
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
h : εClosure nfa pos next currentSave matched saveSlots stack = (matched', next', saveSlots')
⊢ next ⊆ next'
Tactic:
unfold εClosure at h
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
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
h :
  (if hemp : stack.isEmpty = true then (matched, next, saveSlots)
    else
      let entry := stack.back' hemp;
      let stack' := stack.pop;
      let_fun this := ⋯;
      match entry with
      | StackEntry.explore target => exploreεClosure nfa pos next currentSave matched saveSlots target stack'
      | StackEntry.restore save => εClosure nfa pos next save matched saveSlots stack') =
    (matched', next', saveSlots')
⊢ next ⊆ next'

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
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
h :
  (if hemp : stack.isEmpty = true then (matched, next, saveSlots)
    else
      let entry := stack.back' hemp;
      let stack' := stack.pop;
      let_fun this := ⋯;
      match entry with
      | StackEntry.explore target => exploreεClosure nfa pos next currentSave matched saveSlots target stack'
      | StackEntry.restore save => εClosure nfa pos next save matched saveSlots stack') =
    (matched', next', saveSlots')
⊢ next ⊆ next'
Tactic:
split at h
State after:
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
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
h✝ : stack.isEmpty = true
h : (matched, next, saveSlots) = (matched', next', saveSlots')
⊢ next ⊆ 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
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
h✝ : ¬stack.isEmpty = true
h :
  (let entry := stack.back' h✝;
    let stack' := stack.pop;
    let_fun this := ⋯;
    match entry with
    | StackEntry.explore target => exploreεClosure nfa pos next currentSave matched saveSlots target stack'
    | StackEntry.restore save => εClosure nfa pos next save matched saveSlots stack') =
    (matched', next', saveSlots')
⊢ next ⊆ next'

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
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
h✝ : stack.isEmpty = true
h : (matched, next, saveSlots) = (matched', next', saveSlots')
⊢ next ⊆ 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
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
h✝ : ¬stack.isEmpty = true
h :
  (let entry := stack.back' h✝;
    let stack' := stack.pop;
    let_fun this := ⋯;
    match entry with
    | StackEntry.explore target => exploreεClosure nfa pos next currentSave matched saveSlots target stack'
    | StackEntry.restore save => εClosure nfa pos next save matched saveSlots stack') =
    (matched', next', saveSlots')
⊢ next ⊆ next'
Tactic:
next =>
  have : next = next' := by
    simp at h
    simp only [h]
  exact this ▸ SparseSet.subset_self
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
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
h✝ : ¬stack.isEmpty = true
h :
  (let entry := stack.back' h✝;
    let stack' := stack.pop;
    let_fun this := ⋯;
    match entry with
    | StackEntry.explore target => exploreεClosure nfa pos next currentSave matched saveSlots target stack'
    | StackEntry.restore save => εClosure nfa pos next save matched saveSlots stack') =
    (matched', next', saveSlots')
⊢ next ⊆ next'

State before:
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
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
h✝ : ¬stack.isEmpty = true
h :
  (let entry := stack.back' h✝;
    let stack' := stack.pop;
    let_fun this := ⋯;
    match entry with
    | StackEntry.explore target => exploreεClosure nfa pos next currentSave matched saveSlots target stack'
    | StackEntry.restore save => εClosure nfa pos next save matched saveSlots stack') =
    (matched', next', saveSlots')
⊢ next ⊆ next'
Tactic:
next =>
  simp at h
  have : stack.pop.size < stack.size := Array.lt_size_of_pop_of_not_empty _ (by assumption)
  split at h
  next => exact exploreεClosure_subset h
  next => exact εClosure_subset h
State after:
no goals

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
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
h✝ : stack.isEmpty = true
h : (matched, next, saveSlots) = (matched', next', saveSlots')
⊢ next ⊆ next'
Tactic:
have : next = next' := by
  simp at h
  simp only [h]
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
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
h✝ : stack.isEmpty = true
h : (matched, next, saveSlots) = (matched', next', saveSlots')
this : next = next'
⊢ next ⊆ next'

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
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
h✝ : stack.isEmpty = true
h : (matched, next, saveSlots) = (matched', next', saveSlots')
this : next = next'
⊢ next ⊆ next'
Tactic:
exact this ▸ SparseSet.subset_self
State after:
no goals

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
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
h✝ : stack.isEmpty = true
h : (matched, next, saveSlots) = (matched', next', saveSlots')
⊢ next = next'
Tactic:
simp at h
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
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
h✝ : stack.isEmpty = true
h : matched = matched' ∧ next = next' ∧ saveSlots = saveSlots'
⊢ next = next'

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
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
h✝ : stack.isEmpty = true
h : matched = matched' ∧ next = next' ∧ saveSlots = saveSlots'
⊢ next = next'
Tactic:
simp only [h]
State after:
no goals

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
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
h✝ : ¬stack.isEmpty = true
h :
  (let entry := stack.back' h✝;
    let stack' := stack.pop;
    let_fun this := ⋯;
    match entry with
    | StackEntry.explore target => exploreεClosure nfa pos next currentSave matched saveSlots target stack'
    | StackEntry.restore save => εClosure nfa pos next save matched saveSlots stack') =
    (matched', next', saveSlots')
⊢ next ⊆ next'
Tactic:
simp at h
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
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
h✝ : ¬stack.isEmpty = true
h :
  (match stack.back' h✝ with
    | StackEntry.explore target => exploreεClosure nfa pos next currentSave matched saveSlots target stack.pop
    | StackEntry.restore save => εClosure nfa pos next save matched saveSlots stack.pop) =
    (matched', next', saveSlots')
⊢ next ⊆ next'

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
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
h✝ : ¬stack.isEmpty = true
h :
  (match stack.back' h✝ with
    | StackEntry.explore target => exploreεClosure nfa pos next currentSave matched saveSlots target stack.pop
    | StackEntry.restore save => εClosure nfa pos next save matched saveSlots stack.pop) =
    (matched', next', saveSlots')
⊢ next ⊆ next'
Tactic:
have : stack.pop.size < stack.size := Array.lt_size_of_pop_of_not_empty _ (by assumption)
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
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
h✝ : ¬stack.isEmpty = true
h :
  (match stack.back' h✝ with
    | StackEntry.explore target => exploreεClosure nfa pos next currentSave matched saveSlots target stack.pop
    | StackEntry.restore save => εClosure nfa pos next save matched saveSlots stack.pop) =
    (matched', next', saveSlots')
this : stack.pop.size < stack.size
⊢ next ⊆ next'

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
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
h✝ : ¬stack.isEmpty = true
h :
  (match stack.back' h✝ with
    | StackEntry.explore target => exploreεClosure nfa pos next currentSave matched saveSlots target stack.pop
    | StackEntry.restore save => εClosure nfa pos next save matched saveSlots stack.pop) =
    (matched', next', saveSlots')
this : stack.pop.size < stack.size
⊢ next ⊆ next'
Tactic:
split at h
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
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
h✝ : ¬stack.isEmpty = true
this : stack.pop.size < stack.size
entry✝ : StackEntry nfa.nodes.size
target✝ : Fin nfa.nodes.size
heq✝ : stack.back' h✝ = StackEntry.explore target✝
h : exploreεClosure nfa pos next currentSave matched saveSlots target✝ stack.pop = (matched', next', saveSlots')
⊢ next ⊆ 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
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
h✝ : ¬stack.isEmpty = true
this : stack.pop.size < stack.size
entry✝ : StackEntry nfa.nodes.size
save✝ : Array (Option String.Pos)
heq✝ : stack.back' h✝ = StackEntry.restore save✝
h : εClosure nfa pos next save✝ matched saveSlots stack.pop = (matched', next', saveSlots')
⊢ next ⊆ next'

State before:
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
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
h✝ : ¬stack.isEmpty = true
this : stack.pop.size < stack.size
entry✝ : StackEntry nfa.nodes.size
target✝ : Fin nfa.nodes.size
heq✝ : stack.back' h✝ = StackEntry.explore target✝
h : exploreεClosure nfa pos next currentSave matched saveSlots target✝ stack.pop = (matched', next', saveSlots')
⊢ next ⊆ 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
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
h✝ : ¬stack.isEmpty = true
this : stack.pop.size < stack.size
entry✝ : StackEntry nfa.nodes.size
save✝ : Array (Option String.Pos)
heq✝ : stack.back' h✝ = StackEntry.restore save✝
h : εClosure nfa pos next save✝ matched saveSlots stack.pop = (matched', next', saveSlots')
⊢ next ⊆ next'
Tactic:
next => exact exploreεClosure_subset h
State after:
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
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
h✝ : ¬stack.isEmpty = true
this : stack.pop.size < stack.size
entry✝ : StackEntry nfa.nodes.size
save✝ : Array (Option String.Pos)
heq✝ : stack.back' h✝ = StackEntry.restore save✝
h : εClosure nfa pos next save✝ matched saveSlots stack.pop = (matched', next', saveSlots')
⊢ next ⊆ next'

State before:
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
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
h✝ : ¬stack.isEmpty = true
this : stack.pop.size < stack.size
entry✝ : StackEntry nfa.nodes.size
save✝ : Array (Option String.Pos)
heq✝ : stack.back' h✝ = StackEntry.restore save✝
h : εClosure nfa pos next save✝ matched saveSlots stack.pop = (matched', next', saveSlots')
⊢ next ⊆ next'
Tactic:
next => exact εClosure_subset h
State after:
no goals

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
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
h✝ : ¬stack.isEmpty = true
h :
  (match stack.back' h✝ with
    | StackEntry.explore target => exploreεClosure nfa pos next currentSave matched saveSlots target stack.pop
    | StackEntry.restore save => εClosure nfa pos next save matched saveSlots stack.pop) =
    (matched', next', saveSlots')
⊢ ¬stack.isEmpty = true
Tactic:
assumption
State after:
no goals

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
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
h✝ : ¬stack.isEmpty = true
this : stack.pop.size < stack.size
entry✝ : StackEntry nfa.nodes.size
target✝ : Fin nfa.nodes.size
heq✝ : stack.back' h✝ = StackEntry.explore target✝
h : exploreεClosure nfa pos next currentSave matched saveSlots target✝ stack.pop = (matched', next', saveSlots')
⊢ next ⊆ next'
Tactic:
exact exploreεClosure_subset h
State after:
no goals

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
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
h✝ : ¬stack.isEmpty = true
this : stack.pop.size < stack.size
entry✝ : StackEntry nfa.nodes.size
save✝ : Array (Option String.Pos)
heq✝ : stack.back' h✝ = StackEntry.restore save✝
h : εClosure nfa pos next save✝ matched saveSlots stack.pop = (matched', next', saveSlots')
⊢ next ⊆ next'
Tactic:
exact εClosure_subset h
State after:
no goals