internlm/Lean-Github huggingface dataset:
Theorem: Regex.VM.upper_bound_εClosure
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
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 : UpperInvεClosure nfa i stack
hemp : ¬stack.isEmpty = true
h :
  (match stack.back' hemp 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')
⊢ UpperBoundεClosure nfa i next next'
Tactic:
have : stack.pop.size < stack.size := Array.lt_size_of_pop_of_not_empty _ (by assumption)
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
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 : UpperInvεClosure nfa i stack
hemp : ¬stack.isEmpty = true
h :
  (match stack.back' hemp 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
⊢ UpperBoundεClosure nfa i next next'
Theorem: Regex.VM.upper_bound_εClosure
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
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 : UpperInvεClosure nfa i stack
hemp : ¬stack.isEmpty = true
h :
  (match stack.back' hemp 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
⊢ UpperBoundεClosure nfa i next 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
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 : UpperInvεClosure nfa i stack
hemp : ¬stack.isEmpty = true
this : stack.pop.size < stack.size
entry✝ : StackEntry nfa.nodes.size
target✝ : Fin nfa.nodes.size
heq✝ : stack.back' hemp = StackEntry.explore target✝
h : exploreεClosure nfa pos next currentSave matched saveSlots target✝ stack.pop = (matched', next', saveSlots')
⊢ UpperBoundεClosure nfa i 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
i : Fin nfa.nodes.size
inv : UpperInvεClosure nfa i stack
hemp : ¬stack.isEmpty = true
this : stack.pop.size < stack.size
entry✝ : StackEntry nfa.nodes.size
save✝ : Array (Option String.Pos)
heq✝ : stack.back' hemp = StackEntry.restore save✝
h : εClosure nfa pos next save✝ matched saveSlots stack.pop = (matched', next', saveSlots')
⊢ UpperBoundεClosure nfa i next next'
Theorem: Regex.VM.upper_bound_εClosure
Proof 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
i : Fin nfa.nodes.size
inv : UpperInvεClosure nfa i stack
hemp : ¬stack.isEmpty = true
this : stack.pop.size < stack.size
entry✝ : StackEntry nfa.nodes.size
target✝ : Fin nfa.nodes.size
heq✝ : stack.back' hemp = StackEntry.explore target✝
h : exploreεClosure nfa pos next currentSave matched saveSlots target✝ stack.pop = (matched', next', saveSlots')
⊢ UpperBoundεClosure nfa i 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
i : Fin nfa.nodes.size
inv : UpperInvεClosure nfa i stack
hemp : ¬stack.isEmpty = true
this : stack.pop.size < stack.size
entry✝ : StackEntry nfa.nodes.size
save✝ : Array (Option String.Pos)
heq✝ : stack.back' hemp = StackEntry.restore save✝
h : εClosure nfa pos next save✝ matched saveSlots stack.pop = (matched', next', saveSlots')
⊢ UpperBoundεClosure nfa i next next'
Tactic:
next target heq =>
  have inv' : UpperInvExploreεClosure nfa i target stack.pop :=
    ⟨inv target (heq ▸ Array.mem_back' hemp), fun j hj => inv j (Array.mem_of_mem_pop _ _ hj)⟩
  exact upper_bound_exploreεClosure h inv'
Proof 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
i : Fin nfa.nodes.size
inv : UpperInvεClosure nfa i stack
hemp : ¬stack.isEmpty = true
this : stack.pop.size < stack.size
entry✝ : StackEntry nfa.nodes.size
save✝ : Array (Option String.Pos)
heq✝ : stack.back' hemp = StackEntry.restore save✝
h : εClosure nfa pos next save✝ matched saveSlots stack.pop = (matched', next', saveSlots')
⊢ UpperBoundεClosure nfa i next next'
Theorem: Regex.VM.upper_bound_εClosure
Proof 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
i : Fin nfa.nodes.size
inv : UpperInvεClosure nfa i stack
hemp : ¬stack.isEmpty = true
this : stack.pop.size < stack.size
entry✝ : StackEntry nfa.nodes.size
save✝ : Array (Option String.Pos)
heq✝ : stack.back' hemp = StackEntry.restore save✝
h : εClosure nfa pos next save✝ matched saveSlots stack.pop = (matched', next', saveSlots')
⊢ UpperBoundεClosure nfa i next next'
Tactic:
next =>
  have inv' : UpperInvεClosure nfa i stack.pop := fun j hj => inv j (Array.mem_of_mem_pop _ _ hj)
  exact upper_bound_εClosure h inv'
Proof state after:
no goals
Theorem: Regex.VM.upper_bound_εClosure
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
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 : UpperInvεClosure nfa i stack
hemp : ¬stack.isEmpty = true
h :
  (match stack.back' hemp 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
Proof state after:
no goals
Theorem: Regex.VM.upper_bound_εClosure
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
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 : UpperInvεClosure nfa i stack
hemp : ¬stack.isEmpty = true
this : stack.pop.size < stack.size
entry✝ : StackEntry nfa.nodes.size
target : Fin nfa.nodes.size
heq : stack.back' hemp = StackEntry.explore target
h : exploreεClosure nfa pos next currentSave matched saveSlots target stack.pop = (matched', next', saveSlots')
⊢ UpperBoundεClosure nfa i next next'
Tactic:
have inv' : UpperInvExploreεClosure nfa i target stack.pop :=
  ⟨inv target (heq ▸ Array.mem_back' hemp), fun j hj => inv j (Array.mem_of_mem_pop _ _ 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
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 : UpperInvεClosure nfa i stack
hemp : ¬stack.isEmpty = true
this : stack.pop.size < stack.size
entry✝ : StackEntry nfa.nodes.size
target : Fin nfa.nodes.size
heq : stack.back' hemp = StackEntry.explore target
h : exploreεClosure nfa pos next currentSave matched saveSlots target stack.pop = (matched', next', saveSlots')
inv' : UpperInvExploreεClosure nfa i target stack.pop
⊢ UpperBoundεClosure nfa i next next'
Theorem: Regex.VM.upper_bound_εClosure
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
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 : UpperInvεClosure nfa i stack
hemp : ¬stack.isEmpty = true
this : stack.pop.size < stack.size
entry✝ : StackEntry nfa.nodes.size
target : Fin nfa.nodes.size
heq : stack.back' hemp = StackEntry.explore target
h : exploreεClosure nfa pos next currentSave matched saveSlots target stack.pop = (matched', next', saveSlots')
inv' : UpperInvExploreεClosure nfa i target stack.pop
⊢ UpperBoundεClosure nfa i next next'
Tactic:
exact upper_bound_exploreεClosure h inv'
Proof state after:
no goals
Theorem: Regex.VM.upper_bound_εClosure
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
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 : UpperInvεClosure nfa i stack
hemp : ¬stack.isEmpty = true
this : stack.pop.size < stack.size
entry✝ : StackEntry nfa.nodes.size
save✝ : Array (Option String.Pos)
heq✝ : stack.back' hemp = StackEntry.restore save✝
h : εClosure nfa pos next save✝ matched saveSlots stack.pop = (matched', next', saveSlots')
⊢ UpperBoundεClosure nfa i next next'
Tactic:
have inv' : UpperInvεClosure nfa i stack.pop := fun j hj => inv j (Array.mem_of_mem_pop _ _ 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
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 : UpperInvεClosure nfa i stack
hemp : ¬stack.isEmpty = true
this : stack.pop.size < stack.size
entry✝ : StackEntry nfa.nodes.size
save✝ : Array (Option String.Pos)
heq✝ : stack.back' hemp = StackEntry.restore save✝
h : εClosure nfa pos next save✝ matched saveSlots stack.pop = (matched', next', saveSlots')
inv' : UpperInvεClosure nfa i stack.pop
⊢ UpperBoundεClosure nfa i next next'
Theorem: Regex.VM.upper_bound_εClosure
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
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 : UpperInvεClosure nfa i stack
hemp : ¬stack.isEmpty = true
this : stack.pop.size < stack.size
entry✝ : StackEntry nfa.nodes.size
save✝ : Array (Option String.Pos)
heq✝ : stack.back' hemp = StackEntry.restore save✝
h : εClosure nfa pos next save✝ matched saveSlots stack.pop = (matched', next', saveSlots')
inv' : UpperInvεClosure nfa i stack.pop
⊢ UpperBoundεClosure nfa i next next'
Tactic:
exact upper_bound_εClosure h inv'
Proof state after:
no goals
Theorem: Regex.VM.exploreεClosure_spec.mem_next_iff
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
matched' : Option (Array (Option String.Pos))
next' : SparseSet nfa.nodes.size
saveSlots' : Data.Vec (Array (Option String.Pos)) nfa.nodes.size
h : exploreεClosure nfa pos next currentSave matched saveSlots target #[] = (matched', next', saveSlots')
inv : ∀ (i j : Fin nfa.nodes.size), i ∈ next → ↑j ∈ nfa.εStep ↑i → j ∈ next
⊢ ∀ (i : Fin nfa.nodes.size), i ∈ next' ↔ i ∈ next ∨ ↑i ∈ nfa.εClosure ↑target
Tactic:
have lower_inv : LowerInvExploreεClosure nfa next target #[] := by
  intro i j hi hj
  exact .inl (inv i j hi 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
matched' : Option (Array (Option String.Pos))
next' : SparseSet nfa.nodes.size
saveSlots' : Data.Vec (Array (Option String.Pos)) nfa.nodes.size
h : exploreεClosure nfa pos next currentSave matched saveSlots target #[] = (matched', next', saveSlots')
inv : ∀ (i j : Fin nfa.nodes.size), i ∈ next → ↑j ∈ nfa.εStep ↑i → j ∈ next
lower_inv : LowerInvExploreεClosure nfa next target #[]
⊢ ∀ (i : Fin nfa.nodes.size), i ∈ next' ↔ i ∈ next ∨ ↑i ∈ nfa.εClosure ↑target
Theorem: Regex.VM.exploreεClosure_spec.mem_next_iff
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
matched' : Option (Array (Option String.Pos))
next' : SparseSet nfa.nodes.size
saveSlots' : Data.Vec (Array (Option String.Pos)) nfa.nodes.size
h : exploreεClosure nfa pos next currentSave matched saveSlots target #[] = (matched', next', saveSlots')
inv : ∀ (i j : Fin nfa.nodes.size), i ∈ next → ↑j ∈ nfa.εStep ↑i → j ∈ next
lower_inv : LowerInvExploreεClosure nfa next target #[]
⊢ ∀ (i : Fin nfa.nodes.size), i ∈ next' ↔ i ∈ next ∨ ↑i ∈ nfa.εClosure ↑target
Tactic:
have upper_inv : UpperInvExploreεClosure nfa target target #[] := by
  refine ⟨.base, by intros; simp_all⟩
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
matched' : Option (Array (Option String.Pos))
next' : SparseSet nfa.nodes.size
saveSlots' : Data.Vec (Array (Option String.Pos)) nfa.nodes.size
h : exploreεClosure nfa pos next currentSave matched saveSlots target #[] = (matched', next', saveSlots')
inv : ∀ (i j : Fin nfa.nodes.size), i ∈ next → ↑j ∈ nfa.εStep ↑i → j ∈ next
lower_inv : LowerInvExploreεClosure nfa next target #[]
upper_inv : UpperInvExploreεClosure nfa target target #[]
⊢ ∀ (i : Fin nfa.nodes.size), i ∈ next' ↔ i ∈ next ∨ ↑i ∈ nfa.εClosure ↑target
Theorem: Regex.VM.exploreεClosure_spec.mem_next_iff
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
matched' : Option (Array (Option String.Pos))
next' : SparseSet nfa.nodes.size
saveSlots' : Data.Vec (Array (Option String.Pos)) nfa.nodes.size
h : exploreεClosure nfa pos next currentSave matched saveSlots target #[] = (matched', next', saveSlots')
inv : ∀ (i j : Fin nfa.nodes.size), i ∈ next → ↑j ∈ nfa.εStep ↑i → j ∈ next
lower_inv : LowerInvExploreεClosure nfa next target #[]
upper_inv : UpperInvExploreεClosure nfa target target #[]
⊢ ∀ (i : Fin nfa.nodes.size), i ∈ next' ↔ i ∈ next ∨ ↑i ∈ nfa.εClosure ↑target
Tactic:
have lower_bound := lower_bound_exploreεClosure h lower_inv
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
matched' : Option (Array (Option String.Pos))
next' : SparseSet nfa.nodes.size
saveSlots' : Data.Vec (Array (Option String.Pos)) nfa.nodes.size
h : exploreεClosure nfa pos next currentSave matched saveSlots target #[] = (matched', next', saveSlots')
inv : ∀ (i j : Fin nfa.nodes.size), i ∈ next → ↑j ∈ nfa.εStep ↑i → j ∈ next
lower_inv : LowerInvExploreεClosure nfa next target #[]
upper_inv : UpperInvExploreεClosure nfa target target #[]
lower_bound : LowerBoundεClosure nfa target next next'
⊢ ∀ (i : Fin nfa.nodes.size), i ∈ next' ↔ i ∈ next ∨ ↑i ∈ nfa.εClosure ↑target
Theorem: Regex.VM.exploreεClosure_spec.mem_next_iff
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
matched' : Option (Array (Option String.Pos))
next' : SparseSet nfa.nodes.size
saveSlots' : Data.Vec (Array (Option String.Pos)) nfa.nodes.size
h : exploreεClosure nfa pos next currentSave matched saveSlots target #[] = (matched', next', saveSlots')
inv : ∀ (i j : Fin nfa.nodes.size), i ∈ next → ↑j ∈ nfa.εStep ↑i → j ∈ next
lower_inv : LowerInvExploreεClosure nfa next target #[]
upper_inv : UpperInvExploreεClosure nfa target target #[]
lower_bound : LowerBoundεClosure nfa target next next'
⊢ ∀ (i : Fin nfa.nodes.size), i ∈ next' ↔ i ∈ next ∨ ↑i ∈ nfa.εClosure ↑target
Tactic:
have upper_bound := upper_bound_exploreεClosure h upper_inv
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
matched' : Option (Array (Option String.Pos))
next' : SparseSet nfa.nodes.size
saveSlots' : Data.Vec (Array (Option String.Pos)) nfa.nodes.size
h : exploreεClosure nfa pos next currentSave matched saveSlots target #[] = (matched', next', saveSlots')
inv : ∀ (i j : Fin nfa.nodes.size), i ∈ next → ↑j ∈ nfa.εStep ↑i → j ∈ next
lower_inv : LowerInvExploreεClosure nfa next target #[]
upper_inv : UpperInvExploreεClosure nfa target target #[]
lower_bound : LowerBoundεClosure nfa target next next'
upper_bound : UpperBoundεClosure nfa target next next'
⊢ ∀ (i : Fin nfa.nodes.size), i ∈ next' ↔ i ∈ next ∨ ↑i ∈ nfa.εClosure ↑target
Theorem: Regex.VM.exploreεClosure_spec.mem_next_iff
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
matched' : Option (Array (Option String.Pos))
next' : SparseSet nfa.nodes.size
saveSlots' : Data.Vec (Array (Option String.Pos)) nfa.nodes.size
h : exploreεClosure nfa pos next currentSave matched saveSlots target #[] = (matched', next', saveSlots')
inv : ∀ (i j : Fin nfa.nodes.size), i ∈ next → ↑j ∈ nfa.εStep ↑i → j ∈ next
lower_inv : LowerInvExploreεClosure nfa next target #[]
upper_inv : UpperInvExploreεClosure nfa target target #[]
lower_bound : LowerBoundεClosure nfa target next next'
upper_bound : UpperBoundεClosure nfa target next next'
⊢ ∀ (i : Fin nfa.nodes.size), i ∈ next' ↔ i ∈ next ∨ ↑i ∈ nfa.εClosure ↑target
Tactic:
intro i
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
matched' : Option (Array (Option String.Pos))
next' : SparseSet nfa.nodes.size
saveSlots' : Data.Vec (Array (Option String.Pos)) nfa.nodes.size
h : exploreεClosure nfa pos next currentSave matched saveSlots target #[] = (matched', next', saveSlots')
inv : ∀ (i j : Fin nfa.nodes.size), i ∈ next → ↑j ∈ nfa.εStep ↑i → j ∈ next
lower_inv : LowerInvExploreεClosure nfa next target #[]
upper_inv : UpperInvExploreεClosure nfa target target #[]
lower_bound : LowerBoundεClosure nfa target next next'
upper_bound : UpperBoundεClosure nfa target next next'
i : Fin nfa.nodes.size
⊢ i ∈ next' ↔ i ∈ next ∨ ↑i ∈ nfa.εClosure ↑target
Theorem: Regex.VM.exploreεClosure_spec.mem_next_iff
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
matched' : Option (Array (Option String.Pos))
next' : SparseSet nfa.nodes.size
saveSlots' : Data.Vec (Array (Option String.Pos)) nfa.nodes.size
h : exploreεClosure nfa pos next currentSave matched saveSlots target #[] = (matched', next', saveSlots')
inv : ∀ (i j : Fin nfa.nodes.size), i ∈ next → ↑j ∈ nfa.εStep ↑i → j ∈ next
lower_inv : LowerInvExploreεClosure nfa next target #[]
upper_inv : UpperInvExploreεClosure nfa target target #[]
lower_bound : LowerBoundεClosure nfa target next next'
upper_bound : UpperBoundεClosure nfa target next next'
i : Fin nfa.nodes.size
⊢ i ∈ next' ↔ i ∈ next ∨ ↑i ∈ nfa.εClosure ↑target
Tactic:
exact ⟨upper_bound i, lower_bound i⟩
Proof state after:
no goals