internlm/Lean-Github huggingface dataset:
Theorem: Regex.VM.stepChar_spec.mem_next_iff
Proof state before:
case mp
nfa : NFA
c : Char
pos : String.Pos
next : SparseSet nfa.nodes.size
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
inv : ∀ (i j : Fin nfa.nodes.size), i ∈ next → ↑j ∈ nfa.εStep ↑i → j ∈ next
cs : Data.Classes
target' : ℕ
hn : nfa[↑target] = Node.sparse cs target'
hc : c ∈ cs
h :
  exploreεClosure nfa pos next (saveSlots.get ↑target ⋯) none saveSlots ⟨target', ⋯⟩ #[] = (matched', next', saveSlots')
mem_next_iff : ∀ (i : Fin nfa.nodes.size), i ∈ next' ↔ i ∈ next ∨ ↑i ∈ nfa.εClosure ↑⟨target', ⋯⟩
j : Fin nfa.nodes.size
⊢ j ∈ next' → j ∈ next ∨ ∃ i ∈ nfa.charStep (↑target) c, ↑j ∈ nfa.εClosure i

case mpr
nfa : NFA
c : Char
pos : String.Pos
next : SparseSet nfa.nodes.size
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
inv : ∀ (i j : Fin nfa.nodes.size), i ∈ next → ↑j ∈ nfa.εStep ↑i → j ∈ next
cs : Data.Classes
target' : ℕ
hn : nfa[↑target] = Node.sparse cs target'
hc : c ∈ cs
h :
  exploreεClosure nfa pos next (saveSlots.get ↑target ⋯) none saveSlots ⟨target', ⋯⟩ #[] = (matched', next', saveSlots')
mem_next_iff : ∀ (i : Fin nfa.nodes.size), i ∈ next' ↔ i ∈ next ∨ ↑i ∈ nfa.εClosure ↑⟨target', ⋯⟩
j : Fin nfa.nodes.size
⊢ (j ∈ next ∨ ∃ i ∈ nfa.charStep (↑target) c, ↑j ∈ nfa.εClosure i) → j ∈ next'
Tactic:
. intro hj
  cases (mem_next_iff j).mp hj with
  | inl hj => exact .inl hj
  | inr hj =>
    refine .inr ⟨target', ?_, hj⟩
    simp [charStep, Node.charStep, hn, hc]
Proof state after:
case mpr
nfa : NFA
c : Char
pos : String.Pos
next : SparseSet nfa.nodes.size
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
inv : ∀ (i j : Fin nfa.nodes.size), i ∈ next → ↑j ∈ nfa.εStep ↑i → j ∈ next
cs : Data.Classes
target' : ℕ
hn : nfa[↑target] = Node.sparse cs target'
hc : c ∈ cs
h :
  exploreεClosure nfa pos next (saveSlots.get ↑target ⋯) none saveSlots ⟨target', ⋯⟩ #[] = (matched', next', saveSlots')
mem_next_iff : ∀ (i : Fin nfa.nodes.size), i ∈ next' ↔ i ∈ next ∨ ↑i ∈ nfa.εClosure ↑⟨target', ⋯⟩
j : Fin nfa.nodes.size
⊢ (j ∈ next ∨ ∃ i ∈ nfa.charStep (↑target) c, ↑j ∈ nfa.εClosure i) → j ∈ next'
Theorem: Regex.VM.stepChar_spec.mem_next_iff
Proof state before:
case mpr
nfa : NFA
c : Char
pos : String.Pos
next : SparseSet nfa.nodes.size
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
inv : ∀ (i j : Fin nfa.nodes.size), i ∈ next → ↑j ∈ nfa.εStep ↑i → j ∈ next
cs : Data.Classes
target' : ℕ
hn : nfa[↑target] = Node.sparse cs target'
hc : c ∈ cs
h :
  exploreεClosure nfa pos next (saveSlots.get ↑target ⋯) none saveSlots ⟨target', ⋯⟩ #[] = (matched', next', saveSlots')
mem_next_iff : ∀ (i : Fin nfa.nodes.size), i ∈ next' ↔ i ∈ next ∨ ↑i ∈ nfa.εClosure ↑⟨target', ⋯⟩
j : Fin nfa.nodes.size
⊢ (j ∈ next ∨ ∃ i ∈ nfa.charStep (↑target) c, ↑j ∈ nfa.εClosure i) → j ∈ next'
Tactic:
. intro hj
  cases hj with
  | inl hj => exact (mem_next_iff j).mpr (.inl hj)
  | inr hj =>
    simp [charStep, Node.charStep, hn, hc] at hj
    exact (mem_next_iff j).mpr (.inr hj)
Proof state after:
no goals
Theorem: Regex.VM.stepChar_spec.mem_next_iff
Proof state before:
case mp
nfa : NFA
c : Char
pos : String.Pos
next : SparseSet nfa.nodes.size
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
inv : ∀ (i j : Fin nfa.nodes.size), i ∈ next → ↑j ∈ nfa.εStep ↑i → j ∈ next
cs : Data.Classes
target' : ℕ
hn : nfa[↑target] = Node.sparse cs target'
hc : c ∈ cs
h :
  exploreεClosure nfa pos next (saveSlots.get ↑target ⋯) none saveSlots ⟨target', ⋯⟩ #[] = (matched', next', saveSlots')
mem_next_iff : ∀ (i : Fin nfa.nodes.size), i ∈ next' ↔ i ∈ next ∨ ↑i ∈ nfa.εClosure ↑⟨target', ⋯⟩
j : Fin nfa.nodes.size
⊢ j ∈ next' → j ∈ next ∨ ∃ i ∈ nfa.charStep (↑target) c, ↑j ∈ nfa.εClosure i
Tactic:
intro hj
Proof state after:
case mp
nfa : NFA
c : Char
pos : String.Pos
next : SparseSet nfa.nodes.size
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
inv : ∀ (i j : Fin nfa.nodes.size), i ∈ next → ↑j ∈ nfa.εStep ↑i → j ∈ next
cs : Data.Classes
target' : ℕ
hn : nfa[↑target] = Node.sparse cs target'
hc : c ∈ cs
h :
  exploreεClosure nfa pos next (saveSlots.get ↑target ⋯) none saveSlots ⟨target', ⋯⟩ #[] = (matched', next', saveSlots')
mem_next_iff : ∀ (i : Fin nfa.nodes.size), i ∈ next' ↔ i ∈ next ∨ ↑i ∈ nfa.εClosure ↑⟨target', ⋯⟩
j : Fin nfa.nodes.size
hj : j ∈ next'
⊢ j ∈ next ∨ ∃ i ∈ nfa.charStep (↑target) c, ↑j ∈ nfa.εClosure i
Theorem: Regex.VM.stepChar_spec.mem_next_iff
Proof state before:
case mp
nfa : NFA
c : Char
pos : String.Pos
next : SparseSet nfa.nodes.size
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
inv : ∀ (i j : Fin nfa.nodes.size), i ∈ next → ↑j ∈ nfa.εStep ↑i → j ∈ next
cs : Data.Classes
target' : ℕ
hn : nfa[↑target] = Node.sparse cs target'
hc : c ∈ cs
h :
  exploreεClosure nfa pos next (saveSlots.get ↑target ⋯) none saveSlots ⟨target', ⋯⟩ #[] = (matched', next', saveSlots')
mem_next_iff : ∀ (i : Fin nfa.nodes.size), i ∈ next' ↔ i ∈ next ∨ ↑i ∈ nfa.εClosure ↑⟨target', ⋯⟩
j : Fin nfa.nodes.size
hj : j ∈ next'
⊢ j ∈ next ∨ ∃ i ∈ nfa.charStep (↑target) c, ↑j ∈ nfa.εClosure i
Tactic:
cases (mem_next_iff j).mp hj with
| inl hj => exact .inl hj
| inr hj =>
  refine .inr ⟨target', ?_, hj⟩
  simp [charStep, Node.charStep, hn, hc]
Proof state after:
no goals
Theorem: Regex.VM.stepChar_spec.mem_next_iff
Proof state before:
case mp.inl
nfa : NFA
c : Char
pos : String.Pos
next : SparseSet nfa.nodes.size
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
inv : ∀ (i j : Fin nfa.nodes.size), i ∈ next → ↑j ∈ nfa.εStep ↑i → j ∈ next
cs : Data.Classes
target' : ℕ
hn : nfa[↑target] = Node.sparse cs target'
hc : c ∈ cs
h :
  exploreεClosure nfa pos next (saveSlots.get ↑target ⋯) none saveSlots ⟨target', ⋯⟩ #[] = (matched', next', saveSlots')
mem_next_iff : ∀ (i : Fin nfa.nodes.size), i ∈ next' ↔ i ∈ next ∨ ↑i ∈ nfa.εClosure ↑⟨target', ⋯⟩
j : Fin nfa.nodes.size
hj✝ : j ∈ next'
hj : j ∈ next
⊢ j ∈ next ∨ ∃ i ∈ nfa.charStep (↑target) c, ↑j ∈ nfa.εClosure i
Tactic:
exact .inl hj
Proof state after:
no goals
Theorem: Regex.VM.stepChar_spec.mem_next_iff
Proof state before:
case mp.inr
nfa : NFA
c : Char
pos : String.Pos
next : SparseSet nfa.nodes.size
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
inv : ∀ (i j : Fin nfa.nodes.size), i ∈ next → ↑j ∈ nfa.εStep ↑i → j ∈ next
cs : Data.Classes
target' : ℕ
hn : nfa[↑target] = Node.sparse cs target'
hc : c ∈ cs
h :
  exploreεClosure nfa pos next (saveSlots.get ↑target ⋯) none saveSlots ⟨target', ⋯⟩ #[] = (matched', next', saveSlots')
mem_next_iff : ∀ (i : Fin nfa.nodes.size), i ∈ next' ↔ i ∈ next ∨ ↑i ∈ nfa.εClosure ↑⟨target', ⋯⟩
j : Fin nfa.nodes.size
hj✝ : j ∈ next'
hj : ↑j ∈ nfa.εClosure ↑⟨target', ⋯⟩
⊢ j ∈ next ∨ ∃ i ∈ nfa.charStep (↑target) c, ↑j ∈ nfa.εClosure i
Tactic:
refine .inr ⟨target', ?_, hj⟩
Proof state after:
case mp.inr
nfa : NFA
c : Char
pos : String.Pos
next : SparseSet nfa.nodes.size
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
inv : ∀ (i j : Fin nfa.nodes.size), i ∈ next → ↑j ∈ nfa.εStep ↑i → j ∈ next
cs : Data.Classes
target' : ℕ
hn : nfa[↑target] = Node.sparse cs target'
hc : c ∈ cs
h :
  exploreεClosure nfa pos next (saveSlots.get ↑target ⋯) none saveSlots ⟨target', ⋯⟩ #[] = (matched', next', saveSlots')
mem_next_iff : ∀ (i : Fin nfa.nodes.size), i ∈ next' ↔ i ∈ next ∨ ↑i ∈ nfa.εClosure ↑⟨target', ⋯⟩
j : Fin nfa.nodes.size
hj✝ : j ∈ next'
hj : ↑j ∈ nfa.εClosure ↑⟨target', ⋯⟩
⊢ target' ∈ nfa.charStep (↑target) c
Theorem: Regex.VM.stepChar_spec.mem_next_iff
Proof state before:
case mp.inr
nfa : NFA
c : Char
pos : String.Pos
next : SparseSet nfa.nodes.size
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
inv : ∀ (i j : Fin nfa.nodes.size), i ∈ next → ↑j ∈ nfa.εStep ↑i → j ∈ next
cs : Data.Classes
target' : ℕ
hn : nfa[↑target] = Node.sparse cs target'
hc : c ∈ cs
h :
  exploreεClosure nfa pos next (saveSlots.get ↑target ⋯) none saveSlots ⟨target', ⋯⟩ #[] = (matched', next', saveSlots')
mem_next_iff : ∀ (i : Fin nfa.nodes.size), i ∈ next' ↔ i ∈ next ∨ ↑i ∈ nfa.εClosure ↑⟨target', ⋯⟩
j : Fin nfa.nodes.size
hj✝ : j ∈ next'
hj : ↑j ∈ nfa.εClosure ↑⟨target', ⋯⟩
⊢ target' ∈ nfa.charStep (↑target) c
Tactic:
simp [charStep, Node.charStep, hn, hc]
Proof state after:
no goals
Theorem: Regex.VM.stepChar_spec.mem_next_iff
Proof state before:
case mpr
nfa : NFA
c : Char
pos : String.Pos
next : SparseSet nfa.nodes.size
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
inv : ∀ (i j : Fin nfa.nodes.size), i ∈ next → ↑j ∈ nfa.εStep ↑i → j ∈ next
cs : Data.Classes
target' : ℕ
hn : nfa[↑target] = Node.sparse cs target'
hc : c ∈ cs
h :
  exploreεClosure nfa pos next (saveSlots.get ↑target ⋯) none saveSlots ⟨target', ⋯⟩ #[] = (matched', next', saveSlots')
mem_next_iff : ∀ (i : Fin nfa.nodes.size), i ∈ next' ↔ i ∈ next ∨ ↑i ∈ nfa.εClosure ↑⟨target', ⋯⟩
j : Fin nfa.nodes.size
⊢ (j ∈ next ∨ ∃ i ∈ nfa.charStep (↑target) c, ↑j ∈ nfa.εClosure i) → j ∈ next'
Tactic:
intro hj
Proof state after:
case mpr
nfa : NFA
c : Char
pos : String.Pos
next : SparseSet nfa.nodes.size
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
inv : ∀ (i j : Fin nfa.nodes.size), i ∈ next → ↑j ∈ nfa.εStep ↑i → j ∈ next
cs : Data.Classes
target' : ℕ
hn : nfa[↑target] = Node.sparse cs target'
hc : c ∈ cs
h :
  exploreεClosure nfa pos next (saveSlots.get ↑target ⋯) none saveSlots ⟨target', ⋯⟩ #[] = (matched', next', saveSlots')
mem_next_iff : ∀ (i : Fin nfa.nodes.size), i ∈ next' ↔ i ∈ next ∨ ↑i ∈ nfa.εClosure ↑⟨target', ⋯⟩
j : Fin nfa.nodes.size
hj : j ∈ next ∨ ∃ i ∈ nfa.charStep (↑target) c, ↑j ∈ nfa.εClosure i
⊢ j ∈ next'
Theorem: Regex.VM.stepChar_spec.mem_next_iff
Proof state before:
case mpr
nfa : NFA
c : Char
pos : String.Pos
next : SparseSet nfa.nodes.size
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
inv : ∀ (i j : Fin nfa.nodes.size), i ∈ next → ↑j ∈ nfa.εStep ↑i → j ∈ next
cs : Data.Classes
target' : ℕ
hn : nfa[↑target] = Node.sparse cs target'
hc : c ∈ cs
h :
  exploreεClosure nfa pos next (saveSlots.get ↑target ⋯) none saveSlots ⟨target', ⋯⟩ #[] = (matched', next', saveSlots')
mem_next_iff : ∀ (i : Fin nfa.nodes.size), i ∈ next' ↔ i ∈ next ∨ ↑i ∈ nfa.εClosure ↑⟨target', ⋯⟩
j : Fin nfa.nodes.size
hj : j ∈ next ∨ ∃ i ∈ nfa.charStep (↑target) c, ↑j ∈ nfa.εClosure i
⊢ j ∈ next'
Tactic:
cases hj with
| inl hj => exact (mem_next_iff j).mpr (.inl hj)
| inr hj =>
  simp [charStep, Node.charStep, hn, hc] at hj
  exact (mem_next_iff j).mpr (.inr hj)
Proof state after:
no goals
Theorem: Regex.VM.stepChar_spec.mem_next_iff
Proof state before:
case mpr.inl
nfa : NFA
c : Char
pos : String.Pos
next : SparseSet nfa.nodes.size
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
inv : ∀ (i j : Fin nfa.nodes.size), i ∈ next → ↑j ∈ nfa.εStep ↑i → j ∈ next
cs : Data.Classes
target' : ℕ
hn : nfa[↑target] = Node.sparse cs target'
hc : c ∈ cs
h :
  exploreεClosure nfa pos next (saveSlots.get ↑target ⋯) none saveSlots ⟨target', ⋯⟩ #[] = (matched', next', saveSlots')
mem_next_iff : ∀ (i : Fin nfa.nodes.size), i ∈ next' ↔ i ∈ next ∨ ↑i ∈ nfa.εClosure ↑⟨target', ⋯⟩
j : Fin nfa.nodes.size
hj : j ∈ next
⊢ j ∈ next'
Tactic:
exact (mem_next_iff j).mpr (.inl hj)
Proof state after:
no goals
Theorem: Regex.VM.stepChar_spec.mem_next_iff
Proof state before:
case mpr.inr
nfa : NFA
c : Char
pos : String.Pos
next : SparseSet nfa.nodes.size
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
inv : ∀ (i j : Fin nfa.nodes.size), i ∈ next → ↑j ∈ nfa.εStep ↑i → j ∈ next
cs : Data.Classes
target' : ℕ
hn : nfa[↑target] = Node.sparse cs target'
hc : c ∈ cs
h :
  exploreεClosure nfa pos next (saveSlots.get ↑target ⋯) none saveSlots ⟨target', ⋯⟩ #[] = (matched', next', saveSlots')
mem_next_iff : ∀ (i : Fin nfa.nodes.size), i ∈ next' ↔ i ∈ next ∨ ↑i ∈ nfa.εClosure ↑⟨target', ⋯⟩
j : Fin nfa.nodes.size
hj : ∃ i ∈ nfa.charStep (↑target) c, ↑j ∈ nfa.εClosure i
⊢ j ∈ next'
Tactic:
simp [charStep, Node.charStep, hn, hc] at hj
Proof state after:
case mpr.inr
nfa : NFA
c : Char
pos : String.Pos
next : SparseSet nfa.nodes.size
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
inv : ∀ (i j : Fin nfa.nodes.size), i ∈ next → ↑j ∈ nfa.εStep ↑i → j ∈ next
cs : Data.Classes
target' : ℕ
hn : nfa[↑target] = Node.sparse cs target'
hc : c ∈ cs
h :
  exploreεClosure nfa pos next (saveSlots.get ↑target ⋯) none saveSlots ⟨target', ⋯⟩ #[] = (matched', next', saveSlots')
mem_next_iff : ∀ (i : Fin nfa.nodes.size), i ∈ next' ↔ i ∈ next ∨ ↑i ∈ nfa.εClosure ↑⟨target', ⋯⟩
j : Fin nfa.nodes.size
hj : ↑j ∈ nfa.εClosure target'
⊢ j ∈ next'
Theorem: Regex.VM.stepChar_spec.mem_next_iff
Proof state before:
case mpr.inr
nfa : NFA
c : Char
pos : String.Pos
next : SparseSet nfa.nodes.size
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
inv : ∀ (i j : Fin nfa.nodes.size), i ∈ next → ↑j ∈ nfa.εStep ↑i → j ∈ next
cs : Data.Classes
target' : ℕ
hn : nfa[↑target] = Node.sparse cs target'
hc : c ∈ cs
h :
  exploreεClosure nfa pos next (saveSlots.get ↑target ⋯) none saveSlots ⟨target', ⋯⟩ #[] = (matched', next', saveSlots')
mem_next_iff : ∀ (i : Fin nfa.nodes.size), i ∈ next' ↔ i ∈ next ∨ ↑i ∈ nfa.εClosure ↑⟨target', ⋯⟩
j : Fin nfa.nodes.size
hj : ↑j ∈ nfa.εClosure target'
⊢ j ∈ next'
Tactic:
exact (mem_next_iff j).mpr (.inr hj)
Proof state after:
no goals
Theorem: Regex.VM.stepChar_spec.mem_next_iff
Proof state before:
nfa : NFA
c : Char
pos : String.Pos
next : SparseSet nfa.nodes.size
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
inv : ∀ (i j : Fin nfa.nodes.size), i ∈ next → ↑j ∈ nfa.εStep ↑i → j ∈ next
cs : Data.Classes
target' : ℕ
hn : nfa[↑target] = Node.sparse cs target'
hc : c ∉ cs
h : (none, next, saveSlots) = (matched', next', saveSlots')
⊢ ∀ (j : Fin nfa.nodes.size), j ∈ next' ↔ j ∈ next ∨ ∃ i ∈ nfa.charStep (↑target) c, ↑j ∈ nfa.εClosure i
Tactic:
have : ∀ i, ¬i ∈ nfa.charStep target c := by
  intro i
  simp [charStep, Node.charStep, hn, hc]
Proof state after:
nfa : NFA
c : Char
pos : String.Pos
next : SparseSet nfa.nodes.size
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
inv : ∀ (i j : Fin nfa.nodes.size), i ∈ next → ↑j ∈ nfa.εStep ↑i → j ∈ next
cs : Data.Classes
target' : ℕ
hn : nfa[↑target] = Node.sparse cs target'
hc : c ∉ cs
h : (none, next, saveSlots) = (matched', next', saveSlots')
this : ∀ (i : ℕ), i ∉ nfa.charStep (↑target) c
⊢ ∀ (j : Fin nfa.nodes.size), j ∈ next' ↔ j ∈ next ∨ ∃ i ∈ nfa.charStep (↑target) c, ↑j ∈ nfa.εClosure i
Theorem: Regex.VM.stepChar_spec.mem_next_iff
Proof state before:
nfa : NFA
c : Char
pos : String.Pos
next : SparseSet nfa.nodes.size
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
inv : ∀ (i j : Fin nfa.nodes.size), i ∈ next → ↑j ∈ nfa.εStep ↑i → j ∈ next
cs : Data.Classes
target' : ℕ
hn : nfa[↑target] = Node.sparse cs target'
hc : c ∉ cs
h : (none, next, saveSlots) = (matched', next', saveSlots')
this : ∀ (i : ℕ), i ∉ nfa.charStep (↑target) c
⊢ ∀ (j : Fin nfa.nodes.size), j ∈ next' ↔ j ∈ next ∨ ∃ i ∈ nfa.charStep (↑target) c, ↑j ∈ nfa.εClosure i
Tactic:
simp at h
Proof state after:
nfa : NFA
c : Char
pos : String.Pos
next : SparseSet nfa.nodes.size
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
inv : ∀ (i j : Fin nfa.nodes.size), i ∈ next → ↑j ∈ nfa.εStep ↑i → j ∈ next
cs : Data.Classes
target' : ℕ
hn : nfa[↑target] = Node.sparse cs target'
hc : c ∉ cs
this : ∀ (i : ℕ), i ∉ nfa.charStep (↑target) c
h : none = matched' ∧ next = next' ∧ saveSlots = saveSlots'
⊢ ∀ (j : Fin nfa.nodes.size), j ∈ next' ↔ j ∈ next ∨ ∃ i ∈ nfa.charStep (↑target) c, ↑j ∈ nfa.εClosure i
Theorem: Regex.VM.stepChar_spec.mem_next_iff
Proof state before:
nfa : NFA
c : Char
pos : String.Pos
next : SparseSet nfa.nodes.size
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
inv : ∀ (i j : Fin nfa.nodes.size), i ∈ next → ↑j ∈ nfa.εStep ↑i → j ∈ next
cs : Data.Classes
target' : ℕ
hn : nfa[↑target] = Node.sparse cs target'
hc : c ∉ cs
this : ∀ (i : ℕ), i ∉ nfa.charStep (↑target) c
h : none = matched' ∧ next = next' ∧ saveSlots = saveSlots'
⊢ ∀ (j : Fin nfa.nodes.size), j ∈ next' ↔ j ∈ next ∨ ∃ i ∈ nfa.charStep (↑target) c, ↑j ∈ nfa.εClosure i
Tactic:
simp [h, this]
Proof state after:
no goals
Theorem: Regex.VM.stepChar_spec.mem_next_iff
Proof state before:
nfa : NFA
c : Char
pos : String.Pos
next : SparseSet nfa.nodes.size
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
inv : ∀ (i j : Fin nfa.nodes.size), i ∈ next → ↑j ∈ nfa.εStep ↑i → j ∈ next
cs : Data.Classes
target' : ℕ
hn : nfa[↑target] = Node.sparse cs target'
hc : c ∉ cs
h : (none, next, saveSlots) = (matched', next', saveSlots')
⊢ ∀ (i : ℕ), i ∉ nfa.charStep (↑target) c
Tactic:
intro i
Proof state after:
nfa : NFA
c : Char
pos : String.Pos
next : SparseSet nfa.nodes.size
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
inv : ∀ (i j : Fin nfa.nodes.size), i ∈ next → ↑j ∈ nfa.εStep ↑i → j ∈ next
cs : Data.Classes
target' : ℕ
hn : nfa[↑target] = Node.sparse cs target'
hc : c ∉ cs
h : (none, next, saveSlots) = (matched', next', saveSlots')
i : ℕ
⊢ i ∉ nfa.charStep (↑target) c
Theorem: Regex.VM.stepChar_spec.mem_next_iff
Proof state before:
nfa : NFA
c : Char
pos : String.Pos
next : SparseSet nfa.nodes.size
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
inv : ∀ (i j : Fin nfa.nodes.size), i ∈ next → ↑j ∈ nfa.εStep ↑i → j ∈ next
cs : Data.Classes
target' : ℕ
hn : nfa[↑target] = Node.sparse cs target'
hc : c ∉ cs
h : (none, next, saveSlots) = (matched', next', saveSlots')
i : ℕ
⊢ i ∉ nfa.charStep (↑target) c
Tactic:
simp [charStep, Node.charStep, hn, hc]
Proof state after:
no goals
Theorem: Regex.VM.stepChar_spec.mem_next_iff
Proof state before:
nfa : NFA
c : Char
pos : String.Pos
next : SparseSet nfa.nodes.size
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
inv : ∀ (i j : Fin nfa.nodes.size), i ∈ next → ↑j ∈ nfa.εStep ↑i → j ∈ next
hn₁ : ∀ (c' : Char) (target' : ℕ), nfa[target] = Node.char c' target' → False
hn₂ : ∀ (cs : Data.Classes) (target' : ℕ), nfa[target] = Node.sparse cs target' → False
h : (none, next, saveSlots) = (matched', next', saveSlots')
⊢ ∀ (j : Fin nfa.nodes.size), j ∈ next' ↔ j ∈ next ∨ ∃ i ∈ nfa.charStep (↑target) c, ↑j ∈ nfa.εClosure i
Tactic:
have : ∀ i, ¬i ∈ nfa.charStep target c := by
  intro i
  simp at hn₁ hn₂
  simp [charStep, Node.charStep]
Proof state after:
nfa : NFA
c : Char
pos : String.Pos
next : SparseSet nfa.nodes.size
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
inv : ∀ (i j : Fin nfa.nodes.size), i ∈ next → ↑j ∈ nfa.εStep ↑i → j ∈ next
hn₁ : ∀ (c' : Char) (target' : ℕ), nfa[target] = Node.char c' target' → False
hn₂ : ∀ (cs : Data.Classes) (target' : ℕ), nfa[target] = Node.sparse cs target' → False
h : (none, next, saveSlots) = (matched', next', saveSlots')
this : ∀ (i : ℕ), i ∉ nfa.charStep (↑target) c
⊢ ∀ (j : Fin nfa.nodes.size), j ∈ next' ↔ j ∈ next ∨ ∃ i ∈ nfa.charStep (↑target) c, ↑j ∈ nfa.εClosure i