Theorem: Regex.VM.eachStepChar_spec.mem_next_iff.go
Proof state before:
nfa : NFA
c : Char
pos : String.Pos
current : SparseSet nfa.nodes.size
i : ℕ
hle : i ≤ current.count
next : SparseSet nfa.nodes.size
saveSlots : Data.Vec (Array (Option String.Pos)) nfa.nodes.size
matched' : Option (Array (Option String.Pos))
next' : SparseSet nfa.nodes.size
saveSlots' : Data.Vec (Array (Option String.Pos)) nfa.nodes.size
inv₁ :
  ∀ (j : Fin nfa.nodes.size),
    j ∈ next ↔ ∃ i', ∃ (x : i' < current.count), i' < i ∧ ∃ i'' ∈ nfa.charStep (↑current[i']) c, ↑j ∈ nfa.εClosure i''
inv₂ : ∀ (i j : Fin nfa.nodes.size), i ∈ next → ↑j ∈ nfa.εStep ↑i → j ∈ next
hi : ¬i = current.count
hlt : i < current.count
result :
  Option (Array (Option String.Pos)) × SparseSet nfa.nodes.size × Data.Vec (Array (Option String.Pos)) nfa.nodes.size
matched'' : Option (Array (Option String.Pos))
next'' : SparseSet nfa.nodes.size
saveSlots'' : Data.Vec (Array (Option String.Pos)) nfa.nodes.size
hres : stepChar nfa c pos next saveSlots current[i] = (matched'', next'', saveSlots'')
h :
  (match (matched'', next'', saveSlots'').1 with
    | none =>
      eachStepChar.go nfa c pos current (i + 1) ⋯ (matched'', next'', saveSlots'').2.1
        (matched'', next'', saveSlots'').2.2
    | some val => (matched'', next'', saveSlots'')) =
    (matched', next', saveSlots')
⊢ ∃ istop,
    ∀ (j : Fin nfa.nodes.size),
      j ∈ next' ↔
        ∃ i', ∃ (x : i' < current.count), i' < istop ∧ ∃ i'' ∈ nfa.charStep (↑current[i']) c, ↑j ∈ nfa.εClosure i''
Tactic:
have inv₁' : ∀ j, j ∈ next'' ↔ ∃ i', ∃ _ : i' < current.count, i' < i + 1 ∧ ∃ i'' ∈ nfa.charStep current[i'] c, j.val ∈ nfa.εClosure i'' := by
  intro j
  have := stepChar_spec.mem_next_iff hres inv₂
  rw [this j, inv₁ j]
  apply Iff.intro
  . intro h
    cases h with
    | inl h =>
      have ⟨i', hlt', hlt'', hmem⟩ := h
      exact ⟨i', hlt', Nat.le.step hlt'', hmem⟩
    | inr hmem => exact ⟨i, hlt, Nat.lt_succ_self _, hmem⟩
  . intro h
    have ⟨i', hlt', hlt'', hmem⟩ := h
    cases Nat.lt_succ_iff_lt_or_eq.mp hlt'' with
    | inl hlt'' => exact .inl ⟨i', hlt', hlt'', hmem⟩
    | inr heq =>
      simp [heq] at hmem
      exact .inr hmem
Proof state after:
nfa : NFA
c : Char
pos : String.Pos
current : SparseSet nfa.nodes.size
i : ℕ
hle : i ≤ current.count
next : SparseSet nfa.nodes.size
saveSlots : Data.Vec (Array (Option String.Pos)) nfa.nodes.size
matched' : Option (Array (Option String.Pos))
next' : SparseSet nfa.nodes.size
saveSlots' : Data.Vec (Array (Option String.Pos)) nfa.nodes.size
inv₁ :
  ∀ (j : Fin nfa.nodes.size),
    j ∈ next ↔ ∃ i', ∃ (x : i' < current.count), i' < i ∧ ∃ i'' ∈ nfa.charStep (↑current[i']) c, ↑j ∈ nfa.εClosure i''
inv₂ : ∀ (i j : Fin nfa.nodes.size), i ∈ next → ↑j ∈ nfa.εStep ↑i → j ∈ next
hi : ¬i = current.count
hlt : i < current.count
result :
  Option (Array (Option String.Pos)) × SparseSet nfa.nodes.size × Data.Vec (Array (Option String.Pos)) nfa.nodes.size
matched'' : Option (Array (Option String.Pos))
next'' : SparseSet nfa.nodes.size
saveSlots'' : Data.Vec (Array (Option String.Pos)) nfa.nodes.size
hres : stepChar nfa c pos next saveSlots current[i] = (matched'', next'', saveSlots'')
h :
  (match (matched'', next'', saveSlots'').1 with
    | none =>
      eachStepChar.go nfa c pos current (i + 1) ⋯ (matched'', next'', saveSlots'').2.1
        (matched'', next'', saveSlots'').2.2
    | some val => (matched'', next'', saveSlots'')) =
    (matched', next', saveSlots')
inv₁' :
  ∀ (j : Fin nfa.nodes.size),
    j ∈ next'' ↔
      ∃ i', ∃ (x : i' < current.count), i' < i + 1 ∧ ∃ i'' ∈ nfa.charStep (↑current[i']) c, ↑j ∈ nfa.εClosure i''
⊢ ∃ istop,
    ∀ (j : Fin nfa.nodes.size),
      j ∈ next' ↔
        ∃ i', ∃ (x : i' < current.count), i' < istop ∧ ∃ i'' ∈ nfa.charStep (↑current[i']) c, ↑j ∈ nfa.εClosure i''