Package parser-all-thm: Properties of the whole stream parser

Information

Files

• Package tarball parser-all-thm-1.96.tgz
• Theory source file parser-all-thm.thy (included in the package tarball)

Theorems

⊦ ∀p s. length (parseStream p s) ≤ length s

⊦ ∀f p. parseStream (map f p) = map f ∘ parseStream p

⊦ ∀p e l.
    inverse p e ⇒ parseStream p (fromList (concat (map e l))) = fromList l

⊦ ∀p e x s.
    inverse p e ⇒ parseStream p (append (e x) s) = cons x (parseStream p s)

⊦ ∀p e s.
    strongInverse p e ⇒
    case toList (parseStream p s) of
      none → ⊤
    | some l → toList s = some (concat (map e l))

External Type Operators

• →
• bool
• Data
  • List
    • list
  • Option
    • option
  • Pair
    • ×
• Number
  • Natural
    • natural
• Parser
  • parser
  • Stream
    • stream

External Constants

• =
• select
• Data
  • Bool
    • ∀
    • ∧
    • ⇒
    • ∃
    • ∨
    • ¬
    • ⊥
    • ⊤
  • List
    • ::
    • @
    • []
    • concat
    • map
  • Option
    • case
    • none
    • some
  • Pair
    • ,
• Function
  • ∘
• Number
  • Natural
    • ≤
    • suc
    • zero
• Parser
  • destParser
  • inverse
  • map
  • parse
  • parseStream
  • strongInverse
  • Stream
    • append
    • cons
    • eof
    • error
    • fromList
    • isProperSuffix
    • isSuffix
    • length
    • map
    • toList

Assumptions

⊦ ⊤

⊦ length eof = 0

⊦ length error = 0

⊦ concat [] = []

⊦ toList error = none

⊦ ∀t. t ⇒ t

⊦ ∀n. 0 ≤ n

⊦ ∀n. n ≤ n

⊦ ⊥ ⇔ ∀p. p

⊦ toList eof = some []

⊦ ∀x. ¬isProperSuffix x x

⊦ (¬) = λp. p ⇒ ⊥

⊦ ∀t. (λx. t x) = t

⊦ (∀) = λp. p = λx. ⊤

⊦ ∀a. ¬(some a = none)

⊦ ∀t. ⊥ ⇒ t ⇔ ⊤

⊦ ∀t. ⊤ ⇒ t ⇔ t

⊦ ∀t. t ⇒ ⊤ ⇔ ⊤

⊦ ∀s. append [] s = s

⊦ ∀f. map f [] = []

⊦ ∀f. map f eof = eof

⊦ ∀f. map f error = error

⊦ ∀p. parseStream p eof = eof

⊦ ∀p. parseStream p error = error

⊦ ∀p. parse p eof = none

⊦ ∀p. parse p error = none

⊦ ∀l. fromList l = append l eof

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀b f. case b f none = b

⊦ ∀p x. p x ⇒ p ((select) p)

⊦ ∀x y. x = y ⇔ y = x

⊦ ∀a s. length (cons a s) = suc (length s)

⊦ ∀x. x = none ∨ ∃a. x = some a

⊦ (∧) = λp q. (λf. f p q) = λf. f ⊤ ⊤

⊦ (∃) = λp. ∀q. (∀x. p x ⇒ q) ⇒ q

⊦ ∀a a'. some a = some a' ⇔ a = a'

⊦ ∀m n. suc m ≤ suc n ⇔ m ≤ n

⊦ ∀h t. concat (h :: t) = h @ concat t

⊦ ∀x y. isSuffix x y ⇒ length x ≤ length y

⊦ ∀b f a. case b f (some a) = f a

⊦ ∀f g x. (f ∘ g) x = f (g x)

⊦ ∀l. l = [] ∨ ∃h t. l = h :: t

⊦ ∀f g. (∀x. f x = g x) ⇔ f = g

⊦ (∨) = λp q. ∀r. (p ⇒ r) ⇒ (q ⇒ r) ⇒ r

⊦ ∀s s'. isSuffix s s' ⇔ s = s' ∨ isProperSuffix s s'

⊦ ∀f. ∃fn. ∀a b. fn (a, b) = f a b

⊦ ∀p a s. parse p (cons a s) = destParser p a s

⊦ ∀h t s. append (h :: t) s = cons h (append t s)

⊦ ∀m n p. m ≤ n ∧ n ≤ p ⇒ m ≤ p

⊦ ∀x y z. append (x @ y) z = append x (append y z)

⊦ ∀a s.
    toList (cons a s) =
    case toList s of none → none | some l → some (a :: l)

⊦ ∀l s.
    toList (append l s) =
    case toList s of none → none | some ls → some (l @ ls)

⊦ ∀f h t. map f (h :: t) = f h :: map f t

⊦ ∀f a s. map f (cons a s) = cons (f a) (map f s)

⊦ ∀s a s'. isProperSuffix s (cons a s') ⇔ s = s' ∨ isProperSuffix s s'

⊦ ∀x. x = error ∨ x = eof ∨ ∃a₀ a₁. x = cons a₀ a₁

⊦ ∀p. (∀x. (∀y. isProperSuffix y x ⇒ p y) ⇒ p x) ⇒ ∀x. p x

⊦ ∀p q. (∀x. p x ∧ q x) ⇔ (∀x. p x) ∧ ∀x. q x

⊦ ∀p. p [] ∧ (∀h t. p t ⇒ p (h :: t)) ⇒ ∀l. p l

⊦ ∀a b a' b'. (a, b) = (a', b') ⇔ a = a' ∧ b = b'

⊦ ∀p e. inverse p e ⇔ ∀x s. parse p (append (e x) s) = some (x, s)

⊦ ∀p s.
    parse p s = none ∨
    ∃b s'. parse p s = some (b, s') ∧ isProperSuffix s' s

⊦ ∀p a s.
    destParser p a s = none ∨
    ∃b s'. destParser p a s = some (b, s') ∧ isSuffix s' s

⊦ ∀p e.
    strongInverse p e ⇔
    inverse p e ∧ ∀s x s'. parse p s = some (x, s') ⇒ s = append (e x) s'

⊦ ∀p a s.
    parseStream p (cons a s) =
    case destParser p a s of
      none → error
    | some (b, s') → cons b (parseStream p s')

⊦ ∀f p a s.
    destParser (map f p) a s =
    case destParser p a s of none → none | some (b, s') → some (f b, s')

OpenTheory package document generated by opentheory 1.3 (release 20141119)