Package parser-comb-thm: Properties of stream parser combinators

Information

Files

• Package tarball parser-comb-thm-1.87.tgz
• Theory source file parser-comb-thm.thy (included in the package tarball)

Theorems

⊦ isParser parseAll.pa

⊦ isParser parseNone.pn

⊦ destParser parseNone = parseNone.pn

⊦ ∀p. isParser (destParser p)

⊦ inverse parseAll (λa. a :: [])

⊦ strongInverse parseAll (λa. a :: [])

⊦ ∀s. parse parseNone s = none

⊦ ∀pb pc. isParser (parsePair.pbc pb pc)

⊦ ∀f p. isParser (partialMap.pf f p)

⊦ ∀pb pc. destParser (parsePair pb pc) = parsePair.pbc pb pc

⊦ ∀f p. destParser (partialMap f p) = partialMap.pf f p

⊦ parse parseAll = case none none (λa s. some (a, s))

⊦ ∀x s. parse parseAll (cons x s) = some (x, s)

⊦ ∀p.
    parse (parseSome p) =
    case none none (λa s. if p a then some (a, s) else none)

⊦ ∀f.
    parse (parseOption f) =
    case none none (λa s. case f a of none → none | some b → some (b, s))

⊦ ∀p a s. destParser (parseSome p) a s = if p a then some (a, s) else none

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

⊦ ∀f e. (∀b. f (e b) = some b) ⇒ inverse (parseOption f) (λb. e b :: [])

⊦ ∀p a s b s'. destParser p a s = some (b, s') ⇒ isSuffix s' s

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

⊦ ∀f p g e.
    inverse p e ∧ (∀b. f (g b) = b) ⇒ inverse (map f p) (λc. e (g c))

⊦ ∀f p g e.
    inverse p e ∧ (∀b. f (g b) = some b) ⇒
    inverse (partialMap f p) (λc. e (g c))

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

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

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

⊦ ∀pb pc eb ec.
    inverse pb eb ∧ inverse pc ec ⇒
    inverse (parsePair pb pc) (λ(b, c). eb b @ ec c)

⊦ ∀pb pc eb ec.
    strongInverse pb eb ∧ strongInverse pc ec ⇒
    strongInverse (parsePair pb pc) (λ(b, c). eb b @ ec c)

⊦ ∀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')

⊦ ∀f p s.
    parse (partialMap f p) s =
    case parse p s of
      none → none
    | some (b, s') → case f b of none → none | some c → some (c, s')

⊦ ∀f e.
    (∀b. f (e b) = some b) ∧
    (∀a₁ a₂ b. f a₁ = some b ∧ f a₂ = some b ⇒ a₁ = a₂) ⇒
    strongInverse (parseOption f) (λb. e b :: [])

⊦ ∀f p g e.
    strongInverse p e ∧ (∀b. f (g b) = b) ∧
    (∀b₁ b₂ c. f b₁ = c ∧ f b₂ = c ⇒ b₁ = b₂) ⇒
    strongInverse (map f p) (λc. e (g c))

⊦ ∀pb pc s.
    parse (parsePair pb pc) s =
    case parse pb s of
      none → none
    | some (b, s') →
        case parse pc s' of
          none → none
        | some (c, s'') → some ((b, c), s'')

⊦ ∀f p g e.
    strongInverse p e ∧ (∀b. f (g b) = some b) ∧
    (∀b₁ b₂ c. f b₁ = some c ∧ f b₂ = some c ⇒ b₁ = b₂) ⇒
    strongInverse (partialMap f p) (λc. e (g c))

External Type Operators

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

External Constants

• =
• select
• Data
  • Bool
    • ∀
    • ∧
    • ⇒
    • ∃
    • ∨
    • ¬
    • cond
    • ⊥
    • ⊤
  • List
    • ::
    • @
    • []
  • Option
    • case
    • none
    • some
  • Pair
    • ,
• Parser
  • destParser
  • inverse
  • isParser
  • map
  • mkParser
  • parse
  • parseAll
    • parseAll.pa
  • parseNone
    • parseNone.pn
  • parseOption
  • parsePair
    • parsePair.pbc
  • parseSome
  • partialMap
    • partialMap.pf
  • strongInverse
  • Stream
    • append
    • case
    • cons
    • eof
    • error
    • isProperSuffix
    • isSuffix

Assumptions

⊦ ⊤

⊦ parseAll = mkParser parseAll.pa

⊦ parseNone = mkParser parseNone.pn

⊦ ¬⊥ ⇔ ⊤

⊦ ¬⊤ ⇔ ⊥

⊦ ∀t. t ⇒ t

⊦ ∀x. isSuffix x x

⊦ ⊥ ⇔ ∀p. p

⊦ (¬) = λp. p ⇒ ⊥

⊦ ∀t. (∀x. t) ⇔ t

⊦ ∀t. (λx. t x) = t

⊦ (∀) = λp. p = λx. ⊤

⊦ ∀a. ¬(some a = none)

⊦ ∀t. ¬¬t ⇔ t

⊦ ∀t. (⊤ ⇔ t) ⇔ t

⊦ ∀t. ⊥ ∧ t ⇔ ⊥

⊦ ∀t. ⊤ ∧ t ⇔ t

⊦ ∀t. t ∧ ⊤ ⇔ t

⊦ ∀t. ⊥ ⇒ t ⇔ ⊤

⊦ ∀t. ⊤ ⇒ t ⇔ t

⊦ ∀t. t ⇒ ⊤ ⇔ ⊤

⊦ ∀t. ⊥ ∨ t ⇔ t

⊦ ∀t. ⊤ ∨ t ⇔ ⊤

⊦ ∀t. t ∨ ⊤ ⇔ ⊤

⊦ ∀s. append [] s = s

⊦ ∀a. mkParser (destParser a) = a

⊦ ∀p. parse p eof = none

⊦ ∀p. parse p error = none

⊦ ∀t. (⊥ ⇔ t) ⇔ ¬t

⊦ ∀t. t ⇒ ⊥ ⇔ ¬t

⊦ ∀f. parseOption f = partialMap f parseAll

⊦ ∀a s. parseNone.pn a s = none

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀t. (t ⇔ ⊤) ∨ (t ⇔ ⊥)

⊦ ∀t₁ t₂. (if ⊥ then t₁ else t₂) = t₂

⊦ ∀t₁ t₂. (if ⊤ then t₁ else t₂) = t₁

⊦ ∀b f. case b f none = b

⊦ ∀p x. p x ⇒ p ((select) p)

⊦ ∀x. ∃a b. x = (a, b)

⊦ ∀r. isParser r ⇔ destParser (mkParser r) = r

⊦ ∀x y. x = y ⇔ y = x

⊦ ∀t₁ t₂. t₁ ∨ t₂ ⇔ t₂ ∨ t₁

⊦ ∀x y. isProperSuffix x y ⇒ isSuffix x y

⊦ ∀a s. parseAll.pa a s = some (a, s)

⊦ ∀x. x = none ∨ ∃a. x = some a

⊦ ∀pb pc. parsePair pb pc = mkParser (parsePair.pbc pb pc)

⊦ ∀f p. partialMap f p = mkParser (partialMap.pf f p)

⊦ (∧) = λp q. (λf. f p q) = λf. f ⊤ ⊤

⊦ ∀e b f. case e b f eof = b

⊦ ∀e b f. case e b f error = e

⊦ (∃) = λp. ∀q. (∀x. p x ⇒ q) ⇒ q

⊦ ∀a a'. some a = some a' ⇔ a = a'

⊦ ∀p. parseSome p = parseOption (λa. if p a then some a else none)

⊦ ∀b f a. case b f (some a) = f a

⊦ ∀f p. map f p = partialMap (λb. some (f b)) p

⊦ ∀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)

⊦ ∀t₁ t₂ t₃. (t₁ ∨ t₂) ∨ t₃ ⇔ t₁ ∨ t₂ ∨ t₃

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

⊦ ∀x y z. isSuffix x y ∧ isSuffix y z ⇒ isSuffix x z

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

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

⊦ ∀e b f a s. case e b f (cons a s) = f a s

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

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

⊦ ∀a₀ a₁ a0' a1'. cons a₀ a₁ = cons a0' a1' ⇔ a₀ = a0' ∧ a₁ = a1'

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

⊦ ∀p.
    isParser p ⇔
    ∀x xs. case p x xs of none → ⊤ | some (y, xs') → isSuffix xs' xs

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

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

⊦ ∀pb pc a s.
    parsePair.pbc pb pc a s =
    case destParser pb a s of
      none → none
    | some (b, s') →
        case parse pc s' of
          none → none
        | some (c, s'') → some ((b, c), s'')

OpenTheory package document generated by opentheory 1.2 (release 20140721)