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

Information

Files

• Package tarball parser-comb-thm-1.100.tgz
• Theory source file parser-comb-thm.thy (included in the package tarball)

Theorems

⊦ ∀p. invariant (dest p)

⊦ ∀f. invariant (token.prs f)

⊦ ∀p. invariant (sequence.prs p)

⊦ inverse any (λx. x :: [])

⊦ strongInverse any (λx. x :: [])

⊦ ∀xs. apply none xs = none

⊦ ∀p₁ p₂. invariant (orelse.prs p₁ p₂)

⊦ ∀p f. invariant (mapPartial.prs p f)

⊦ ∀p. filter any p = some p

⊦ ∀f. mapPartial any f = token f

⊦ ∀f. dest (token f) = token.prs f

⊦ ∀p. dest (sequence p) = sequence.prs p

⊦ ∀x xs. dest none x xs = none

⊦ ∀p₁ p₂. dest (orelse p₁ p₂) = orelse.prs p₁ p₂

⊦ ∀p f. dest (mapPartial p f) = mapPartial.prs p f

⊦ apply any = case none none (λx xs. some (x, xs))

⊦ ∀x xs. dest any x xs = some (x, xs)

⊦ ∀p₁ p₂. dest p₁ = dest p₂ ⇔ p₁ = p₂

⊦ ∀p.
    apply (some p) =
    case none none (λx xs. if p x then some (x, xs) else none)

⊦ ∀f.
    apply (token f) =
    case none none (λx xs. case f x of none → none | some y → some (y, xs))

⊦ ∀p x xs. dest (some p) x xs = if p x then some (x, xs) else none

⊦ ∀f e. (∀x. f (e x) = some x) ⇒ inverse (token f) (λx. e x :: [])

⊦ ∀p₁ p₂ xs.
    apply (orelse p₁ p₂) xs =
    case apply p₁ xs of none → apply p₂ xs | some yys → some yys

⊦ ∀p x xs y ys. dest p x xs = some (y, ys) ⇒ isSuffix ys xs

⊦ ∀p xs.
    apply p xs = none ∨
    ∃y ys. apply p xs = some (y, ys) ∧ isProperSuffix ys xs

⊦ ∀p f g e.
    inverse p e ∧ (∀x. f (g x) = x) ⇒ inverse (map p f) (λx. e (g x))

⊦ ∀p xs.
    apply (sequence p) xs =
    case apply p xs of none → none | some (y, ys) → apply y ys

⊦ ∀p f g e.
    inverse p e ∧ (∀x. f (g x) = some x) ⇒
    inverse (mapPartial p f) (λx. e (g x))

⊦ ∀p x xs.
    dest p x xs = none ∨ ∃y ys. dest p x xs = some (y, ys) ∧ isSuffix ys xs

⊦ ∀p x xs.
    invariant p ⇒
    p x xs = none ∨ ∃y ys. p x xs = some (y, ys) ∧ isSuffix ys xs

⊦ (∀p. apply p error = none) ∧ (∀p. apply p eof = none) ∧
  ∀p x xs. apply p (cons x xs) = dest p x xs

⊦ ∀p f xs.
    apply (map p f) xs =
    case apply p xs of none → none | some (y, ys) → some (f y, ys)

⊦ ∀p₁ p₂ e₁ e₂.
    inverse p₁ e₁ ∧ inverse p₂ e₂ ⇒
    inverse (pair p₁ p₂) (λ(x₁, x₂). e₁ x₁ @ e₂ x₂)

⊦ ∀p₁ p₂ e₁ e₂.
    strongInverse p₁ e₁ ∧ strongInverse p₂ e₂ ⇒
    strongInverse (pair p₁ p₂) (λ(x₁, x₂). e₁ x₁ @ e₂ x₂)

⊦ ∀p f xs.
    apply (filter p f) xs =
    case apply p xs of
      none → none
    | some (y, ys) → if f y then some (y, ys) else none

⊦ ∀p f x xs.
    dest (map p f) x xs =
    case dest p x xs of none → none | some (y, ys) → some (f y, ys)

⊦ ∀p f xs.
    apply (mapPartial p f) xs =
    case apply p xs of
      none → none
    | some (y, ys) → case f y of none → none | some z → some (z, ys)

⊦ ∀p f x xs.
    dest (filter p f) x xs =
    case dest p x xs of
      none → none
    | some (y, ys) → if f y then some (y, ys) else none

⊦ ∀f e.
    (∀x. f (e x) = some x) ∧
    (∀y₁ y₂ x. f y₁ = some x ∧ f y₂ = some x ⇒ y₁ = y₂) ⇒
    strongInverse (token f) (λx. e x :: [])

⊦ ∀p f g e.
    strongInverse p e ∧ (∀x. f (g x) = x) ∧
    (∀y₁ y₂ x. f y₁ = x ∧ f y₂ = x ⇒ y₁ = y₂) ⇒
    strongInverse (map p f) (λx. e (g x))

⊦ ∀p₁ p₂ xs.
    apply (pair p₁ p₂) xs =
    case apply p₁ xs of
      none → none
    | some (y, ys) →
        case apply p₂ ys of none → none | some (z, zs) → some ((y, z), zs)

⊦ ∀p f g e.
    strongInverse p e ∧ (∀x. f (g x) = some x) ∧
    (∀y₁ y₂ x. f y₁ = some x ∧ f y₂ = some x ⇒ y₁ = y₂) ⇒
    strongInverse (mapPartial p f) (λx. e (g x))

⊦ ∀p₁ p₂ x xs.
    dest (pair p₁ p₂) x xs =
    case dest p₁ x xs of
      none → none
    | some (y, ys) →
        case apply p₂ ys of none → none | some (z, zs) → some ((y, z), zs)

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
    • ,
• Function
  • Function.const
• Parser
  • any
  • apply
  • dest
  • filter
  • invariant
  • inverse
  • map
  • mapPartial
    • mapPartial.prs
  • mk
  • none
  • orelse
    • orelse.prs
  • pair
  • sequence
    • sequence.prs
  • some
  • strongInverse
  • token
    • token.prs
  • Stream
    • append
    • case
    • cons
    • eof
    • error
    • isProperSuffix
    • isSuffix

Assumptions

⊦ ⊤

⊦ ∀t. t ⇒ t

⊦ ∀xs. isSuffix xs xs

⊦ ⊥ ⇔ ∀p. p

⊦ any = some (Function.const ⊤)

⊦ none = token (Function.const none)

⊦ (¬) = 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 ⇔ ⊤

⊦ ∀a. mk (dest a) = a

⊦ ∀p. apply p eof = none

⊦ ∀p. apply p error = none

⊦ ∀f. token f = mk (token.prs f)

⊦ ∀p. sequence p = mk (sequence.prs p)

⊦ ∀x y. Function.const x y = x

⊦ (⇒) = 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)

⊦ ∀f y. (let x ← y in f x) = f y

⊦ ∀x. ∃a b. x = (a, b)

⊦ ∀r. invariant r ⇔ dest (mk r) = r

⊦ ∀x y. x = y ⇔ y = x

⊦ ∀xs ys. isProperSuffix xs ys ⇒ isSuffix xs ys

⊦ ∀x. x = none ∨ ∃a. x = some a

⊦ ∀p₁ p₂. orelse p₁ p₂ = mk (orelse.prs p₁ p₂)

⊦ ∀p f. mapPartial p f = mk (mapPartial.prs p f)

⊦ (∧) = λp q. (λf. f p q) = λf. f ⊤ ⊤

⊦ (∃) = λp. ∀q. (∀x. p x ⇒ q) ⇒ q

⊦ ∀a b. some a = some b ⇔ a = b

⊦ ∀p. some p = token (λx. if p x then some x else none)

⊦ ∀b f a. case b f (some a) = f a

⊦ ∀p f. map p f = mapPartial p (λx. some (f x))

⊦ ∀xs y ys. isProperSuffix xs (cons y ys) ⇔ isSuffix xs ys

⊦ ∀f g. (∀x. f x = g x) ⇔ f = g

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

⊦ ∀f. ∃fn. ∀a b. fn (a, b) = f a b

⊦ ∀p x xs. apply p (cons x xs) = dest p x xs

⊦ ∀xs ys zs. append (xs @ ys) zs = append xs (append ys zs)

⊦ ∀xs ys zs. isSuffix xs ys ∧ isSuffix ys zs ⇒ isSuffix xs zs

⊦ ∀p f. filter p f = mapPartial p (λx. if f x then some x else none)

⊦ ∀p₁ p₂. pair p₁ p₂ = sequence (map p₁ (λx. map p₂ (λy. (x, y))))

⊦ ∀xs. xs = error ∨ xs = eof ∨ ∃x xt. xs = cons x xt

⊦ ∀f x xs.
    token.prs f x xs = case f x of none → none | some y → some (y, xs)

⊦ ∀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'

⊦ ∀x xs y ys. cons x xs = cons y ys ⇔ x = y ∧ xs = ys

⊦ ∀p e. inverse p e ⇔ ∀x xs. apply p (append (e x) xs) = some (x, xs)

⊦ ∀p c x y. p (if c then x else y) ⇔ (c ⇒ p x) ∧ (¬c ⇒ p y)

⊦ (∀xs. append [] xs = xs) ∧
  ∀h t xs. append (h :: t) xs = cons h (append t xs)

⊦ ∀p₁ p₂ x xs.
    orelse.prs p₁ p₂ x xs =
    case dest p₁ x xs of none → dest p₂ x xs | some yys → some yys

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

⊦ ∀p x xs.
    sequence.prs p x xs =
    case dest p x xs of none → none | some (q, ys) → apply q ys

⊦ ∀p e.
    strongInverse p e ⇔
    inverse p e ∧
    ∀xs y ys. apply p xs = some (y, ys) ⇒ xs = append (e y) ys

⊦ ∀p f x xs.
    mapPartial.prs p f x xs =
    case dest p x xs of
      none → none
    | some (y, ys) → case f y of none → none | some z → some (z, ys)

⊦ (∀e b f. case e b f error = e) ∧ (∀e b f. case e b f eof = b) ∧
  ∀e b f x xs. case e b f (cons x xs) = f x xs

OpenTheory package document generated by opentheory 1.3 (release 20150429)