Package parser: Stream parsers

Information

name	parser
version	1.100
description	Stream parsers
author	Joe Leslie-Hurd <joe@gilith.com>
license	MIT
requires	bool function list natural option pair probability relation
show	Data.Bool Data.List Data.Option Data.Pair Function Number.Natural Parser Parser.Stream Probability.Random Relation

Files

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

Defined Type Operators

Parser
- parser
- Stream
  - stream

Defined Constants

Parser
- destParser
- inverse
- isParser
- map
- mapToken
  - mapToken.pf
- mkParser
- parse
- parseAll
  - parseAll.pa
- parseNone
  - parseNone.pn
- parseOption
- parsePair
  - parsePair.pbc
- parseSome
- parseStream
- partialMap
  - partialMap.pf
- strongInverse
- Stream
  - append
  - case
  - cons
  - eof
  - error
  - fromList
  - fromRandom
  - isProperSuffix
  - isSuffix
  - length
  - map
  - toList

Theorems

⊦ isParser parseAll.pa

⊦ isParser parseNone.pn

⊦ wellFounded isProperSuffix

⊦ ¬(error = eof)

⊦ parseAll = mkParser parseAll.pa

⊦ parseNone = mkParser parseNone.pn

⊦ length eof = 0

⊦ length error = 0

⊦ toList error = none

⊦ destParser parseNone = parseNone.pn

⊦ ∀x. isSuffix x x

⊦ ∀p. isParser (destParser p)

⊦ toList eof = some []

⊦ ∀s. ¬isProperSuffix s eof

⊦ ∀s. ¬isProperSuffix s error

⊦ ∀x. ¬isProperSuffix x x

⊦ inverse parseAll (λa. a :: [])

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

⊦ ∀s. append [] s = s

⊦ ∀s. parse parseNone s = none

⊦ ∀f. map f eof = eof

⊦ ∀f. map f error = error

⊦ ∀p. parseStream p eof = eof

⊦ ∀p. parseStream p error = error

⊦ ∀a. mkParser (destParser a) = a

⊦ ∀p. parse p eof = none

⊦ ∀p. parse p error = none

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

⊦ ∀f p. isParser (partialMap.pf f p)

⊦ ∀l. fromList l = append l eof

⊦ ∀l. length (fromList l) = length l

⊦ ∀l. toList (fromList l) = some l

⊦ ∀f. parseOption f = partialMap f parseAll

⊦ ∀a s. parseNone.pn a s = none

⊦ ∀a0' a1'. ¬(eof = cons a0' a1')

⊦ ∀a0' a1'. ¬(error = cons a0' a1')

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

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

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

⊦ ∀a s. parseAll.pa a s = some (a, s)

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

⊦ ∀s. case toList s of none → ⊤ | some l → length l = length s

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

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

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

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

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

⊦ ∀e b f. case e b f eof = b

⊦ ∀e b f. case e b f error = e

⊦ ∀x y. isProperSuffix x y ⇒ length x < length y

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

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

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

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

⊦ ∀l s. length (append l s) = length l + length s

⊦ ∀f s. toList (map f s) = map (map f) (toList s)

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

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

⊦ ∀f g p. mapToken f g p = mkParser (mapToken.pf f g p)

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

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

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

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

⊦ ∀x y z. isSuffix x y ∧ isSuffix y z ⇒ isSuffix x 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 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.
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. (∀x. (∀y. isProperSuffix y x ⇒ p y) ⇒ p x) ⇒ ∀x. p x

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

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

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

⊦ ∀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

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

⊦ ∀P. P error ∧ P eof ∧ (∀a₀ a₁. P a₁ ⇒ P (cons a₀ a₁)) ⇒ ∀x. P x

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

⊦ ∀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))

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

⊦ ∀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

⊦ ∀f₀ f₁ f₂.
    ∃fn.
      fn error = f₀ ∧ fn eof = f₁ ∧
      ∀a₀ a₁. fn (cons a₀ a₁) = f₂ a₀ a₁ (fn a₁)

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

⊦ ∀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')

⊦ ∀h.
(∀f g s. (∀s'. isProperSuffix s' s ⇒ f s' = g s') ⇒ h f s = h g s) ⇒
∃f. ∀s. f s = h f 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')

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

⊦ ∀d r.
    fromRandom d r =
    let (l, r') ← Geometric.fromRandom d r in
    let (b, r'') ← bit r' in
    (append l (if b then error else eof), r'')

⊦ ∀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))

⊦ ∀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'')

External Type Operators

→
bool
Data
- List
  - list
- Option
  - option
- Pair
  - ×
Number
- Natural
  - natural
Probability
- Random
  - random

External Constants

=
select
Data
- Bool
  - ∀
  - ∧
  - ⇒
  - ∃
  - ∃!
  - ∨
  - ¬
  - cond
  - ⊥
  - ⊤
- List
  - ::
  - @
  - []
  - concat
  - length
  - map
  - Geometric
    - Geometric.fromRandom
- Option
  - case
  - map
  - none
  - some
- Pair
  - ,
Function
- id
- ∘
Number
- Natural
  - *
  - +
  - <
  - ≤
  - ↑
  - bit0
  - bit1
  - even
  - suc
  - zero
Probability
- Random
  - bit
Relation
- irreflexive
- measure
- subrelation
- wellFounded

Assumptions

⊦ ⊤

⊦ ¬⊥ ⇔ ⊤

⊦ ¬⊤ ⇔ ⊥

⊦ length [] = 0

⊦ bit0 0 = 0

⊦ concat [] = []

⊦ ∀t. t ⇒ t

⊦ ∀n. 0 ≤ n

⊦ ∀n. n ≤ n

⊦ ∀m. wellFounded (measure m)

⊦ ⊥ ⇔ ∀p. p

⊦ ∀x. id x = x

⊦ (¬) = λp. p ⇒ ⊥

⊦ (∃) = λp. p ((select) p)

⊦ ∀a. ∃!x. x = a

⊦ ∀t. (∀x. t) ⇔ t

⊦ ∀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 ⇔ t

⊦ ∀t. ⊤ ∨ t ⇔ ⊤

⊦ ∀t. t ∨ ⊥ ⇔ t

⊦ ∀t. t ∨ ⊤ ⇔ ⊤

⊦ ∀n. ¬(suc n = 0)

⊦ ∀n. 0 + n = n

⊦ ∀m. m + 0 = m

⊦ ∀l. [] @ l = l

⊦ ∀f. map f [] = []

⊦ ∀f. map f none = none

⊦ ∀r. wellFounded r ⇒ irreflexive r

⊦ ∀t. (⊥ ⇔ t) ⇔ ¬t

⊦ ∀t. (t ⇔ ⊥) ⇔ ¬t

⊦ ∀t. t ⇒ ⊥ ⇔ ¬t

⊦ ∀n. even (2 * n)

⊦ ∀n. bit1 n = suc (bit0 n)

⊦ ∀m. m ↑ 0 = 1

⊦ ∀x. case none some x = x

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀t. (t ⇔ ⊤) ∨ (t ⇔ ⊥)

⊦ ∀n. even (suc n) ⇔ ¬even n

⊦ ∀m. m ≤ 0 ⇔ m = 0

⊦ ∀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)

⊦ ∀n. bit0 (suc n) = suc (suc (bit0 n))

⊦ ∀f y. (let x ← y in f x) = f y

⊦ ∀x. ∃a b. x = (a, b)

⊦ ∀x y. x = y ⇔ y = x

⊦ ∀t₁ t₂. t₁ ∨ t₂ ⇔ t₂ ∨ t₁

⊦ ∀m n. m < n ⇒ m ≤ n

⊦ ∀r x. irreflexive r ⇒ ¬r x x

⊦ ∀n. 2 * n = n + n

⊦ ∀h t. length (h :: t) = suc (length t)

⊦ ∀m n. ¬(m < n ∧ n ≤ m)

⊦ ∀m n. ¬(m ≤ n ∧ n < m)

⊦ ∀m n. m < suc n ⇔ m ≤ n

⊦ ∀m n. suc m ≤ n ⇔ m < n

⊦ ∀x. x = none ∨ ∃a. x = some a

⊦ ∀f a. map f (some a) = some (f a)

⊦ (∧) = λp q. (λf. f p q) = λf. f ⊤ ⊤

⊦ ∀p. ¬(∀x. p x) ⇔ ∃x. ¬p x

⊦ ∀p. ¬(∃x. p x) ⇔ ∀x. ¬p x

⊦ (∃) = λp. ∀q. (∀x. p x ⇒ q) ⇒ q

⊦ ∀a a'. some a = some a' ⇔ a = a'

⊦ ∀t₁ t₂. ¬(t₁ ⇒ t₂) ⇔ t₁ ∧ ¬t₂

⊦ ∀m n. m + suc n = suc (m + n)

⊦ ∀m n. suc m + n = suc (m + n)

⊦ ∀m n. suc m = suc n ⇔ m = n

⊦ ∀m n. suc m ≤ suc n ⇔ m ≤ n

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

⊦ ∀r s. subrelation r s ∧ wellFounded s ⇒ wellFounded r

⊦ ∀b f a. case b f (some a) = f a

⊦ ∀f g x. (f ∘ g) x = f (g x)

⊦ ∀t₁ t₂. ¬(t₁ ∧ t₂) ⇔ ¬t₁ ∨ ¬t₂

⊦ ∀m n. even (m * n) ⇔ even m ∨ even n

⊦ ∀m n. even (m + n) ⇔ even m ⇔ even n

⊦ ∀m n. m ↑ suc n = m * m ↑ n

⊦ ∀l. l = [] ∨ ∃h t. l = h :: t

⊦ ∀f g. (∀x. f x = g x) ⇔ f = g

⊦ ∀p a. (∃x. a = x ∧ p x) ⇔ p a

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

⊦ ∀m n. m ≤ n ⇔ m < n ∨ m = n

⊦ ∀m n. m ≤ n ∧ n ≤ m ⇔ m = n

⊦ ∀f. ∃fn. ∀a b. fn (a, b) = f a b

⊦ ∀m x y. measure m x y ⇔ m x < m y

⊦ ∀p q. (∀x. p ∨ q x) ⇔ p ∨ ∀x. q x

⊦ ∀p q. (∃x. p ∧ q x) ⇔ p ∧ ∃x. q x

⊦ ∀p q. p ∧ (∀x. q x) ⇔ ∀x. p ∧ q x

⊦ ∀p q. p ∧ (∃x. q x) ⇔ ∃x. p ∧ q x

⊦ ∀p q. p ∨ (∀x. q x) ⇔ ∀x. p ∨ q x

⊦ ∀p q. p ∨ (∃x. q x) ⇔ ∃x. p ∨ q x

⊦ ∀p q. (∃x. p x) ∧ q ⇔ ∃x. p x ∧ q

⊦ ∀p q. (∃x. p x) ∨ q ⇔ ∃x. p x ∨ q

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

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

⊦ ∀m n p. m * (n * p) = m * n * p

⊦ ∀p m n. m + p = n + p ⇔ m = n

⊦ ∀m n p. m ≤ n ∧ n ≤ p ⇒ m ≤ p

⊦ ∀l h t. (h :: t) @ l = h :: t @ l

⊦ ∀p x. (∀y. p y ⇔ y = x) ⇒ (select) p = x

⊦ ∀r. (∀x. ∃y. r x y) ⇔ ∃f. ∀x. r x (f x)

⊦ ∀m n. m ≤ suc n ⇔ m = suc n ∨ m ≤ n

⊦ ∀m n. m * n = 0 ⇔ m = 0 ∨ n = 0

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

⊦ ∀p. p 0 ∧ (∀n. p n ⇒ p (suc n)) ⇒ ∀n. p n

⊦ ∀m n. m ↑ n = 0 ⇔ m = 0 ∧ ¬(n = 0)

⊦ ∀r s. subrelation r s ⇔ ∀x y. r x y ⇒ s x y

⊦ (∃!) = λp. (∃) p ∧ ∀x y. p x ∧ p y ⇒ x = y

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

⊦ ∀p q. (∀x. p x ⇒ q x) ⇒ (∀x. p x) ⇒ ∀x. q x

⊦ ∀p q. (∀x. p x ⇒ q x) ⇒ (∃x. p x) ⇒ ∃x. q x

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

⊦ ∀e f. ∃!fn. fn 0 = e ∧ ∀n. fn (suc n) = f (fn n) n

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

⊦ ∀m n p. m * n = m * p ⇔ m = 0 ∨ n = p

⊦ ∀m n p. m * n ≤ m * p ⇔ m = 0 ∨ n ≤ p

⊦ ∀m n p. m * n < m * p ⇔ ¬(m = 0) ∧ n < p

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

⊦ ∀p₁ p₂ q₁ q₂. (p₁ ⇒ p₂) ∧ (q₁ ⇒ q₂) ⇒ p₁ ∧ q₁ ⇒ p₂ ∧ q₂

⊦ ∀p₁ p₂ q₁ q₂. (p₁ ⇒ p₂) ∧ (q₁ ⇒ q₂) ⇒ p₁ ∨ q₁ ⇒ p₂ ∨ q₂

⊦ ∀p₁ p₂ q₁ q₂. (p₂ ⇒ p₁) ∧ (q₁ ⇒ q₂) ⇒ (p₁ ⇒ q₁) ⇒ p₂ ⇒ q₂

⊦ ∀p. (∀x. ∃!y. p x y) ⇔ ∃f. ∀x y. p x y ⇔ f x = y

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

⊦ ∀b f. ∃fn. fn [] = b ∧ ∀h t. fn (h :: t) = f h t (fn t)

⊦ ∀p. (∃!x. p x) ⇔ (∃x. p x) ∧ ∀x x'. p x ∧ p x' ⇒ x = x'

⊦ ∀r. wellFounded r ⇔ ∀p. (∀x. (∀y. r y x ⇒ p y) ⇒ p x) ⇒ ∀x. p x