Package char: Unicode characters

Information

namechar
version1.145
descriptionUnicode characters
authorJoe Leslie-Hurd <joe@gilith.com>
licenseMIT
homepagehttp://opentheory.gilith.com/?pkg=char
hol-light-int-filehol-light.int
hol-light-thm-filehol-light.art
haskell-nameopentheory-unicode
haskell-categoryText
haskell-int-filehaskell.int
haskell-src-filehaskell.art
haskell-test-filehaskell-test.art
haskell-equality-type"Data.Char.char"
haskell-arbitrary-type"Data.Char.char"
checksum89d95d8a6246be390595fd84777cfa6f5e7d6ac5
requiresbase
byte
natural-bits
parser
probability
showData.Bool
Data.Byte
Data.Char
Data.List
Data.Option
Data.Pair
Data.Sum
Function
Number.Natural
Parser
Parser.Stream
Probability.Random
Set

Files

Defined Type Operator

Defined Constants

Theorems

surjective random

finite universe

inverse UTF8.parseChar UTF8.encodeChar

strongInverse UTF8.parseChar UTF8.encodeChar

plane = destPlane dest

position = destPosition dest

UTF8.parseNatural = orelse UTF8.parseAscii UTF8.parseMultibyte

a. mk (dest a) = a

b. UTF8.reencode (UTF8.decode b) = b

c. r. random r = c

c. plane c = destPlane (dest c)

c. position c = destPosition (dest c)

c. 1 length (UTF8.encodeChar c)

c. UTF8.reencodeChar (right c) = UTF8.encodeChar c

n. length (UTF8.encodeAscii n) = 1

b. length (UTF8.decode b) length b

b. UTF8.reencode (map left b) = b

c. length c length (UTF8.encode c)

c. 1 length (UTF8.reencodeChar c)

c. length c length (UTF8.reencode c)

b. UTF8.reencodeChar (left b) = b :: []

n. UTF8.encodeAscii n = fromNatural n :: []

n. length (UTF8.encodeChar.encode2 n) = 2

n. length (UTF8.encodeChar.encode3 n) = 3

c. UTF8.encode c = concat (map UTF8.encodeChar c)

c. UTF8.decode (UTF8.encode c) = map right c

c. UTF8.reencode (map right c) = UTF8.encode c

c. UTF8.reencode c = concat (map UTF8.reencodeChar c)

UTF8.parse = orelse (map UTF8.parseChar right) (map any left)

r. invariant r dest (mk r) = r

n. length (UTF8.encodeChar.encode4 n) = 4

c. n. invariant n c = mk n

finite { n. n | invariant n }

c. plane c < 17

b. UTF8.decode b = fst (toList (parse UTF8.parse (fromList b)))

n b. apply UTF8.parseAscii (append (UTF8.encodeChar.encode2 n) b) = none

n b. apply UTF8.parseAscii (append (UTF8.encodeChar.encode3 n) b) = none

n b. apply UTF8.parseAscii (append (UTF8.encodeChar.encode4 n) b) = none

UTF8.parseChar =
  mapPartial UTF8.parseNatural
    (λn. if invariant n then some (mk n) else none)

c1 c2. dest c1 = dest c2 c1 = c2

n. destPlane n = Bits.shiftRight n 16

n. destPosition n = Bits.bound n 16

x.
    UTF8.reencodeChar x =
    case x of left b b :: [] | right c UTF8.encodeChar c

image mk { n. n | invariant n } = universe

n. invariant n Bits.width n 21

UTF8.parseAscii =
  token (λb. if bit b 7 then none else some (toNatural b))

c. n. invariant n c = mk n dest c = n

c. dest c = position c + Bits.shiftLeft (plane c) 16

b. UTF8.isContinuationByte b bit b 7 ¬bit b 6

n. UTF8.isContinuationByte (or 128 (fromNatural (Bits.bound n 6)))

c. position c < 65534

size universe = 1111998

n b.
    n < 128
    apply UTF8.parseAscii (append (UTF8.encodeAscii n) b) = some (n, b)

c. dest c < 1114110

n. destPlane n = 0 n < 65536

n. invariant n n < 1114110

b n.
    UTF8.parseMultibyte.addContinuationByte b n =
    if UTF8.isContinuationByte b then
      some (toNatural (and b 63) + Bits.shiftLeft n 6)
    else none

b.
    UTF8.parseMultibyte.parse2 b =
    filter
      (foldN UTF8.parseMultibyte.addContinuationByte 0
         (toNatural (and b 31))) (λn. 128 n)

b.
    UTF8.parseMultibyte.parse3 b =
    filter
      (foldN UTF8.parseMultibyte.addContinuationByte 1
         (toNatural (and b 15))) (λn. 2048 n)

n m.
    UTF8.parseMultibyte.addContinuationByte
      (or 128 (fromNatural (Bits.bound n 6))) m =
    some (Bits.bound n 6 + Bits.shiftLeft m 6)

b.
    UTF8.parseMultibyte.parse4 b =
    filter
      (foldN UTF8.parseMultibyte.addContinuationByte 2
         (toNatural (and b 7))) (λn. 65536 n)

n b.
    128 n n < 2048
    apply UTF8.parseMultibyte (append (UTF8.encodeChar.encode2 n) b) =
    some (n, b)

UTF8.parseMultibyte =
  sequence
    (token
       (λb.
          if bit b 6 then
            if bit b 5 then
              if bit b 4 then
                if bit b 3 then none
                else some (UTF8.parseMultibyte.parse4 b)
              else some (UTF8.parseMultibyte.parse3 b)
            else some (UTF8.parseMultibyte.parse2 b)
          else none))

n b.
    65536 n Bits.width n 21
    apply UTF8.parseMultibyte (append (UTF8.encodeChar.encode4 n) b) =
    some (n, b)

n.
    UTF8.encodeChar.encode2 n =
    let n1 Bits.shiftRight n 6 in
    let b0 or 192 (fromNatural n1) in
    let b1 or 128 (fromNatural (Bits.bound n 6)) in
    b0 :: b1 :: []

n b.
    2048 n n < 65536
    apply UTF8.parseMultibyte (append (UTF8.encodeChar.encode3 n) b) =
    some (n, b)

UTF8.encodeChar =
  λc.
    let n dest c in
    if n < 128 then UTF8.encodeAscii n
    else if n < 2048 then UTF8.encodeChar.encode2 n
    else if n < 65536 then UTF8.encodeChar.encode3 n
    else UTF8.encodeChar.encode4 n

c.
    UTF8.encodeChar c =
    let n dest c in
    if n < 128 then UTF8.encodeAscii n
    else if n < 2048 then UTF8.encodeChar.encode2 n
    else if n < 65536 then UTF8.encodeChar.encode3 n
    else UTF8.encodeChar.encode4 n

n.
    UTF8.encodeChar.encode3 n =
    let n1 Bits.shiftRight n 6 in
    let n2 Bits.shiftRight n1 6 in
    let b0 or 224 (fromNatural n2) in
    let b1 or 128 (fromNatural (Bits.bound n1 6)) in
    let b2 or 128 (fromNatural (Bits.bound n 6)) in
    b0 :: b1 :: b2 :: []

n.
    UTF8.encodeChar.encode4 n =
    let n1 Bits.shiftRight n 6 in
    let n2 Bits.shiftRight n1 6 in
    let n3 Bits.shiftRight n2 6 in
    let b0 or 240 (fromNatural n3) in
    let b1 or 128 (fromNatural (Bits.bound n2 6)) in
    let b2 or 128 (fromNatural (Bits.bound n1 6)) in
    let b3 or 128 (fromNatural (Bits.bound n 6)) in
    b0 :: b1 :: b2 :: b3 :: []

n.
    invariant n
    let pl destPlane n in
    let pos destPosition n in
    pl < 17 pos < 65534
    (pl = 0 ¬(55296 pos pos < 57344) ¬(64976 pos pos < 65008))

n.
    invariant n
    let pl destPlane n in
    let pos destPosition n in
    pos < 65534
    if ¬(pl = 0) then pl < 17
    else ¬(55296 pos pos < 57344) ¬(64976 pos pos < 65008)

r.
    random r =
    let n0 Uniform.random 1111998 r in
    let n1 if n0 < 55296 then n0 else n0 + 2048 in
    let n2 if n1 < 64976 then n1 else n1 + 32 in
    let pl n2 div 65534 in
    let pos n2 mod 65534 in
    let n pos + Bits.shiftLeft pl 16 in
    mk n

External Type Operators

External Constants

Assumptions

Bits.head 1

¬

¬

length [] = 0

bit0 0 = 0

Bits.fromList [] = 0

concat [] = []

fromList [] = eof

x. x universe

t. t t

n. 0 n

n. n n

xs. isSuffix xs xs

p. p

x. ¬(x )

t. t ¬t

m. ¬(m < 0)

n. ¬(n < n)

n. 0 < suc n

n. n < suc n

(¬) = λp. p

() = λp. p ((select) p)

t. (x. t) t

t. (λx. t x) = t

() = λp. p = λx.

a. ¬(some a = none)

x. replicate x 0 = []

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. t t

t. t

x. fromNatural (toNatural x) = x

n. ¬(suc n = 0)

n. 0 * n = 0

m. m * 0 = 0

n. 0 + n = n

m. m + 0 = m

m. m - 0 = m

n. n - n = 0

n. min n n = n

k. Bits.bound 0 k = 0

n. Bits.or 0 n = n

k. Bits.shiftLeft 0 k = 0

k. Bits.shiftRight 0 k = 0

l. [] @ l = l

f. map f [] = []

f. map f none = none

p. parse p eof = eof

p. apply p eof = none

p. apply p error = none

width = 8

t. ( t) ¬t

t. (t ) ¬t

t. t ¬t

q. Bits.compare q [] [] q

n. bit1 n = suc (bit0 n)

n. Bits.bit n 0 Bits.head n

m. m 0 = 1

m. m * 1 = m

m. 1 * m = m

l. length (fromList l) = length l

l. Bits.toByte l = fromNatural (Bits.fromList l)

m n. m m + n

n k. Bits.bound n k n

() = λp q. p q p

x. size (insert x ) = 1

t. (t ) (t )

w. Bits.bound (toNatural w) width = toNatural w

n. even (suc n) ¬even n

m. m 0 m = 0

n. toNatural (fromNatural n) = Bits.bound n width

l. toList (fromList l) = (l, )

x. (fst x, snd x) = x

t1 t2. (if then t1 else t2) = t2

t1 t2. (if then t1 else t2) = t1

a b. fst (a, b) = a

b f. case b f none = b

m n. ¬(n + m < m)

n k. Bits.width (Bits.bound n k) k

p x. p x p ((select) p)

n. 0 < n ¬(n = 0)

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

x y. x = y y = x

h t. nth (h :: t) 0 = h

t1 t2. t1 t2 t2 t1

m n. m * n = n * m

m n. m + n = n + m

m n. m < n m n

m n. m + n - m = n

m n. m + n - n = m

n k. Bits.bound (Bits.shiftLeft n k) k = 0

n k. Bits.shiftRight (Bits.bound n k) k = 0

s t. finite s finite (s \ t)

f l. length (map f l) = length l

f s. finite s finite (image f s)

p xs. length (parse p xs) length xs

n. Bits.cons n = 2 * n

n. n 2 = n * n

n. 2 * n = n + n

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

w n. bit w n Bits.bit (toNatural w) n

m n. ¬(m < n n m)

m n. ¬(m n n < m)

n i. Bits.bit n i i < Bits.width n

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

f a. map f (some a) = some (f a)

() = λp q. (λf. f p q) = λf. f

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

k. Bits.shiftLeft 1 k = 2 k

f. surjective f y. x. y = f x

p. ¬(x. p x) x. ¬p x

p. ¬(x. p x) x. ¬p x

() = λp. q. (x. p x q) q

x y. x insert y x = y

a b. some a = some b a = b

h t. (h :: []) @ t = h :: t

t1 t2. ¬t1 ¬t2 t2 t1

h t. Bits.fromList (h :: t) = Bits.cons h (Bits.fromList t)

x y. x < y toNatural x < toNatural y

m n. m < n m div n = 0

m n. m < n m mod n = m

n1 n2. n1 n2 Bits.width n1 Bits.width n2

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

m n. suc m suc n m n

m n. m + n = m n = 0

k n. Bits.shiftLeft n k = 0 n = 0

n k. Bits.bound (Bits.bound n k) k = Bits.bound n k

l1 l2. fromList (l1 @ l2) = append l1 (fromList l2)

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

s t. disjoint s t s t =

s t. finite t s t finite s

b f a. case b f (some a) = f a

n. finite { m. m | m < n }

k. Bits.width (2 k - 1) = k

f g a. case f g (left a) = f a

f g b. case f g (right b) = g b

f g x. (f g) x = f (g x)

t1 t2. ¬(t1 t2) ¬t1 ¬t2

t1 t2. ¬(t1 t2) ¬t1 ¬t2

w1 w2. and w1 w2 = fromNatural (Bits.and (toNatural w1) (toNatural w2))

w1 w2. or w1 w2 = fromNatural (Bits.or (toNatural w1) (toNatural w2))

w1 w2. Bits.compare (Bits.fromByte w1) (Bits.fromByte w2) w1 < w2

m n. min m n = if m n then m else n

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

m n. m suc n = m * m n

m n. suc m * n = m * n + n

n k. Bits.bound n k = n Bits.width n k

n k. Bits.shiftRight n k = 0 Bits.width n k

l1 l2. length (l1 @ l2) = length l1 + length l2

s t. finite (s t) finite s finite t

s t. finite s finite t finite (cross s t)

p. (x. p x) a b. p (a, b)

p. (x. p x) a b. p (a, b)

m n. m n d. n = m + d

n i. i < n r. Uniform.random n r = i

n. Bits.cons n = 1 + 2 * n

xs y ys. isProperSuffix xs (cons y ys) isSuffix xs ys

f g. (x. f x = g x) f = g

x. (a. x = left a) b. x = right b

() = λp q. r. (p r) (q r) r

m n. m n m < n m = n

n k. Bits.shiftLeft n k = 2 k * n

m n. m n n m m = n

n k. Bits.bound n k + Bits.shiftLeft (Bits.shiftRight n k) k = n

f. fn. a b. fn (a, b) = f a b

m n. m < n d. n = m + suc d

n. size { m. m | m < n } = n

p q. p (x. q x) x. p q x

n k. Bits.width n k n < 2 k

x y z. x = y y = z x = z

t1 t2 t3. (t1 t2) t3 t1 t2 t3

n k i. Bits.bit n (k + i) Bits.bit (Bits.shiftRight n k) i

m n p. m * (n * p) = n * (m * p)

m n p. m * (n * p) = m * n * p

m n p. m + (n + p) = m + n + p

n k1 k2.
    Bits.shiftRight n (k1 + k2) = Bits.shiftRight (Bits.shiftRight n k1) k2

n j k. Bits.bound (Bits.bound n j) k = Bits.bound n (min j k)

m n p. m + n = m + p n = p

p m n. m + p = n + p m = n

m n p. n + m < p + m n < p

m n p. m + n m + p n p

m n p. n + m p + m n p

k n1 n2. Bits.shiftLeft n1 k Bits.shiftLeft n2 k n1 n2

m n p. (m * n + p) mod n = p mod n

n1 n2 k. Bits.bound (n1 + Bits.shiftLeft n2 k) k = Bits.bound n1 k

m n p. m < n n p m < p

m n p. m n n p m p

l h t. (h :: t) @ l = h :: t @ l

s t. s t x. x s x t

s t. (x. x s x t) s = t

m n. m suc n m = suc n m n

m n. m * n = 0 m = 0 n = 0

m n. m + n = 0 m = 0 n = 0

l i. Bits.bit (Bits.fromList l) i i < length l nth l i

xs ys e. toList xs = (ys, e) length xs = length ys

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

p. p 0 (n. p n p (suc n)) n. p n

n k. Bits.and n (2 k - 1) = Bits.bound n k

p x. x { y. y | p y } p x

f s. apply (fold f s) = case none none (λx xs. fold.prs f s x xs)

k w1 w2. bit (or w1 w2) k bit w1 k bit w2 k

n i k. Bits.bit (Bits.bound n k) i i < k Bits.bit n i

m n i. Bits.bit (Bits.or m n) i Bits.bit m i Bits.bit n i

m n p. m * (n + p) = m * n + m * p

m n p. m (n + p) = m n * m p

m n p. (m + n) * p = m * p + n * p

m n k.
    Bits.bound (Bits.and m n) k =
    Bits.and (Bits.bound m k) (Bits.bound n k)

m n k.
    Bits.bound (Bits.or m n) k = Bits.or (Bits.bound m k) (Bits.bound n k)

n j k. Bits.width n j + k Bits.width (Bits.shiftRight n k) j

n1 n2 k.
    Bits.shiftRight (n1 + Bits.shiftLeft n2 k) k =
    Bits.shiftRight n1 k + n2

xs. xs = error xs = eof x xt. xs = cons x xt

s t x. x s t x s x t

s t x. x s t x s x t

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

p. (xs. (ys. isProperSuffix ys xs p ys) p xs) xs. p xs

s t x. x s \ t x s ¬(x t)

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

s t. finite s finite t size (cross s t) = size s * size t

p. p [] (h t. p t p (h :: t)) l. p l

h t n. n < length t nth (h :: t) (suc n) = nth t n

w1 w2. (i. i < width (bit w1 i bit w2 i)) w1 = w2

n k i. Bits.bit (Bits.shiftLeft n k) i k i Bits.bit n (i - k)

m n p. m * n = m * p m = 0 n = p

m n p. m * p = n * p m = n p = 0

m n p. m * n m * p m = 0 n p

y s f. y image f s x. y = f x x s

l n. bit (Bits.toByte l) n n < width n < length l nth l n

s t. finite s t s size (s \ t) = size s - size t

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

m n p. m * n < m * p ¬(m = 0) n < p

h1 h2 t1 t2. h1 :: t1 = h2 :: t2 h1 = h2 t1 = t2

a b a' b'. (a, b) = (a', b') a = a' b = b'

x y s t. (x, y) cross s t x s y t

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)

l.
    Bits.fromByte (Bits.toByte l) =
    if length l width then l @ replicate (width - length l)
    else take width l

p c x y. p (if c then x else y) (c p x) (¬c p y)

a b n. ¬(n = 0) (a * n + b) div n = a + b div n

m n j k.
    Bits.width (Bits.bound m k + Bits.shiftLeft n k) j + k
    Bits.width n j

l xs ys e. toList (append l xs) = (l @ ys, e) toList xs = (ys, e)

s t. finite s finite t disjoint s t size (s t) = size s + size t

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

p xs.
    apply p xs = none
    y ys. apply p xs = some (y, ys) isProperSuffix ys xs

x xs. toList (cons x xs) = let (l, e) toList xs in (x :: l, e)

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

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

p e.
    strongInverse p e
    inverse p e
    xs y ys. apply p xs = some (y, ys) xs = append (e y) ys

q h1 h2 t1 t2.
    Bits.compare q (h1 :: t1) (h2 :: t2)
    Bits.compare (¬h1 h2 ¬(h1 ¬h2) q) t1 t2

p xs.
    parse p xs =
    case apply p xs of
      none case xs of error error | eof eof | cons y ys error
    | some (y, ys) cons y (parse p ys)

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

b.
    x0 x1 x2 x3 x4 x5 x6 x7.
      b = Bits.toByte (x0 :: x1 :: x2 :: x3 :: x4 :: x5 :: x6 :: x7 :: [])

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)

f s.
    (x y. x s y s f x = f y x = y) finite s
    size (image f s) = size s

f s x xs.
    fold.prs f s x xs =
    case f x s of
      none none
    | some y
        case y of
          left z some (z, xs)
        | right t
            case xs of
              error none
            | eof none
            | cons z zs fold.prs f t z zs

toList error = ([], ) toList eof = ([], )
  x xs. toList (cons x xs) = let (l, e) toList xs in (x :: l, e)

f n s.
    foldN f n s =
    fold
      (λx (m, t).
         map (λu. if m = 0 then left u else right (m - 1, u)) (f x t))
      (n, s)

(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