Package word16-bytes: Basic theory of 16-bit words as pairs of bytes

Information

name	word16-bytes
version	1.2
description	Basic theory of 16-bit words as pairs of bytes
author	Joe Hurd <joe@gilith.com>
license	MIT
show	Data.Bool Data.List Data.Word16 Number.Numeral

Files

Package tarball word16-bytes-1.2.tgz
Theory file word16-bytes.thy (included in the package tarball)

Defined Constants

Data
- Word16
  - fromBytes
  - toBytes

Theorems

⊦ ∀b.
fromNatural (Data.Byte.toNatural b) =
Bits.toWord (Data.Byte.Bits.fromWord b)

⊦ ∀w.
Data.Byte.fromNatural (toNatural w) =
Data.Byte.Bits.toWord (Bits.fromWord w)

⊦ ∀w. ∃b0 b1. w = fromBytes b0 b1

⊦ ∀b0 b1.
Bits.toWord (Data.Byte.Bits.fromWord b1 @ Data.Byte.Bits.fromWord b0) =
fromBytes b0 b1

⊦ ∀w. ∃b0 b1. w = fromBytes b0 b1 ∧ toBytes w = Data.Pair., b0 b1

⊦ ∀b1 b2.
    fromBytes b1 b2 =
    or (shiftLeft (fromNatural (Data.Byte.toNatural b1)) 8)
      (fromNatural (Data.Byte.toNatural b2))

⊦ ∀w.
Data.Pair., (Data.Byte.Bits.toWord (drop 8 (Bits.fromWord w)))
(Data.Byte.Bits.toWord (take 8 (Bits.fromWord w))) = toBytes w

⊦ ∀w.
    toBytes w =
    Data.Pair., (Data.Byte.fromNatural (toNatural (shiftRight w 8)))
      (Data.Byte.fromNatural (toNatural (and w (fromNatural 255))))

Input Type Operators

→
bool
Data
- Byte
  - Data.Byte.byte
- List
  - list
- Pair
  - Data.Pair.*
- Word16
  - word16
Number
- Natural
  - Number.Natural.natural

Input Constants

=
Data
- Bool
  - ∀
  - ∧
  - ⇒
  - ∃
  - ∨
  - ¬
  - cond
  - select
  - F
  - T
- Byte
  - Data.Byte.bit
  - Data.Byte.fromNatural
  - Data.Byte.modulus
  - Data.Byte.toNatural
  - Data.Byte.width
  - Bits
    - Data.Byte.Bits.fromWord
    - Data.Byte.Bits.toWord
- List
  - ::
  - @
  - []
  - drop
  - length
  - nth
  - replicate
  - take
  - zipWith
- Pair
  - Data.Pair.,
  - Data.Pair.fst
  - Data.Pair.snd
- Word16
  - +
  - and
  - bit
  - fromNatural
  - modulus
  - or
  - shiftLeft
  - shiftRight
  - toNatural
  - width
  - Bits
    - Bits.fromWord
    - Bits.toWord
Function
- Function.id
Number
- Natural
  - Number.Natural.*
  - Number.Natural.+
  - Number.Natural.-
  - Number.Natural.<
  - Number.Natural.≤
  - Number.Natural.div
  - Number.Natural.even
  - Number.Natural.exp
  - Number.Natural.mod
  - Number.Natural.odd
  - Number.Natural.suc
- Numeral
  - bit0
  - bit1
  - zero

Assumptions

⊦ T

⊦ ∀x. x = x

⊦ ∀n. Number.Natural.≤ 0 n

⊦ ∀n. Number.Natural.≤ n n

⊦ F ⇔ ∀p. p

⊦ ∀x. Function.id x = x

⊦ ∀t. t ∨ ¬t

⊦ (¬) = λp. p ⇒ F

⊦ (∃) = λP. P ((select) P)

⊦ ∀a. ∃x. x = a

⊦ ∀t. (∀x. t) ⇔ t

⊦ ∀t. (∃x. t) ⇔ t

⊦ ∀t. (λx. t x) = t

⊦ (∀) = λP. P = λx. T

⊦ ∀x. x = x ⇔ T

⊦ ∀w. Data.Byte.Bits.toWord (Data.Byte.Bits.fromWord w) = w

⊦ ∀w. length (Data.Byte.Bits.fromWord w) = Data.Byte.width

⊦ ∀w. Bits.toWord (Bits.fromWord w) = w

⊦ ∀w. length (Bits.fromWord w) = width

⊦ ∀n. ¬(Number.Natural.suc n = 0)

⊦ ∀n. Number.Natural.- n n = 0

⊦ Data.Byte.modulus = Number.Natural.exp 2 Data.Byte.width

⊦ Data.Byte.width = 8

⊦ modulus = Number.Natural.exp 2 width

⊦ ∀n. bit0 n = Number.Natural.+ n n

⊦ ∀n. Number.Natural.mod n 1 = 0

⊦ ∀x. (select y. y = x) = x

⊦ ∀m n. Number.Natural.≤ m (Number.Natural.+ m n)

⊦ ∀m n. Number.Natural.≤ n (Number.Natural.+ m n)

⊦ width = 16

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀t. (t ⇔ T) ∨ (t ⇔ F)

⊦ ∀m. Number.Natural.suc m = Number.Natural.+ m 1

⊦ ∀n. bit1 n = Number.Natural.suc (Number.Natural.+ n n)

⊦ ∀x.
Data.Byte.toNatural (Data.Byte.fromNatural x) =
Number.Natural.mod x Data.Byte.modulus

⊦ ∀x. toNatural (fromNatural x) = Number.Natural.mod x modulus

⊦ ∀x. Data.Pair., (Data.Pair.fst x) (Data.Pair.snd x) = x

⊦ ∀x y. Data.Pair.fst (Data.Pair., x y) = x

⊦ ∀x y. Data.Pair.snd (Data.Pair., x y) = y

⊦ (¬T ⇔ F) ∧ (¬F ⇔ T)

⊦ ∀x y. x = y ⇔ y = x

⊦ ∀t1 t2. t1 ∨ t2 ⇔ t2 ∨ t1

⊦ ∀m n. m = n ⇒ Number.Natural.≤ m n

⊦ ∀m n. Number.Natural.< m n ⇒ Number.Natural.≤ m n

⊦ ∀m n. Number.Natural.< m n ∨ Number.Natural.≤ n m

⊦ ∀m n. Number.Natural.≤ m n ∨ Number.Natural.≤ n m

⊦ ∀m n. Number.Natural.- m (Number.Natural.+ m n) = 0

⊦ ∀m n. Number.Natural.- (Number.Natural.+ m n) m = n

⊦ ∀m n. Number.Natural.- (Number.Natural.+ m n) n = m

⊦ ∀n. Number.Natural.* 2 n = Number.Natural.+ n n

⊦ ∀m n. ¬(Number.Natural.< m n ∧ Number.Natural.≤ n m)

⊦ ∀m n. ¬(Number.Natural.≤ m n ∧ Number.Natural.< n m)

⊦ ∀m n. ¬Number.Natural.< m n ⇔ Number.Natural.≤ n m

⊦ ∀m n. ¬Number.Natural.≤ m n ⇔ Number.Natural.< n m

⊦ ∀m n. Number.Natural.< m (Number.Natural.suc n) ⇔ Number.Natural.≤ m n

⊦ ∀m n. Number.Natural.≤ (Number.Natural.suc m) n ⇔ Number.Natural.< m n

⊦ ∀m. m = 0 ∨ ∃n. m = Number.Natural.suc n

⊦ (∧) = λp q. (λf. f p q) = λf. f T T

⊦ ∀P. ¬(∀x. P x) ⇔ ∃x. ¬P x

⊦ ∀P. ¬(∃x. P x) ⇔ ∀x. ¬P x

⊦ (∃) = λP. ∀q. (∀x. P x ⇒ q) ⇒ q

⊦ ∀w1 w2. Data.Byte.Bits.fromWord w1 = Data.Byte.Bits.fromWord w2 ⇔ w1 = w2

⊦ ∀w1 w2. Bits.fromWord w1 = Bits.fromWord w2 ⇔ w1 = w2

⊦ ∀m n. Number.Natural.< m n ⇒ Number.Natural.div m n = 0

⊦ ∀m n. Number.Natural.suc m = Number.Natural.suc n ⇔ m = n

⊦ ∀m n. Number.Natural.+ m n = m ⇔ n = 0

⊦ ∀m n. Number.Natural.- m n = 0 ⇔ Number.Natural.≤ m n

⊦ ∀n. Number.Natural.odd n ⇔ Number.Natural.mod n 2 = 1

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

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

⊦ ∀m n. ¬(n = 0) ⇒ Number.Natural.< (Number.Natural.mod m n) n

⊦ ∀P. (∀p. P p) ⇔ ∀p1 p2. P (Data.Pair., p1 p2)

⊦ ∀m n. Number.Natural.≤ m n ⇔ ∃d. n = Number.Natural.+ m d

⊦ ∀f g. f = g ⇔ ∀x. f x = g x

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

⊦ (Number.Natural.even 0 ⇔ T) ∧
∀n. Number.Natural.even (Number.Natural.suc n) ⇔ ¬Number.Natural.even n

⊦ (Number.Natural.odd 0 ⇔ F) ∧
∀n. Number.Natural.odd (Number.Natural.suc n) ⇔ ¬Number.Natural.odd n

⊦ ∀w1 w2.
and w1 w2 =
Bits.toWord (zipWith (∧) (Bits.fromWord w1) (Bits.fromWord w2))

⊦ ∀w1 w2.
or w1 w2 =
Bits.toWord (zipWith (∨) (Bits.fromWord w1) (Bits.fromWord w2))

⊦ ∀m n. Number.Natural.≤ m n ⇔ Number.Natural.< m n ∨ m = n

⊦ ∀m n.
Number.Natural.odd (Number.Natural.+ m n) ⇔
¬(Number.Natural.odd m ⇔ Number.Natural.odd n)

⊦ ∀m n. Number.Natural.≤ m n ∧ Number.Natural.≤ n m ⇔ m = n

⊦ ∀l n. shiftLeft (Bits.toWord l) n = Bits.toWord (replicate n F @ l)

⊦ ∀m n.
Number.Natural.< m n ⇔
∃d. n = Number.Natural.+ m (Number.Natural.suc d)

⊦ ∀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

⊦ ∀P Q. (∀x. P x) ∨ Q ⇔ ∀x. P x ∨ Q

⊦ ∀P Q. (∃x. P x) ∨ Q ⇔ ∃x. P x ∨ Q

⊦ ∀x y z. x = y ∧ y = z ⇒ x = z

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

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

⊦ ∀p q r. p ∧ q ⇒ r ⇔ p ⇒ q ⇒ r

⊦ ∀m n p.
Number.Natural.* m (Number.Natural.* n p) =
Number.Natural.* (Number.Natural.* m n) p

⊦ ∀m n p.
Number.Natural.+ m (Number.Natural.+ n p) =
Number.Natural.+ (Number.Natural.+ m n) p

⊦ ∀m n p. Number.Natural.+ m n = Number.Natural.+ m p ⇔ n = p

⊦ ∀m n p.
Number.Natural.≤ (Number.Natural.+ m n) (Number.Natural.+ m p) ⇔
Number.Natural.≤ n p

⊦ ∀m n p.
Number.Natural.≤ (Number.Natural.+ m p) (Number.Natural.+ n p) ⇔
Number.Natural.≤ m n

⊦ ∀m n p.
Number.Natural.- (Number.Natural.+ m n) (Number.Natural.+ m p) =
Number.Natural.- n p

⊦ ∀m n p.
Number.Natural.< m n ∧ Number.Natural.≤ n p ⇒ Number.Natural.< m p

⊦ ∀m n p.
Number.Natural.≤ m n ∧ Number.Natural.≤ n p ⇒ Number.Natural.≤ m p

⊦ ∀n.
    Bits.toWord
      (Number.Natural.odd n ::
       Bits.fromWord (fromNatural (Number.Natural.div n 2))) =
    fromNatural n

⊦ ∀P. (∀x. ∃y. P x y) ⇔ ∃y. ∀x. P x (y x)

⊦ ∀t1 t2. (if T then t1 else t2) = t1 ∧ (if F then t1 else t2) = t2

⊦ ∀w n.
    Data.Byte.bit w n ⇔
    Number.Natural.odd
      (Number.Natural.div (Data.Byte.toNatural w) (Number.Natural.exp 2 n))

⊦ ∀w n.
    bit w n ⇔
    Number.Natural.odd
      (Number.Natural.div (toNatural w) (Number.Natural.exp 2 n))

⊦ ∀m n. Number.Natural.* m n = 0 ⇔ m = 0 ∨ n = 0

⊦ ∀m n. Number.Natural.+ m n = 0 ⇔ m = 0 ∧ n = 0

⊦ length [] = 0 ∧ ∀h t. length (h :: t) = Number.Natural.suc (length t)

⊦ ∀P. P 0 ∧ (∀n. P n ⇒ P (Number.Natural.suc n)) ⇒ ∀n. P n

⊦ (∀t. ¬¬t ⇔ t) ∧ (¬T ⇔ F) ∧ (¬F ⇔ T)

⊦ ∀m n. Number.Natural.exp m n = 0 ⇔ m = 0 ∧ ¬(n = 0)

⊦ ∀m n p.
Number.Natural.* m (Number.Natural.+ n p) =
Number.Natural.+ (Number.Natural.* m n) (Number.Natural.* m p)

⊦ ∀m n p.
Number.Natural.* m (Number.Natural.- n p) =
Number.Natural.- (Number.Natural.* m n) (Number.Natural.* m p)

⊦ ∀m n p.
Number.Natural.exp m (Number.Natural.+ n p) =
Number.Natural.* (Number.Natural.exp m n) (Number.Natural.exp m p)

⊦ ∀m n p.
Number.Natural.* (Number.Natural.+ m n) p =
Number.Natural.+ (Number.Natural.* m p) (Number.Natural.* n p)

⊦ ∀m n p.
Number.Natural.* (Number.Natural.- m n) p =
Number.Natural.- (Number.Natural.* m p) (Number.Natural.* n p)

⊦ ∀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

⊦ ∀P Q. (∃x. P x) ∨ (∃x. Q x) ⇔ ∃x. P x ∨ Q x

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

⊦ ∀m n.
    ¬(n = 0) ⇒
    Number.Natural.+ (Number.Natural.* (Number.Natural.div m n) n)
      (Number.Natural.mod m n) = m

⊦ ∀w1 w2.
    (∀i.
       Number.Natural.< i Data.Byte.width ⇒
       (Data.Byte.bit w1 i ⇔ Data.Byte.bit w2 i)) ⇒ w1 = w2

⊦ ∀w1 w2. (∀i. Number.Natural.< i width ⇒ (bit w1 i ⇔ bit w2 i)) ⇒ w1 = w2

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

⊦ ∀m n p.
Number.Natural.≤ (Number.Natural.* m n) (Number.Natural.* m p) ⇔
m = 0 ∨ Number.Natural.≤ n p

⊦ ∀l n.
    Data.Byte.bit (Data.Byte.Bits.toWord l) n ⇔
    Number.Natural.< n Data.Byte.width ∧ Number.Natural.< n (length l) ∧
    nth n l

⊦ ∀l n.
bit (Bits.toWord l) n ⇔
Number.Natural.< n width ∧ Number.Natural.< n (length l) ∧ nth n l

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

⊦ (∀x. replicate 0 x = []) ∧
∀n x. replicate (Number.Natural.suc n) x = x :: replicate n x

⊦ ∀h1 h2 t1 t2. h1 :: t1 = h2 :: t2 ⇔ h1 = h2 ∧ t1 = t2

⊦ ∀x y a b. Data.Pair., x y = Data.Pair., a b ⇔ x = a ∧ y = b

⊦ ∀m n p q.
Number.Natural.< m p ∧ Number.Natural.< n q ⇒
Number.Natural.< (Number.Natural.+ m n) (Number.Natural.+ p q)

⊦ ∀m n p q.
Number.Natural.≤ m p ∧ Number.Natural.≤ n q ⇒
Number.Natural.≤ (Number.Natural.+ m n) (Number.Natural.+ p q)

⊦ ∀l.
    Data.Byte.Bits.fromWord (Data.Byte.Bits.toWord l) =
    if Number.Natural.≤ (length l) Data.Byte.width then
      l @ replicate (Number.Natural.- Data.Byte.width (length l)) F
    else take Data.Byte.width l

⊦ ∀l.
    Bits.fromWord (Bits.toWord l) =
    if Number.Natural.≤ (length l) width then
      l @ replicate (Number.Natural.- width (length l)) F
    else take width l

⊦ (∀m. Number.Natural.exp m 0 = 1) ∧
  ∀m n.
    Number.Natural.exp m (Number.Natural.suc n) =
    Number.Natural.* m (Number.Natural.exp m n)

⊦ (∀l. drop 0 l = l) ∧
∀n h t. drop (Number.Natural.suc n) (h :: t) = drop n t

⊦ ∀m n p.
    ¬(Number.Natural.* n p = 0) ⇒
    Number.Natural.mod (Number.Natural.mod m (Number.Natural.* n p)) n =
    Number.Natural.mod m n

⊦ (∀l. [] @ l = l) ∧ ∀h t l. (h :: t) @ l = h :: t @ l

⊦ ∀t1 t2. (¬(t1 ∧ t2) ⇔ ¬t1 ∨ ¬t2) ∧ (¬(t1 ∨ t2) ⇔ ¬t1 ∧ ¬t2)

⊦ (∀l. take 0 l = []) ∧
∀n h t. take (Number.Natural.suc n) (h :: t) = h :: take n t

⊦ ∀m n p.
    ¬(Number.Natural.* n p = 0) ⇒
    Number.Natural.mod (Number.Natural.div m n) p =
    Number.Natural.div (Number.Natural.mod m (Number.Natural.* n p)) n

⊦ (∀m. Number.Natural.≤ m 0 ⇔ m = 0) ∧
  ∀m n.
    Number.Natural.≤ m (Number.Natural.suc n) ⇔
    m = Number.Natural.suc n ∨ Number.Natural.≤ m n

⊦ ∀t. ((T ⇔ t) ⇔ t) ∧ ((t ⇔ T) ⇔ t) ∧ ((F ⇔ t) ⇔ ¬t) ∧ ((t ⇔ F) ⇔ ¬t)

⊦ ∀m n q r.
m = Number.Natural.+ (Number.Natural.* q n) r ∧ Number.Natural.< r n ⇒
Number.Natural.div m n = q ∧ Number.Natural.mod m n = r

⊦ Bits.toWord [] = fromNatural 0 ∧
  ∀h t.
    Bits.toWord (h :: t) =
    if h then shiftLeft (Bits.toWord t) 1 + fromNatural 1
    else shiftLeft (Bits.toWord t) 1

⊦ ∀t. (T ∧ t ⇔ t) ∧ (t ∧ T ⇔ t) ∧ (F ∧ t ⇔ F) ∧ (t ∧ F ⇔ F) ∧ (t ∧ t ⇔ t)

⊦ ∀t. (T ∨ t ⇔ T) ∧ (t ∨ T ⇔ T) ∧ (F ∨ t ⇔ t) ∧ (t ∨ F ⇔ t) ∧ (t ∨ t ⇔ t)

⊦ ∀t. (T ⇒ t ⇔ t) ∧ (t ⇒ T ⇔ T) ∧ (F ⇒ t ⇔ T) ∧ (t ⇒ t ⇔ T) ∧ (t ⇒ F ⇔ ¬t)

⊦ ∀x m n.
Number.Natural.≤ (Number.Natural.exp x m) (Number.Natural.exp x n) ⇔
if x = 0 then m = 0 ⇒ n = 0 else x = 1 ∨ Number.Natural.≤ m n

⊦ (∀f. zipWith f [] [] = []) ∧
∀f h1 h2 t1 t2.
zipWith f (h1 :: t1) (h2 :: t2) = f h1 h2 :: zipWith f t1 t2

⊦ ∀b.
    ∃x0 x1 x2 x3 x4 x5 x6 x7.
      b =
      Data.Byte.Bits.toWord
        (x0 :: x1 :: x2 :: x3 :: x4 :: x5 :: x6 :: x7 :: [])

⊦ ∀m n p.
    Number.Natural.* m n = Number.Natural.* n m ∧
    Number.Natural.* (Number.Natural.* m n) p =
    Number.Natural.* m (Number.Natural.* n p) ∧
    Number.Natural.* m (Number.Natural.* n p) =
    Number.Natural.* n (Number.Natural.* m p)

⊦ ∀m n p.
    Number.Natural.+ m n = Number.Natural.+ n m ∧
    Number.Natural.+ (Number.Natural.+ m n) p =
    Number.Natural.+ m (Number.Natural.+ n p) ∧
    Number.Natural.+ m (Number.Natural.+ n p) =
    Number.Natural.+ n (Number.Natural.+ m p)

⊦ ∀l n.
    shiftRight (Bits.toWord l) n =
    if Number.Natural.≤ (length l) width then
      if Number.Natural.≤ (length l) n then Bits.toWord []
      else Bits.toWord (drop n l)
    else if Number.Natural.≤ width n then Bits.toWord []
    else Bits.toWord (drop n (take width l))

⊦ (∀n. Number.Natural.+ 0 n = n) ∧ (∀m. Number.Natural.+ m 0 = m) ∧
  (∀m n.
     Number.Natural.+ (Number.Natural.suc m) n =
     Number.Natural.suc (Number.Natural.+ m n)) ∧
  ∀m n.
    Number.Natural.+ m (Number.Natural.suc n) =
    Number.Natural.suc (Number.Natural.+ m n)

⊦ ∀p q r.
(p ∨ q ⇔ q ∨ p) ∧ ((p ∨ q) ∨ r ⇔ p ∨ q ∨ r) ∧ (p ∨ q ∨ r ⇔ q ∨ p ∨ r) ∧
(p ∨ p ⇔ p) ∧ (p ∨ p ∨ q ⇔ p ∨ q)

⊦ (∀n. Number.Natural.* 0 n = 0) ∧ (∀m. Number.Natural.* m 0 = 0) ∧
  (∀n. Number.Natural.* 1 n = n) ∧ (∀m. Number.Natural.* m 1 = m) ∧
  (∀m n.
     Number.Natural.* (Number.Natural.suc m) n =
     Number.Natural.+ (Number.Natural.* m n) n) ∧
  ∀m n.
    Number.Natural.* m (Number.Natural.suc n) =
    Number.Natural.+ m (Number.Natural.* m n)

⊦ ∀w.
    ∃x0 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15.
      w =
      Bits.toWord
        (x0 :: x1 :: x2 :: x3 :: x4 :: x5 :: x6 :: x7 :: x8 :: x9 :: x10 ::
         x11 :: x12 :: x13 :: x14 :: x15 :: [])