Package byte-def: Definition of bytes

Information

name	byte-def
version	1.58
description	Definition of bytes
author	Joe Leslie-Hurd <joe@gilith.com>
license	MIT
provenance	HOL Light theory extracted on 2011-07-25
requires	bool list natural natural-bits natural-divides pair probability
show	Data.Bool Data.Byte Data.Byte.Bits Data.List Data.Pair Number.Natural Probability.Random

Files

Package tarball byte-def-1.58.tgz
Theory file byte-def.thy (included in the package tarball)

Defined Type Operator

Data
- Byte
  - byte

Defined Constants

Data
- Byte
  - *
  - +
  - -
  - <
  - ≤
  - ↑
  - ~
  - and
  - bit
  - fromNatural
  - fromRandom
  - modulus
  - not
  - or
  - shiftLeft
  - shiftRight
  - toNatural
  - width
  - Bits
    - compare
    - fromByte
    - normal
    - toByte

Theorems

⊦ ¬(modulus = 0)

⊦ ∀w. normal (fromByte w)

⊦ ∀x. x ≤ x

⊦ fromNatural modulus = 0

⊦ toByte [] = 0

⊦ toNatural (toByte []) = 0

⊦ modulus mod modulus = 0

⊦ 0 mod modulus = 0

⊦ ∀x. ¬(x < x)

⊦ ∀x. toNatural x < modulus

⊦ ~0 = 0

⊦ ∀x. ~~x = x

⊦ ∀x. fromNatural (toNatural x) = x

⊦ ∀w. toByte (fromByte w) = w

⊦ ∀w. length (fromByte w) = width

⊦ ∀n. n mod modulus < modulus

⊦ ∀n. n mod modulus ≤ n

⊦ modulus = 2 ↑ width

⊦ width = 8

⊦ ∀q. compare q [] [] ⇔ q

⊦ ∀x. x + 0 = x

⊦ ∀x. x ↑ 1 = x

⊦ ∀x. 0 + x = x

⊦ ∀x. toNatural x div modulus = 0

⊦ ∀l. fromNatural (Bits.toNatural l) = toByte l

⊦ ∀x. x ↑ 0 = 1

⊦ ∀x. x * 0 = 0

⊦ ∀x. x + ~x = 0

⊦ ∀x. 0 * x = 0

⊦ ∀x. ~x + x = 0

⊦ ∀x. toNatural x mod modulus = toNatural x

⊦ ∀x. x * 1 = x

⊦ ∀x. 1 * x = x

⊦ ∀n. toNatural (fromNatural n) = n mod modulus

⊦ ∀l. normal l ⇔ length l = width

⊦ ∀x. ~x = fromNatural (modulus - toNatural x)

⊦ ∀w. not w = toByte (map (¬) (fromByte w))

⊦ ∀x y. x * y = y * x

⊦ ∀x y. x + y = y + x

⊦ ∀w. fromByte w = map (bit w) (interval 0 width)

⊦ ∀n. divides modulus n ⇔ n mod modulus = 0

⊦ ∀n. n < modulus ⇒ toNatural (fromNatural n) = n

⊦ ∀n. n < modulus ⇒ n mod modulus = n

⊦ ∀x. fromNatural x = 0 ⇔ divides modulus x

⊦ ∀n. n mod modulus mod modulus = n mod modulus

⊦ ∀x y. x - y = x + ~y

⊦ ∀x y. ¬(x < y) ⇔ y ≤ x

⊦ ∀x y. ¬(x ≤ y) ⇔ y < x

⊦ ∀x. ~x = 0 ⇔ x = 0

⊦ ∀l. toNatural (toByte l) < 2 ↑ length l

⊦ ∀l. length l = width ⇔ fromByte (toByte l) = l

⊦ ∀x y. x < y ⇔ toNatural x < toNatural y

⊦ ∀x y. x ≤ y ⇔ toNatural x ≤ toNatural y

⊦ ∀x y. x * ~y = ~(x * y)

⊦ ∀x y. ~x * y = ~(x * y)

⊦ ∀w₁ w₂. fromByte w₁ = fromByte w₂ ⇔ w₁ = w₂

⊦ ∀x y. ~x = ~y ⇒ x = y

⊦ ∀x y. toNatural x = toNatural y ⇒ x = y

⊦ ∀w₁ w₂. fromByte w₁ = fromByte w₂ ⇒ w₁ = w₂

⊦ ∀w n. bit w n ⇔ odd (toNatural (shiftRight w n))

⊦ ∀m n. fromNatural (m ↑ n) = fromNatural m ↑ n

⊦ ∀x y. x + y = x ⇔ y = 0

⊦ ∀x y. y + x = x ⇔ y = 0

⊦ ∀x y. ~x + ~y = ~(x + y)

⊦ ∀w₁ w₂. compare ⊥ (fromByte w₁) (fromByte w₂) ⇔ w₁ < w₂

⊦ ∀w₁ w₂. compare ⊤ (fromByte w₁) (fromByte w₂) ⇔ w₁ ≤ w₂

⊦ ∀x n. x ↑ suc n = x * x ↑ n

⊦ ∀x₁ y₁. fromNatural (x₁ * y₁) = fromNatural x₁ * fromNatural y₁

⊦ ∀x₁ y₁. fromNatural (x₁ + y₁) = fromNatural x₁ + fromNatural y₁

⊦ ∀l. width ≤ length l ⇒ fromByte (toByte l) = take width l

⊦ ∀w₁ w₂. and w₁ w₂ = toByte (zipWith (∧) (fromByte w₁) (fromByte w₂))

⊦ ∀w₁ w₂. or w₁ w₂ = toByte (zipWith (∨) (fromByte w₁) (fromByte w₂))

⊦ ∀l n. shiftLeft (toByte l) n = toByte (replicate ⊥ n @ l)

⊦ ∀x y. toNatural (x * y) = toNatural x * toNatural y mod modulus

⊦ ∀x y. toNatural (x + y) = (toNatural x + toNatural y) mod modulus

⊦ ∀x y z. x * y * z = x * (y * z)

⊦ ∀x y z. x + y + z = x + (y + z)

⊦ ∀x y z. x + y = x + z ⇔ y = z

⊦ ∀x y z. y + x = z + x ⇔ y = z

⊦ ∀x₁ x₂ x₃. x₁ < x₂ ∧ x₂ < x₃ ⇒ x₁ < x₃

⊦ ∀x₁ x₂ x₃. x₁ < x₂ ∧ x₂ ≤ x₃ ⇒ x₁ < x₃

⊦ ∀x₁ x₂ x₃. x₁ ≤ x₂ ∧ x₂ < x₃ ⇒ x₁ < x₃

⊦ ∀x₁ x₂ x₃. x₁ ≤ x₂ ∧ x₂ ≤ x₃ ⇒ x₁ ≤ x₃

⊦ ∀n. 0 ↑ n = if n = 0 then 1 else 0

⊦ ∀n. toByte (odd n :: fromByte (fromNatural (n div 2))) = fromNatural n

⊦ ∀w n. bit w n ⇔ odd (toNatural w div 2 ↑ n)

⊦ ∀w n. shiftLeft w n = fromNatural (2 ↑ n * toNatural w)

⊦ ∀w n. shiftRight w n = fromNatural (toNatural w div 2 ↑ n)

⊦ ∀x y. fromNatural x = fromNatural y ⇔ x mod modulus = y mod modulus

⊦ ∀x y z. x * (y + z) = x * y + x * z

⊦ ∀x y z. (y + z) * x = y * x + z * x

⊦ ∀x m n. x ↑ m * x ↑ n = x ↑ (m + n)

⊦ ∀m n. (m mod modulus) * (n mod modulus) mod modulus = m * n mod modulus

⊦ ∀m n. (m mod modulus + n mod modulus) mod modulus = (m + n) mod modulus

⊦ ∀l.
length l ≤ width ⇒
fromByte (toByte l) = l @ replicate ⊥ (width - length l)

⊦ ∀q w₁ w₂.
compare q (fromByte w₁) (fromByte w₂) ⇔ if q then w₁ ≤ w₂ else w₁ < w₂

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

⊦ ∀l. 2 * toNatural (toByte l) + 1 < 2 ↑ suc (length l)

⊦ ∀x y. x < modulus ∧ y < modulus ∧ fromNatural x = fromNatural y ⇒ x = y

⊦ ∀l n. bit (toByte l) n ⇔ n < width ∧ n < length l ∧ nth l n

⊦ ∀n.
    fromNatural n =
    toByte
      (if n = 0 then [] else odd n :: fromByte (fromNatural (n div 2)))

⊦ ∀r.
fromRandom r =
let (n, r') ← Uniform.fromRandom modulus r in (fromNatural n, r')

⊦ ∀l.
    fromByte (toByte l) =
    if length l ≤ width then l @ replicate ⊥ (width - length l)
    else take width l

⊦ ∀h t.
toByte (h :: t) =
if h then shiftLeft (toByte t) 1 + 1 else shiftLeft (toByte t) 1

⊦ ∀h t.
toNatural (toByte (h :: t)) =
(2 * toNatural (toByte t) + if h then 1 else 0) mod modulus

⊦ ∀h t.
2 * toNatural (toByte t) + (if h then 1 else 0) < 2 ↑ suc (length t)

⊦ ∀l n.
    length l ≤ width ⇒
    shiftRight (toByte l) n =
    if length l ≤ n then toByte [] else toByte (drop n l)

⊦ ∀l n.
    width ≤ length l ⇒
    shiftRight (toByte l) n =
    if width ≤ n then toByte [] else toByte (drop n (take width l))

⊦ ∀q h₁ h₂ t₁ t₂.
compare q (h₁ :: t₁) (h₂ :: t₂) ⇔
compare (¬h₁ ∧ h₂ ∨ ¬(h₁ ∧ ¬h₂) ∧ q) t₁ t₂

⊦ ∀r.
    fromRandom r =
    let (r₁, r₂) ← split r in
    let (l, r1') ← bits width r₁ in
    (toByte l, r₂)

⊦ ∀l n.
    shiftRight (toByte l) n =
    if length l ≤ width then
      if length l ≤ n then toByte [] else toByte (drop n l)
    else if width ≤ n then toByte []
    else toByte (drop n (take width l))

Input Type Operators

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

Input Constants

=
select
Data
- Bool
  - ∀
  - ∧
  - ⇒
  - ∃
  - ∃!
  - ∨
  - ¬
  - cond
  - ⊥
  - ⊤
- List
  - ::
  - @
  - []
  - drop
  - head
  - interval
  - length
  - map
  - nth
  - replicate
  - tail
  - take
  - zipWith
- Pair
  - ,
  - fst
Number
- Natural
  - *
  - +
  - -
  - <
  - ≤
  - ↑
  - bit0
  - bit1
  - div
  - divides
  - even
  - mod
  - odd
  - suc
  - zero
  - Bits
    - Bits.toNatural
    - Bits.width
  - Uniform
    - Uniform.fromRandom
      - Uniform.fromRandom.loop
Probability
- Random
  - bits
  - split

Assumptions

⊦ ⊤

⊦ even 0

⊦ ¬odd 0

⊦ ¬⊥ ⇔ ⊤

⊦ ¬⊤ ⇔ ⊥

⊦ length [] = 0

⊦ bit0 0 = 0

⊦ Bits.toNatural [] = 0

⊦ ∀t. t ⇒ t

⊦ ∀n. 0 ≤ n

⊦ ∀n. n ≤ n

⊦ ⊥ ⇔ ∀p. p

⊦ ∀t. t ∨ ¬t

⊦ ∀m. ¬(m < 0)

⊦ ∀n. 0 < suc n

⊦ ∀n. n < suc n

⊦ ∀n. n ≤ suc n

⊦ (¬) = λp. p ⇒ ⊥

⊦ (∃) = λp. p ((select) p)

⊦ ∀t. (∀x. t) ⇔ t

⊦ ∀t. (λx. t x) = t

⊦ (∀) = λp. p = λx. ⊤

⊦ ∀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 ∨ ⊤ ⇔ ⊤

⊦ ∀n. ¬(suc n = 0)

⊦ ∀n. even n ∨ odd n

⊦ ∀n. 0 * n = 0

⊦ ∀n. 0 + n = n

⊦ ∀m. m + 0 = m

⊦ ∀m. m - 0 = m

⊦ ∀n. n - n = 0

⊦ ∀m. interval m 0 = []

⊦ ∀l. [] @ l = l

⊦ ∀l. drop 0 l = l

⊦ ∀f. map f [] = []

⊦ ∀t. (⊥ ⇔ t) ⇔ ¬t

⊦ ∀t. (t ⇔ ⊥) ⇔ ¬t

⊦ ∀t. t ⇒ ⊥ ⇔ ¬t

⊦ ∀n. even (2 * n)

⊦ ∀n. bit1 n = suc (bit0 n)

⊦ ∀n. ¬even n ⇔ odd n

⊦ ∀n. ¬odd n ⇔ even n

⊦ ∀m. m ↑ 0 = 1

⊦ ∀m. m * 1 = m

⊦ ∀n. n ↑ 1 = n

⊦ ∀n. n div 1 = n

⊦ ∀n. n mod 1 = 0

⊦ ∀m. 1 * m = m

⊦ ∀l. take (length l) l = l

⊦ ∀m n. m ≤ m + n

⊦ ∀m n. n ≤ m + n

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀t. (t ⇔ ⊤) ∨ (t ⇔ ⊥)

⊦ ∀n. odd (suc (2 * n))

⊦ ∀m. suc m = m + 1

⊦ ∀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

⊦ ∀x y. fst (x, y) = x

⊦ ∀x n. length (replicate x n) = n

⊦ ∀h t. head (h :: t) = h

⊦ ∀h t. tail (h :: t) = t

⊦ ∀m n. length (interval m n) = n

⊦ ∀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

⊦ ∀xy. ∃x y. xy = (x, y)

⊦ ∀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

⊦ ∀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

⊦ ∀m n. m + n - m = n

⊦ ∀n r. length (fst (bits n r)) = n

⊦ ∀f l. length (map f l) = length l

⊦ ∀n. 2 * n = n + n

⊦ ∀l. Bits.toNatural l < 2 ↑ length l

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

⊦ ∀m n. ¬(m < n ∧ n ≤ m)

⊦ ∀m n. ¬(m ≤ n ∧ n < m)

⊦ ∀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

⊦ ∀m. m = 0 ∨ ∃n. m = suc n

⊦ (∧) = λp q. (λf. f p q) = λf. f ⊤ ⊤

⊦ ∀n. even n ⇔ n mod 2 = 0

⊦ ∀n. ¬(n = 0) ⇒ 0 mod n = 0

⊦ ∀n. ¬(n = 0) ⇒ n mod n = 0

⊦ ∀p. ¬(∀x. p x) ⇔ ∃x. ¬p x

⊦ (∃) = λp. ∀q. (∀x. p x ⇒ q) ⇒ q

⊦ ∀t₁ t₂. ¬t₁ ⇒ ¬t₂ ⇔ t₂ ⇒ t₁

⊦ ∀m n. m < n ⇒ m div n = 0

⊦ ∀m n. m < n ⇒ m mod n = m

⊦ ∀m n. m + suc n = suc (m + n)

⊦ ∀m n. suc m + n = suc (m + n)

⊦ ∀m n. m < m + n ⇔ 0 < 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

⊦ ∀n. odd n ⇔ n mod 2 = 1

⊦ ∀n. 0 ↑ n = if n = 0 then 1 else 0

⊦ ∀m. Bits.width (2 ↑ m - 1) = m

⊦ ∀x n. replicate x (suc n) = x :: replicate x 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. ¬(n = 0) ⇒ m mod n < n

⊦ ∀m n. ¬(n = 0) ⇒ m div n ≤ m

⊦ ∀m n. ¬(n = 0) ⇒ m mod n ≤ m

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

⊦ ∀m n. m ≤ n ⇔ ∃d. n = m + d

⊦ ∀f g. (∀x. f x = g x) ⇔ f = g

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

⊦ ∀m n. m ≤ n ⇔ m < n ∨ m = n

⊦ ∀m n. n ≤ m ⇒ n + (m - n) = m

⊦ ∀m n. n ≤ m ⇒ m - n + n = m

⊦ ∀m n. interval m (suc n) = m :: interval (suc m) n

⊦ ∀m n. m ≤ n ∧ n ≤ m ⇔ m = n

⊦ ∀n l. n ≤ length l ⇒ length (take n l) = n

⊦ ∀f. ∃fn. ∀x y. fn (x, y) = f x y

⊦ ∀p. (∀x y. p x y) ⇔ ∀y x. p x y

⊦ ∀p q. p ∨ (∃x. q x) ⇔ ∃x. p ∨ q x

⊦ ∀m n. ¬(m = 0) ⇒ m * n div m = n

⊦ ∀m n. ¬(m = 0) ⇒ m * n mod m = 0

⊦ ∀x y z. x = y ∧ y = z ⇒ x = z

⊦ ∀x n i. i < n ⇒ nth (replicate x n) i = x

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

⊦ ∀m n p. m * (n * p) = m * n * p

⊦ ∀m n p. m + (n + p) = m + n + p

⊦ ∀m n p. m + n < m + p ⇔ n < p

⊦ ∀m n p. n + m < p + m ⇔ n < p

⊦ ∀m n p. m + n ≤ m + p ⇔ n ≤ p

⊦ ∀m n p. m < n ∧ n < p ⇒ m < p

⊦ ∀m n p. m < n ∧ n ≤ p ⇒ m < p

⊦ ∀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

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

⊦ ∀m n. n < m ⇒ suc (m - suc n) = m - n

⊦ ∀m n. n ≤ m ⇒ (m - n = 0 ⇔ m = n)

⊦ ∀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

⊦ ∀a b. ¬(a = 0) ⇒ (divides a b ⇔ b mod a = 0)

⊦ ∀m n. ¬(n = 0) ⇒ (m div n = 0 ⇔ m < n)

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

⊦ ∀m n. m ↑ n = 0 ⇔ m = 0 ∧ ¬(n = 0)

⊦ ∀m n i. i < n ⇒ nth (interval m n) i = m + 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

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

⊦ ∀b f x y. f (if b then x else y) = if b then f x else f y

⊦ ∀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

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

⊦ ∀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

⊦ ∀n h t. n ≤ length t ⇒ drop (suc n) (h :: t) = drop n t

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

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

⊦ ∀f l i. i < length l ⇒ nth (map f l) i = f (nth l i)

⊦ ∀h t. Bits.toNatural (h :: t) = 2 * Bits.toNatural t + if h then 1 else 0

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

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

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

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

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

⊦ ∀n l i. n ≤ length l ∧ i < n ⇒ nth (take n l) i = nth l i

⊦ ∀m n q r. m = q * n + r ∧ r < n ⇒ m div n = q

⊦ ∀m n q r. m = q * n + r ∧ r < n ⇒ m mod n = r

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

⊦ ∀a b n. ¬(n = 0) ⇒ (a mod n + b mod n) mod n = (a + b) mod n

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

⊦ ∀l₁ l₂.
length l₁ = length l₂ ∧ (∀i. i < length l₁ ⇒ nth l₁ i = nth l₂ i) ⇒
l₁ = l₂

⊦ ∀x m n. x ↑ m ≤ x ↑ n ⇔ if x = 0 then m = 0 ⇒ n = 0 else x = 1 ∨ m ≤ n

⊦ ∀n r.
    Uniform.fromRandom n r =
    let w ← Bits.width (n - 1) in
    let (r₁, r₂) ← split r in
    (Uniform.fromRandom.loop n w r₁ mod n, r₂)

⊦ ∀l₁ l₂ k.
    k < length l₁ + length l₂ ⇒
    nth (l₁ @ l₂) k =
    if k < length l₁ then nth l₁ k else nth l₂ (k - length l₁)

⊦ ∀n w r.
    Uniform.fromRandom.loop n w r =
    let (l, r') ← bits w r in
    let m ← Bits.toNatural l in
    if m < n then m else Uniform.fromRandom.loop n w r'

⊦ ∀a b n.
¬(n = 0) ⇒
((a + b) mod n = a mod n + b mod n ⇔ (a + b) div n = a div n + b div n)