Package probability-thm: Properties of probability

Information

name	probability-thm
version	1.20
description	Properties of probability
author	Joe Leslie-Hurd <joe@gilith.com>
license	MIT
provenance	HOL Light theory extracted on 2015-04-28
checksum	273c4473bc1871452598224e52ad2659944977cd
requires	base probability-def stream
show	Data.Bool Data.List Data.Pair Data.Stream Function Number.Natural Probability.Random

Files

Package tarball probability-thm-1.20.tgz
Theory source file probability-thm.thy (included in the package tarball)

Theorems

⊦ ∀b. ∃r. bit r ⇔ b

⊦ ∀n r. length (bits n r) = n

⊦ ∀r₁ r₂. ∃r. split r = (r₁, r₂)

⊦ ∀f n r. length (random f n r) = n

⊦ ∀n l. length l = n ⇒ ∃r. bits n r = l

⊦ ∀f n l. surjective f ∧ length l = n ⇒ ∃r. random f n r = l

⊦ ∀n r.
    Geometric.random.loop n r =
    let (r₁, r₂) ← split r in
    if bit r₁ then n else Geometric.random.loop (n + 1) r₂

⊦ ∀f n r.
    random f n r =
    if n = 0 then []
    else let (r₁, r₂) ← split r in f r₁ :: random f (n - 1) r₂

External Type Operators

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

External Constants

=
select
Data
- Bool
  - ∀
  - ∧
  - ⇒
  - ∃
  - ∨
  - ¬
  - cond
  - ⊥
  - ⊤
- List
  - ::
  - []
  - length
  - random
- Pair
  - ,
- Stream
  - head
  - interleave
  - nth
  - replicate
  - split
Function
- surjective
Number
- Natural
  - +
  - -
  - bit1
  - suc
  - zero
  - Geometric
    - random
      - Geometric.random.loop
Probability
- Random
  - bit
  - bits
  - fromStream
  - split
  - toStream

Assumptions

⊦ ⊤

⊦ bits = random bit

⊦ length [] = 0

⊦ ⊥ ⇔ ∀p. p

⊦ (¬) = λp. p ⇒ ⊥

⊦ ∀t. (∀x. t) ⇔ t

⊦ ∀t. (∃x. t) ⇔ t

⊦ ∀t. (λx. t x) = t

⊦ (∀) = λp. p = λx. ⊤

⊦ ∀t. ⊤ ∧ t ⇔ t

⊦ ∀t. t ⇒ ⊤ ⇔ ⊤

⊦ ∀n. ¬(suc n = 0)

⊦ ∀r. fromStream (toStream r) = r

⊦ ∀s. toStream (fromStream s) = s

⊦ ∀r. bit r ⇔ head (toStream r)

⊦ ∀s. head s = nth s 0

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀m. suc m = m + 1

⊦ ∀n. suc n - 1 = n

⊦ ∀t₁ t₂. (if ⊥ then t₁ else t₂) = t₂

⊦ ∀t₁ t₂. (if ⊤ then t₁ else t₂) = t₁

⊦ ∀a n. nth (replicate a) n = a

⊦ ∀p x. p x ⇒ p ((select) p)

⊦ ∀f r. random f 0 r = []

⊦ ∀x. ∃a b. x = (a, b)

⊦ ∀x y. x = y ⇔ y = x

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

⊦ ∀m. m = 0 ∨ ∃n. m = suc n

⊦ ∀s₁ s₂. split (interleave s₁ s₂) = (s₁, s₂)

⊦ (∧) = λp q. (λf. f p q) = λf. f ⊤ ⊤

⊦ ∀f. surjective f ⇔ ∀y. ∃x. y = f x

⊦ (∃) = λp. ∀q. (∀x. p x ⇒ q) ⇒ q

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

⊦ ∀f. ∃fn. ∀a b. fn (a, b) = f a b

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

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

⊦ ∀r.
split r =
let (s₁, s₂) ← split (toStream r) in (fromStream s₁, fromStream s₂)

⊦ ∀n r.
    Geometric.random.loop n r =
    let (r₁, r₂) ← split r in
    if bit r₁ then n else Geometric.random.loop (suc n) r₂

⊦ ∀f r n.
random f (suc n) r = let (r₁, r₂) ← split r in f r₁ :: random f n r₂