Package probability: Probability

Information

name	probability
version	1.31
description	Probability
author	Joe Leslie-Hurd <joe@gilith.com>
license	MIT
requires	bool list natural pair relation stream
show	Data.Bool Data.List Data.Pair Data.Stream Number.Natural Probability.Random

Files

Package tarball probability-1.31.tgz
Theory source file probability.thy (included in the package tarball)

Defined Type Operator

Probability
- Random
  - random

Defined Constants

Data
- List
  - fromRandom
  - Geometric
    - Geometric.fromRandom
Number
- Natural
  - Geometric
    - Geometric.fromRandom
      - Geometric.fromRandom.loop
Probability
- Random
  - bit
  - bits
  - fromStream
  - split
  - toStream

Theorems

⊦ bits = fromRandom bit

⊦ ∀r. fromStream (toStream r) = r

⊦ ∀s. toStream (fromStream s) = s

⊦ ∀n r. length (fst (bits n r)) = n

⊦ ∀d r. fromRandom d 0 r = ([], r)

⊦ ∀r. bit r = (head (toStream r), fromStream (tail (toStream r)))

⊦ ∀d n r. length (fst (fromRandom d n r)) = n

⊦ ∀r.
Geometric.fromRandom r =
let (r₁, r₂) ← split r in (Geometric.fromRandom.loop 0 r₁, r₂)

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

⊦ ∀d r.
Geometric.fromRandom d r =
let (n, r') ← Geometric.fromRandom r in fromRandom d n r'

⊦ ∀n r.
    Geometric.fromRandom.loop n r =
    let (b, r') ← bit r in
    if b then n else Geometric.fromRandom.loop (suc n) r'

⊦ ∀d r n.
fromRandom d (suc n) r =
let (h, r') ← d r in let (t, r'') ← fromRandom d n r' in (h :: t, r'')

External Type Operators

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

External Constants

=
select
Data
- Bool
  - ∀
  - ∧
  - ⇒
  - ∃
  - ∃!
  - ∨
  - ¬
  - cond
  - ⊥
  - ⊤
- List
  - ::
  - []
  - length
- Pair
  - ,
  - fst
- Stream
  - head
  - split
  - tail
Number
- Natural
  - suc
  - zero

Assumptions

⊦ ⊤

⊦ ¬⊥ ⇔ ⊤

⊦ ¬⊤ ⇔ ⊥

⊦ length [] = 0

⊦ ∀t. t ⇒ t

⊦ (∃) = λp. p ((select) p)

⊦ ∀t. (∀x. t) ⇔ t

⊦ ∀t. (∃x. t) ⇔ t

⊦ ∀t. (λx. t x) = t

⊦ (∀) = λp. p = λx. ⊤

⊦ ∀t. (⊤ ⇔ t) ⇔ t

⊦ ∀t. t ⇒ ⊤ ⇔ ⊤

⊦ (⇒) = λp q. p ∧ q ⇔ p

⊦ ∀t. (t ⇔ ⊤) ∨ (t ⇔ ⊥)

⊦ ∀t₁ t₂. (if ⊥ then t₁ else t₂) = t₂

⊦ ∀t₁ t₂. (if ⊤ then t₁ else t₂) = t₁

⊦ ∀a b. fst (a, b) = a

⊦ ∀p x. p x ⇒ p ((select) p)

⊦ ∀x. ∃a b. x = (a, b)

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

⊦ (∧) = λp q. (λf. f p q) = λf. f ⊤ ⊤

⊦ (∃) = λp. ∀q. (∀x. p x ⇒ q) ⇒ q

⊦ ∀m n. suc m = suc n ⇔ m = n

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

⊦ ∀f. ∃fn. ∀a b. fn (a, b) = f a b

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

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

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

⊦ ∀p g h. ∃f. ∀x. f x = if p x then f (g x) else h x

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