Package probability: Probability
Information
| name | probability |
| version | 1.49 |
| description | Probability |
| author | Joe Leslie-Hurd <joe@gilith.com> |
| license | MIT |
| homepage | http://opentheory.gilith.com/?pkg=probability |
| haskell-int-file | haskell.int |
| haskell-src-file | haskell.art |
| haskell-arbitrary-type | "Probability.Random.random" |
| checksum | 3ec7e1a12000492fc348de4d4df4a667c4c21204 |
| requires | base stream |
| show | Data.Bool Data.List Data.Pair Data.Stream Function Number.Natural Probability.Random |
Files
- Package tarball probability-1.49.tgz
- Theory source file probability.thy (included in the package tarball)
Defined Type Operator
- Probability
- Random
- random
- Random
Defined Constants
- Data
- List
- random
- Geometric
- Geometric.random
- List
- Number
- Natural
- Geometric
- Geometric.random
- Geometric.random.loop
- Geometric.random
- Geometric
- Natural
- Probability
- Random
- bit
- bits
- fromStream
- split
- toStream
- Random
Theorems
⊦ Geometric.random = Geometric.random.loop 0
⊦ bits = random bit
⊦ ∀r. fromStream (toStream r) = r
⊦ ∀s. toStream (fromStream s) = s
⊦ ∀b. ∃r. bit r ⇔ b
⊦ ∀r. bit r ⇔ head (toStream r)
⊦ ∀n r. length (bits n r) = n
⊦ ∀f r. random f 0 r = []
⊦ ∀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
⊦ ∀r.
split r =
let (s1, s2) ← split (toStream r) in (fromStream s1, fromStream s2)
⊦ ∀f r.
Geometric.random f r =
let (r1, r2) ← split r in random f (Geometric.random r1) r2
⊦ ∀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
⊦ ∀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 r n.
random f (suc n) r = let (r1, r2) ← split r in f r1 :: random f n 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
- List
- Number
- Natural
- natural
- Natural
External Constants
- =
- select
- Data
- Bool
- ∀
- ∧
- ⇒
- ∃
- ∃!
- ∨
- ¬
- cond
- ⊥
- ⊤
- List
- ::
- []
- length
- Pair
- ,
- Stream
- head
- interleave
- nth
- replicate
- split
- Bool
- Function
- surjective
- Number
- Natural
- +
- -
- bit1
- suc
- zero
- Natural
Assumptions
⊦ ⊤
⊦ ¬⊥ ⇔ ⊤
⊦ ¬⊤ ⇔ ⊥
⊦ length [] = 0
⊦ ⊥ ⇔ ∀p. p
⊦ (¬) = λp. p ⇒ ⊥
⊦ (∃) = 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 ⇔ t
⊦ ∀t. t ⇒ ⊤ ⇔ ⊤
⊦ ∀n. ¬(suc n = 0)
⊦ ∀s. head s = nth s 0
⊦ (⇒) = λp q. p ∧ q ⇔ p
⊦ ∀t. (t ⇔ ⊤) ∨ (t ⇔ ⊥)
⊦ ∀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)
⊦ ∀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
⊦ (∃!) = 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
⊦ ∀p. p [] ∧ (∀h t. p t ⇒ p (h :: t)) ⇒ ∀l. p l