Copyright	(c) Levent Erkok
License	BSD3
Maintainer	erkokl@gmail.com
Stability	experimental
Safe Haskell	None
Language	Haskell2010

Documentation.SBV.Examples.ProofTools.Fibonacci

Contents

System state

Description

Example inductive proof to show partial correctness of the for-loop based fibonacci algorithm:

    i = 0
    k = 1
    m = 0
    while i < n:
       m, k = k, m + k
       i++

We do the proof against an axiomatized fibonacci implementation using an uninterpreted function.

Synopsis

data S a = S {
- i :: a
- k :: a
- m :: a
- n :: a
}
fibCorrect :: IO (InductionResult (S Integer))

System state

data S a Source #

System state. We simply have two components, parameterized over the type so we can put in both concrete and symbolic values.

Constructors

S
Fields i :: a k :: a m :: a n :: a

Instances

Instances details

Functor S Source #
Instance details Defined in Documentation.SBV.Examples.ProofTools.Fibonacci Methods fmap :: (a -> b) -> S a -> S b # (<$) :: a -> S b -> S a #
Foldable S Source #
Instance details Defined in Documentation.SBV.Examples.ProofTools.Fibonacci Methods fold :: Monoid m => S m -> m # foldMap :: Monoid m => (a -> m) -> S a -> m # foldMap' :: Monoid m => (a -> m) -> S a -> m # foldr :: (a -> b -> b) -> b -> S a -> b # foldr' :: (a -> b -> b) -> b -> S a -> b # foldl :: (b -> a -> b) -> b -> S a -> b # foldl' :: (b -> a -> b) -> b -> S a -> b # foldr1 :: (a -> a -> a) -> S a -> a # foldl1 :: (a -> a -> a) -> S a -> a # toList :: S a -> [a] # null :: S a -> Bool # length :: S a -> Int # elem :: Eq a => a -> S a -> Bool # maximum :: Ord a => S a -> a # minimum :: Ord a => S a -> a # sum :: Num a => S a -> a # product :: Num a => S a -> a #
Traversable S Source #
Instance details Defined in Documentation.SBV.Examples.ProofTools.Fibonacci Methods traverse :: Applicative f => (a -> f b) -> S a -> f (S b) # sequenceA :: Applicative f => S (f a) -> f (S a) # mapM :: Monad m => (a -> m b) -> S a -> m (S b) # sequence :: Monad m => S (m a) -> m (S a) #
Fresh IO (S SInteger) Source #	`Fresh` instance for our state
Instance details Defined in Documentation.SBV.Examples.ProofTools.Fibonacci Methods fresh :: QueryT IO (S SInteger) Source #
Show a => Show (S a) Source #
Instance details Defined in Documentation.SBV.Examples.ProofTools.Fibonacci Methods showsPrec :: Int -> S a -> ShowS # show :: S a -> String # showList :: [S a] -> ShowS #
Generic (S a) Source #
Instance details Defined in Documentation.SBV.Examples.ProofTools.Fibonacci Associated Types type Rep (S a) :: Type -> Type # Methods from :: S a -> Rep (S a) x # to :: Rep (S a) x -> S a #
Mergeable a => Mergeable (S a) Source #
Instance details Defined in Documentation.SBV.Examples.ProofTools.Fibonacci Methods symbolicMerge :: Bool -> SBool -> S a -> S a -> S a Source # select :: (Ord b, SymVal b, Num b) => [S a] -> S a -> SBV b -> S a Source #
type Rep (S a) Source #
Instance details Defined in Documentation.SBV.Examples.ProofTools.Fibonacci type Rep (S a) = D1 ('MetaData "S" "Documentation.SBV.Examples.ProofTools.Fibonacci" "sbv-8.12-86sGFtUuaUAKiWFK6rsFH6" 'False) (C1 ('MetaCons "S" 'PrefixI 'True) ((S1 ('MetaSel ('Just "i") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 a) :: S1 ('MetaSel ('Just "k") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 a)) :: (S1 ('MetaSel ('Just "m") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 a) :*: S1 ('MetaSel ('Just "n") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 a))))

fibCorrect :: IO (InductionResult (S Integer)) Source #

Encoding partial correctness of the sum algorithm. We have:

>>> fibCorrect
Q.E.D.

NB. In my experiments, I found that this proof is quite fragile due to the use of quantifiers: If you make a mistake in your algorithm or the coding, z3 pretty much spins forever without finding a counter-example. However, with the correct coding, the proof is almost instantaneous!