Documentation.SBV.Examples.ProofTools.Fibonacci

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.

System state

data S a

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

Foldable S

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

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)

Functor S

Defined in Documentation.SBV.Examples.ProofTools.Fibonacci

Methods

fmap :: (a -> b) -> S a -> S b

(<$) :: a -> S b -> S a

Fresh IO (S SInteger)

Fresh instance for our state

Defined in Documentation.SBV.Examples.ProofTools.Fibonacci

Methods

fresh :: QueryT IO (S SInteger)

Generic (S a)

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

Show a => Show (S a)

Defined in Documentation.SBV.Examples.ProofTools.Fibonacci

Methods

showsPrec :: Int -> S a -> ShowS

show :: S a -> String

showList :: [S a] -> ShowS

Mergeable a => Mergeable (S a)

Defined in Documentation.SBV.Examples.ProofTools.Fibonacci

Methods

symbolicMerge :: Bool -> SBool -> S a -> S a -> S a

select :: (Ord b, SymVal b, Num b) => [S a] -> S a -> SBV b -> S a

type Rep (S a)

Defined in Documentation.SBV.Examples.ProofTools.Fibonacci

type Rep (S a) = D1 ('MetaData "S" "Documentation.SBV.Examples.ProofTools.Fibonacci" "sbv-9.1-K2obF3Nc7RH75SD7QUXzc3" '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))

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!