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

Documentation.SBV.Examples.ProofTools.Sum

Contents

System state

Description

Example inductive proof to show partial correctness of the traditional for-loop sum algorithm:

    s = 0
    i = 0
    while i <= n:
       s += i
       i++

We prove the loop invariant and establish partial correctness that s is the sum of all numbers up to and including n upon termination.

Synopsis

data S a = S {
- s :: a
- i :: a
- n :: a
}
sumCorrect :: 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 s :: a i :: a n :: a

Instances

Instances details

Functor S Source #
Instance details Defined in Documentation.SBV.Examples.ProofTools.Sum Methods fmap :: (a -> b) -> S a -> S b # (<$) :: a -> S b -> S a #
Foldable S Source #
Instance details Defined in Documentation.SBV.Examples.ProofTools.Sum 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.Sum 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.Sum Methods fresh :: QueryT IO (S SInteger) Source #
Show a => Show (S a) Source #
Instance details Defined in Documentation.SBV.Examples.ProofTools.Sum 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.Sum 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.Sum 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.Sum type Rep (S a) = D1 ('MetaData "S" "Documentation.SBV.Examples.ProofTools.Sum" "sbv-8.9-ATrzvBVsNws6CzdQ7iCv1r" 'False) (C1 ('MetaCons "S" 'PrefixI 'True) (S1 ('MetaSel ('Just "s") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 a) :: (S1 ('MetaSel ('Just "i") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 a) :: S1 ('MetaSel ('Just "n") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 a))))

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

Encoding partial correctness of the sum algorithm. We have:

>>> sumCorrect
Q.E.D.