Documentation.SBV.Examples.WeakestPreconditions.IntSqrt

Proof of correctness of an imperative integer square-root algorithm, using weakest preconditions. The algorithm computes the floor of the square-root of a given non-negative integer by keeping a running some of all odd numbers starting from 1. Recall that 1+3+5+...+(2n+1) = (n+1)^2, thus we can stop the counting when we exceed the input number.

Program state

data SqrtS a

The state for the division program, parameterized over a base type a.

Constructors

SqrtS

Fields

• x :: a

  The input

• sqrt :: a

  The floor of the square root

• i :: a

  Successive squares, as the sum of j's

• j :: a

  Successive odds

Instances

Functor SqrtS

Defined in Documentation.SBV.Examples.WeakestPreconditions.IntSqrt

Methods

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

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

Foldable SqrtS

Defined in Documentation.SBV.Examples.WeakestPreconditions.IntSqrt

Methods

fold :: Monoid m => SqrtS m -> m

foldMap :: Monoid m => (a -> m) -> SqrtS a -> m

foldMap' :: Monoid m => (a -> m) -> SqrtS a -> m

foldr :: (a -> b -> b) -> b -> SqrtS a -> b

foldr' :: (a -> b -> b) -> b -> SqrtS a -> b

foldl :: (b -> a -> b) -> b -> SqrtS a -> b

foldl' :: (b -> a -> b) -> b -> SqrtS a -> b

foldr1 :: (a -> a -> a) -> SqrtS a -> a

foldl1 :: (a -> a -> a) -> SqrtS a -> a

toList :: SqrtS a -> [a]

null :: SqrtS a -> Bool

length :: SqrtS a -> Int

elem :: Eq a => a -> SqrtS a -> Bool

maximum :: Ord a => SqrtS a -> a

minimum :: Ord a => SqrtS a -> a

sum :: Num a => SqrtS a -> a

product :: Num a => SqrtS a -> a

Traversable SqrtS

Defined in Documentation.SBV.Examples.WeakestPreconditions.IntSqrt

Methods

traverse :: Applicative f => (a -> f b) -> SqrtS a -> f (SqrtS b)

sequenceA :: Applicative f => SqrtS (f a) -> f (SqrtS a)

mapM :: Monad m => (a -> m b) -> SqrtS a -> m (SqrtS b)

sequence :: Monad m => SqrtS (m a) -> m (SqrtS a)

SymVal a => Fresh IO (SqrtS (SBV a))

Fresh instance for the program state

Defined in Documentation.SBV.Examples.WeakestPreconditions.IntSqrt

Methods

fresh :: QueryT IO (SqrtS (SBV a))

Show a => Show (SqrtS a)

Defined in Documentation.SBV.Examples.WeakestPreconditions.IntSqrt

Methods

showsPrec :: Int -> SqrtS a -> ShowS

show :: SqrtS a -> String

showList :: [SqrtS a] -> ShowS

(SymVal a, Show a) => Show (SqrtS (SBV a))

Show instance for SqrtS. The above deriving clause would work just as well, but we want it to be a little prettier here, and hence the OVERLAPS directive.

Defined in Documentation.SBV.Examples.WeakestPreconditions.IntSqrt

Methods

showsPrec :: Int -> SqrtS (SBV a) -> ShowS

show :: SqrtS (SBV a) -> String

showList :: [SqrtS (SBV a)] -> ShowS

Generic (SqrtS a)

Defined in Documentation.SBV.Examples.WeakestPreconditions.IntSqrt

Associated Types

type Rep (SqrtS a) :: Type -> Type

Methods

from :: SqrtS a -> Rep (SqrtS a) x

to :: Rep (SqrtS a) x -> SqrtS a

Mergeable a => Mergeable (SqrtS a)

Defined in Documentation.SBV.Examples.WeakestPreconditions.IntSqrt

Methods

symbolicMerge :: Bool -> SBool -> SqrtS a -> SqrtS a -> SqrtS a

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

type Rep (SqrtS a)

Defined in Documentation.SBV.Examples.WeakestPreconditions.IntSqrt

type Rep (SqrtS a) = D1 ('MetaData "SqrtS" "Documentation.SBV.Examples.WeakestPreconditions.IntSqrt" "sbv-8.9-ATrzvBVsNws6CzdQ7iCv1r" 'False) (C1 ('MetaCons "SqrtS" 'PrefixI 'True) ((S1 ('MetaSel ('Just "x") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 a) :*: S1 ('MetaSel ('Just "sqrt") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 a)) :*: (S1 ('MetaSel ('Just "i") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 a) :*: S1 ('MetaSel ('Just "j") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 a))))

type S = SqrtS SInteger

Helper type synonym

The algorithm

algorithm :: Invariant S -> Maybe (Measure S) -> Stmt S

The imperative square-root algorithm, assuming non-negative x

   sqrt = 0                  -- set sqrt to 0
   i    = 1                  -- set i to 1, sum of j's so far
   j    = 1                  -- set j to be the first odd number i
   while i <= x              -- while the sum hasn't exceeded x yet
     sqrt = sqrt + 1              -- increase the sqrt
     j    = j + 2                 -- next odd number
     i    = i + j                 -- running sum of j's

Note that we need to explicitly annotate each loop with its invariant and the termination measure. For convenience, we take those two as parameters for simplicity.

pre :: S -> SBool

Precondition for our program: x must be non-negative. Note that there is an explicit call to abort in our program to protect against this case, so if we do not have this precondition, all programs will fail.

post :: S -> SBool

Postcondition for our program: The sqrt squared must be less than or equal to x, and sqrt+1 squared must strictly exceed x.

noChange :: Stable S

Stability condition: Program must leave x unchanged.

imperativeSqrt :: Invariant S -> Maybe (Measure S) -> Program S

A program is the algorithm, together with its pre- and post-conditions.

Correctness

invariant :: Invariant S

The invariant is that at each iteration of the loop sqrt remains below or equal to the actual square-root, and i tracks the square of the next value. We also have that j is the sqrt'th odd value. Coming up with this invariant is not for the faint of heart, for details I would strongly recommend looking at Manna's seminal Mathematical Theory of Computation book (chapter 3). The j .> 0 part is needed to establish the termination.

measure :: Measure S

The measure. In each iteration i strictly increases, thus reducing the differential x - i

correctness :: IO ()

Check that the program terminates and the post condition holds. We have:

>>> correctness
Total correctness is established.
Q.E.D.