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

Documentation.SBV.Examples.WeakestPreconditions.IntSqrt

Contents

Program state
The algorithm
Correctness

Description

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.

Synopsis

data SqrtS a = SqrtS {
- x :: a
- sqrt :: a
- i :: a
- j :: a
}
type S = SqrtS SInteger
algorithm :: Invariant S -> Maybe (Measure S) -> Stmt S
pre :: S -> SBool
post :: S -> SBool
noChange :: Stable S
imperativeSqrt :: Invariant S -> Maybe (Measure S) -> Program S
invariant :: Invariant S
measure :: Measure S
correctness :: IO ()

Program state

data SqrtS a Source #

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 Source #
Instance details Defined in Documentation.SBV.Examples.WeakestPreconditions.IntSqrt Methods fmap :: (a -> b) -> SqrtS a -> SqrtS b # (<$) :: a -> SqrtS b -> SqrtS a #
Foldable SqrtS Source #
Instance details Defined in Documentation.SBV.Examples.WeakestPreconditions.IntSqrt Methods fold :: Monoid m => SqrtS m -> 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 Source #
Instance details 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, SMTValue a) => Fresh IO (SqrtS (SBV a)) Source #	`Fresh` instance for the program state
Instance details Defined in Documentation.SBV.Examples.WeakestPreconditions.IntSqrt Methods fresh :: QueryT IO (SqrtS (SBV a)) Source #
Show a => Show (SqrtS a) Source #
Instance details 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)) Source #	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.
Instance details 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) Source #
Instance details 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) Source #
Instance details Defined in Documentation.SBV.Examples.WeakestPreconditions.IntSqrt Methods symbolicMerge :: Bool -> SBool -> SqrtS a -> SqrtS a -> SqrtS a Source # select :: (Ord b, SymVal b, Num b) => [SqrtS a] -> SqrtS a -> SBV b -> SqrtS a Source #
type Rep (SqrtS a) Source #
Instance details Defined in Documentation.SBV.Examples.WeakestPreconditions.IntSqrt type Rep (SqrtS a) = D1 (MetaData "SqrtS" "Documentation.SBV.Examples.WeakestPreconditions.IntSqrt" "sbv-8.4-LKcvIZGSCFCGr6QyM4W6l5" 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 Source #

Helper type synonym

The algorithm

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

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 Source #

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 Source #

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 Source #

Stability condition: Program must leave x unchanged.

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

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

Correctness

invariant :: Invariant S Source #

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 Source #

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

correctness :: IO () Source #

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

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