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

Documentation.SBV.Examples.WeakestPreconditions.IntDiv

Contents

Program state
The algorithm
Correctness

Description

Proof of correctness of an imperative integer division algorithm, using weakest preconditions. The algorithm simply keeps subtracting the divisor until the desired quotient and the remainder is found.

Synopsis

data DivS a = DivS {
- x :: a
- y :: a
- q :: a
- r :: a
}
type D = DivS SInteger
algorithm :: Invariant D -> Maybe (Measure D) -> Stmt D
pre :: D -> SBool
post :: D -> SBool
noChange :: Stable D
imperativeDiv :: Invariant D -> Maybe (Measure D) -> Program D
invariant :: Invariant D
measure :: Measure D
correctness :: IO ()

Program state

data DivS a Source #

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

Constructors

DivS
Fields x :: a The dividend y :: a The divisor q :: a The quotient r :: a The remainder

Instances

Functor DivS Source #
Instance details Defined in Documentation.SBV.Examples.WeakestPreconditions.IntDiv Methods fmap :: (a -> b) -> DivS a -> DivS b # (<$) :: a -> DivS b -> DivS a #
Foldable DivS Source #
Instance details Defined in Documentation.SBV.Examples.WeakestPreconditions.IntDiv Methods fold :: Monoid m => DivS m -> m # foldMap :: Monoid m => (a -> m) -> DivS a -> m # foldr :: (a -> b -> b) -> b -> DivS a -> b # foldr' :: (a -> b -> b) -> b -> DivS a -> b # foldl :: (b -> a -> b) -> b -> DivS a -> b # foldl' :: (b -> a -> b) -> b -> DivS a -> b # foldr1 :: (a -> a -> a) -> DivS a -> a # foldl1 :: (a -> a -> a) -> DivS a -> a # toList :: DivS a -> [a] # null :: DivS a -> Bool # length :: DivS a -> Int # elem :: Eq a => a -> DivS a -> Bool # maximum :: Ord a => DivS a -> a # minimum :: Ord a => DivS a -> a # sum :: Num a => DivS a -> a # product :: Num a => DivS a -> a #
Traversable DivS Source #
Instance details Defined in Documentation.SBV.Examples.WeakestPreconditions.IntDiv Methods traverse :: Applicative f => (a -> f b) -> DivS a -> f (DivS b) # sequenceA :: Applicative f => DivS (f a) -> f (DivS a) # mapM :: Monad m => (a -> m b) -> DivS a -> m (DivS b) # sequence :: Monad m => DivS (m a) -> m (DivS a) #
(SymVal a, SMTValue a) => Fresh IO (DivS (SBV a)) Source #	`Fresh` instance for the program state
Instance details Defined in Documentation.SBV.Examples.WeakestPreconditions.IntDiv Methods fresh :: QueryT IO (DivS (SBV a)) Source #
Show a => Show (DivS a) Source #
Instance details Defined in Documentation.SBV.Examples.WeakestPreconditions.IntDiv Methods showsPrec :: Int -> DivS a -> ShowS # show :: DivS a -> String # showList :: [DivS a] -> ShowS #
(SymVal a, Show a) => Show (DivS (SBV a)) Source #	Show instance for `DivS`. 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.IntDiv Methods showsPrec :: Int -> DivS (SBV a) -> ShowS # show :: DivS (SBV a) -> String # showList :: [DivS (SBV a)] -> ShowS #
Generic (DivS a) Source #
Instance details Defined in Documentation.SBV.Examples.WeakestPreconditions.IntDiv Associated Types type Rep (DivS a) :: Type -> Type # Methods from :: DivS a -> Rep (DivS a) x # to :: Rep (DivS a) x -> DivS a #
Mergeable a => Mergeable (DivS a) Source #
Instance details Defined in Documentation.SBV.Examples.WeakestPreconditions.IntDiv Methods symbolicMerge :: Bool -> SBool -> DivS a -> DivS a -> DivS a Source # select :: (Ord b, SymVal b, Num b) => [DivS a] -> DivS a -> SBV b -> DivS a Source #
type Rep (DivS a) Source #
Instance details Defined in Documentation.SBV.Examples.WeakestPreconditions.IntDiv type Rep (DivS a) = D1 (MetaData "DivS" "Documentation.SBV.Examples.WeakestPreconditions.IntDiv" "sbv-8.4-LKcvIZGSCFCGr6QyM4W6l5" False) (C1 (MetaCons "DivS" PrefixI True) ((S1 (MetaSel (Just "x") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 a) :: S1 (MetaSel (Just "y") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 a)) :: (S1 (MetaSel (Just "q") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 a) :*: S1 (MetaSel (Just "r") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 a))))

type D = DivS SInteger Source #

Helper type synonym

The algorithm

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

The imperative division algorithm, assuming non-negative x and strictly positive y:

   r = x                     -- set remainder to x
   q = 0                     -- set quotient  to 0
   while y <= r              -- while we can still subtract
     r = r - y                    -- reduce the remainder
     q = q + 1                    -- increase the quotient

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 :: D -> SBool Source #

Precondition for our program: x must non-negative and y must be strictly positive. 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 :: D -> SBool Source #

Postcondition for our program: Remainder must be non-negative and less than y, and it must hold that x = q*y + r:

noChange :: Stable D Source #

Stability: x and y must remain unchanged.

imperativeDiv :: Invariant D -> Maybe (Measure D) -> Program D Source #

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

Correctness

invariant :: Invariant D Source #

The invariant is simply that x = q * y + r holds at all times and r is strictly positive. We need the y > 0 part of the invariant to establish the measure decreases, which is guaranteed by our precondition.

measure :: Measure D Source #

The measure. In each iteration r decreases, but always remains positive. Since y is strictly positive, r can serve as a measure for the loop.

correctness :: IO () Source #

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

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