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

Documentation.SBV.Examples.WeakestPreconditions.Length

Contents

Program state
The algorithm
Correctness

Description

Proof of correctness of an imperative list-length algorithm, using weakest preconditions. Illustrates the use of SBV's symbolic lists together with the WP algorithm.

Synopsis

data LenS a = LenS {
- xs :: SList a
- ys :: SList a
- l :: SInteger
}
data LenC a = LenC [a] [a] Integer
type S = LenS Integer
algorithm :: Invariant S -> Maybe (Measure S) -> Stmt S
pre :: S -> SBool
post :: S -> SBool
noChange :: Stable S
imperativeLength :: Invariant S -> Maybe (Measure S) -> Program S
invariant :: Invariant S
measure :: Measure S
correctness :: IO ()

Program state

data LenS a Source #

The state of the length program, paramaterized over the element type a

Constructors

LenS
Fields xs :: SList a The input list ys :: SList a Copy of input l :: SInteger Running length

Instances

Queriable IO (LenS Integer) (LenC Integer) Source #	We have to write the bijection between `LenS` and `LenC` explicitly. Luckily, the definition is more or less boilerplate.
Instance details Defined in Documentation.SBV.Examples.WeakestPreconditions.Length Methods create :: QueryT IO (LenS Integer) Source # project :: LenS Integer -> QueryT IO (LenC Integer) Source # embed :: LenC Integer -> QueryT IO (LenS Integer) Source #
(SymVal a, Show a) => Show (LenS a) Source #	Show instance: A simplified version of what would otherwise be generated.
Instance details Defined in Documentation.SBV.Examples.WeakestPreconditions.Length Methods showsPrec :: Int -> LenS a -> ShowS # show :: LenS a -> String # showList :: [LenS a] -> ShowS #
Generic (LenS a) Source #
Instance details Defined in Documentation.SBV.Examples.WeakestPreconditions.Length Associated Types type Rep (LenS a) :: Type -> Type # Methods from :: LenS a -> Rep (LenS a) x # to :: Rep (LenS a) x -> LenS a #
SymVal a => Mergeable (LenS a) Source #
Instance details Defined in Documentation.SBV.Examples.WeakestPreconditions.Length Methods symbolicMerge :: Bool -> SBool -> LenS a -> LenS a -> LenS a Source # select :: (Ord b, SymVal b, Num b) => [LenS a] -> LenS a -> SBV b -> LenS a Source #
type Rep (LenS a) Source #
Instance details Defined in Documentation.SBV.Examples.WeakestPreconditions.Length type Rep (LenS a) = D1 (MetaData "LenS" "Documentation.SBV.Examples.WeakestPreconditions.Length" "sbv-8.4-LKcvIZGSCFCGr6QyM4W6l5" False) (C1 (MetaCons "LenS" PrefixI True) (S1 (MetaSel (Just "xs") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 (SList a)) :: (S1 (MetaSel (Just "ys") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 (SList a)) :: S1 (MetaSel (Just "l") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 SInteger))))

data LenC a Source #

The concrete counterpart to LenS. Note that we can no longer use the duality between SBV a and a as in other examples and just use one datatype for both. This is because SList a and [a] are fundamentally different types. This can be a bit confusing at first, but the point is that it is the list that is symbolic in case of an SList a, not that we have a concrete list with symbolic elements in it. Subtle difference, but it is important to keep these two separate.

Constructors

LenC [a] [a] Integer

Instances

Queriable IO (LenS Integer) (LenC Integer) Source #	We have to write the bijection between `LenS` and `LenC` explicitly. Luckily, the definition is more or less boilerplate.
Instance details Defined in Documentation.SBV.Examples.WeakestPreconditions.Length Methods create :: QueryT IO (LenS Integer) Source # project :: LenS Integer -> QueryT IO (LenC Integer) Source # embed :: LenC Integer -> QueryT IO (LenS Integer) Source #
Show a => Show (LenC a) Source #	Show instance: Similarly, we want to be a bit more concise here.
Instance details Defined in Documentation.SBV.Examples.WeakestPreconditions.Length Methods showsPrec :: Int -> LenC a -> ShowS # show :: LenC a -> String # showList :: [LenC a] -> ShowS #

type S = LenS Integer Source #

Helper type synonym

The algorithm

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

The imperative length algorithm:

   ys = xs
   l  = 0
   while not (null ys)
     l  = l+1
     ys = tail ys

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

pre :: S -> SBool Source #

Precondition for our program. Nothing! It works for all lists.

post :: S -> SBool Source #

Postcondition for our program: l must be the length of the input list.

noChange :: Stable S Source #

Stability condition: Program must leave xs unchanged.

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

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

invariant :: Invariant S Source #

The invariant simply relates the length of the input to the length of the current suffix and the length of the prefix traversed so far.

measure :: Measure S Source #

The measure is obviously the length of ys, as we peel elements off of it through the loop.

Correctness

correctness :: IO () Source #

We check that l is the length of the input list xs upon termination. Note that even though this is an inductive proof, it is fairly easy to prove with our SMT based technology, which doesn't really handle induction at all! The usual inductive proof steps are baked into the invariant establishment phase of the WP proof. We have:

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