Documentation.SBV.Examples.WeakestPreconditions.Basics

Some basic aspects of weakest preconditions, demonstrating programs that do not use while loops. We use a simple increment program as an example.

Program state

data IncS a

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

Constructors

IncS

Fields

• x :: a

  Input value

• y :: a

  Output

Instances

Functor IncS

Defined in Documentation.SBV.Examples.WeakestPreconditions.Basics

Methods

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

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

Foldable IncS

Defined in Documentation.SBV.Examples.WeakestPreconditions.Basics

Methods

fold :: Monoid m => IncS m -> m

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

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

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

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

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

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

foldr1 :: (a -> a -> a) -> IncS a -> a

foldl1 :: (a -> a -> a) -> IncS a -> a

toList :: IncS a -> [a]

null :: IncS a -> Bool

length :: IncS a -> Int

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

maximum :: Ord a => IncS a -> a

minimum :: Ord a => IncS a -> a

sum :: Num a => IncS a -> a

product :: Num a => IncS a -> a

Traversable IncS

Defined in Documentation.SBV.Examples.WeakestPreconditions.Basics

Methods

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

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

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

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

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

Fresh instance for the program state

Defined in Documentation.SBV.Examples.WeakestPreconditions.Basics

Methods

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

Show a => Show (IncS a)

Defined in Documentation.SBV.Examples.WeakestPreconditions.Basics

Methods

showsPrec :: Int -> IncS a -> ShowS

show :: IncS a -> String

showList :: [IncS a] -> ShowS

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

Show instance for IncS. 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.Basics

Methods

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

show :: IncS (SBV a) -> String

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

Generic (IncS a)

Defined in Documentation.SBV.Examples.WeakestPreconditions.Basics

Associated Types

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

Methods

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

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

Mergeable a => Mergeable (IncS a)

Defined in Documentation.SBV.Examples.WeakestPreconditions.Basics

Methods

symbolicMerge :: Bool -> SBool -> IncS a -> IncS a -> IncS a

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

type Rep (IncS a)

Defined in Documentation.SBV.Examples.WeakestPreconditions.Basics

type Rep (IncS a) = D1 ('MetaData "IncS" "Documentation.SBV.Examples.WeakestPreconditions.Basics" "sbv-8.14-15M8sp7rKYE8iwne1Q2gA" 'False) (C1 ('MetaCons "IncS" 'PrefixI 'True) (S1 ('MetaSel ('Just "x") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 a) :*: S1 ('MetaSel ('Just "y") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 a)))

type I = IncS SInteger

Helper type synonym

The algorithm

algorithm :: Stmt I -> Stmt I -> Stmt I

The increment algorithm:

   y = x+1

The point here isn't really that this program is interesting, but we want to demonstrate various aspects of WP proofs. So, we take a before and after program to annotate our algorithm so we can experiment later.

pre :: I -> SBool

Precondition for our program. Strictly speaking, we don't really need any preconditions, but for example purposes, we'll require x to be non-negative.

post :: I -> SBool

Postcondition for our program: y must x+1.

noChange :: Stable I

Stability: x must remain unchanged.

imperativeInc :: Stmt I -> Stmt I -> Program I

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

Correctness

correctness :: Stmt I -> Stmt I -> IO (ProofResult (IncS Integer))

State the correctness with respect to before/after programs. In the simple case of nothing prior/after, we have the obvious proof:

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

Example proof attempts

It is instructive to look at how the proof changes as we put in different pre and post values.

Violating the post condition

If we stick in an extra increment for y after, we can easily break the postcondition:

>>> :set -XNamedFieldPuns
>>> import Control.Monad (void)
>>> void $ correctness Skip $ Assign $ \st@IncS{y} -> st{y = y+1}
Following proof obligation failed:
==================================
  Postcondition fails:
    Start: IncS {x = 0, y = 0}
    End  : IncS {x = 0, y = 2}

We're told that the program ends up in a state where x=0 and y=2, violating the requirement y=x+1, as expected.

Using assert

There are two main use cases for assert, which merely ends up being a call to Abort. One is making sure the inputs are well formed. And the other is the user putting in their own invariants into the code.

Let's assume that we only want to accept strictly positive values of x. We can try:

>>> void $ correctness (assert "x > 0" (\st@IncS{x} -> x .> 0)) Skip
Following proof obligation failed:
==================================
  Abort "x > 0" condition is satisfiable:
    Before: IncS {x = 0, y = 0}
    After : IncS {x = 0, y = 0}

Recall that our precondition (pre) required x to be non-negative. So, we can put in something weaker and it would be fine:

>>> void $ correctness (assert "x > -5" (\st@IncS{x} -> x .> -5)) Skip
Total correctness is established.

In this case the precondition to our program ensures that the assert will always be satisfied.

As another example, let us put a post assertion that y is even:

>>> void $ correctness Skip (assert "y is even" (\st@IncS{y} -> y `sMod` 2 .== 0))
Following proof obligation failed:
==================================
  Abort "y is even" condition is satisfiable:
    Before: IncS {x = 0, y = 0}
    After : IncS {x = 0, y = 1}

It is important to emphasize that you can put whatever invariant you might want:

>>> void $ correctness Skip (assert "y > x" (\st@IncS{x, y} -> y .> x))
Total correctness is established.

Violating stability

What happens if our program modifies x? After all, we can simply set x=10 and y=11 and our post condition would be satisfied:

>>> void $ correctness Skip (Assign $ \st -> st{x = 10, y = 11})
Following proof obligation failed:
==================================
  Stability fails for "x":
    Before: IncS {x = 0, y = 1}
    After : IncS {x = 10, y = 11}

So, the stability condition prevents programs from cheating!