sbv-8.7: SMT Based Verification: Symbolic Haskell theorem prover using SMT solving.

Documentation.SBV.Examples.WeakestPreconditions.Basics

Description

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

Synopsis

# Program state

data IncS a Source #

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

Constructors

 IncS Fieldsx :: aInput valuey :: aOutput
Instances
 Source # Instance details Methodsfmap :: (a -> b) -> IncS a -> IncS b #(<$) :: a -> IncS b -> IncS a # Source # Instance details Methodsfold :: Monoid m => IncS m -> 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 # Source # Instance details Methodstraverse :: 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)) Source # Fresh instance for the program state Instance details Methodsfresh :: QueryT IO (IncS (SBV a)) Source # Show a => Show (IncS a) Source # Instance details MethodsshowsPrec :: Int -> IncS a -> ShowS #show :: IncS a -> String #showList :: [IncS a] -> ShowS # (SymVal a, Show a) => Show (IncS (SBV a)) Source # 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. Instance details MethodsshowsPrec :: Int -> IncS (SBV a) -> ShowS #show :: IncS (SBV a) -> String #showList :: [IncS (SBV a)] -> ShowS # Generic (IncS a) Source # Instance details Associated Typestype Rep (IncS a) :: Type -> Type # Methodsfrom :: IncS a -> Rep (IncS a) x #to :: Rep (IncS a) x -> IncS a # Mergeable a => Mergeable (IncS a) Source # Instance details MethodssymbolicMerge :: Bool -> SBool -> IncS a -> IncS a -> IncS a Source #select :: (Ord b, SymVal b, Num b) => [IncS a] -> IncS a -> SBV b -> IncS a Source # type Rep (IncS a) Source # Instance details type Rep (IncS a) = D1 (MetaData "IncS" "Documentation.SBV.Examples.WeakestPreconditions.Basics" "sbv-8.7-DbQHjiKtor73WzWR2O4MT3" 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 Source # Helper type synonym # The algorithm 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 Source # 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. Postcondition for our program: y must x+1. Stability: x must remain unchanged. A program is the algorithm, together with its pre- and post-conditions. # Correctness 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!