-----------------------------------------------------------------------------
-- |
-- Module    : Documentation.SBV.Examples.WeakestPreconditions.Basics
-- Copyright : (c) Levent Erkok
-- License   : BSD3
-- Maintainer: erkokl@gmail.com
-- Stability : experimental
--
-- Some basic aspects of weakest preconditions, demonstrating programs
-- that do not use while loops. We use a simple increment program as
-- an example.
-----------------------------------------------------------------------------

{-# LANGUAGE CPP                   #-}
{-# LANGUAGE DeriveAnyClass        #-}
{-# LANGUAGE DeriveGeneric         #-}
{-# LANGUAGE DeriveTraversable     #-}
{-# LANGUAGE FlexibleInstances     #-}
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE NamedFieldPuns        #-}
{-# LANGUAGE TypeFamilies          #-}

{-# OPTIONS_GHC -Wall -Werror #-}

module Documentation.SBV.Examples.WeakestPreconditions.Basics where

import Data.SBV
import Data.SBV.Tools.WeakestPreconditions

import GHC.Generics (Generic)

#ifndef HADDOCK
-- $setup
-- >>> -- For doctest purposes only:
-- >>> import Data.SBV
-- >>> import Data.SBV.Control
-- >>> import Data.SBV.Tools.WeakestPreconditions
#endif

-- * Program state

-- | The state for the swap program, parameterized over a base type @a@.
data IncS a = IncS { forall a. IncS a -> a
x :: a    -- ^ Input value
                   , forall a. IncS a -> a
y :: a    -- ^ Output
                   }
                   deriving (Int -> IncS a -> ShowS
[IncS a] -> ShowS
IncS a -> String
(Int -> IncS a -> ShowS)
-> (IncS a -> String) -> ([IncS a] -> ShowS) -> Show (IncS a)
forall a. Show a => Int -> IncS a -> ShowS
forall a. Show a => [IncS a] -> ShowS
forall a. Show a => IncS a -> String
forall a.
(Int -> a -> ShowS) -> (a -> String) -> ([a] -> ShowS) -> Show a
$cshowsPrec :: forall a. Show a => Int -> IncS a -> ShowS
showsPrec :: Int -> IncS a -> ShowS
$cshow :: forall a. Show a => IncS a -> String
show :: IncS a -> String
$cshowList :: forall a. Show a => [IncS a] -> ShowS
showList :: [IncS a] -> ShowS
Show, (forall x. IncS a -> Rep (IncS a) x)
-> (forall x. Rep (IncS a) x -> IncS a) -> Generic (IncS a)
forall x. Rep (IncS a) x -> IncS a
forall x. IncS a -> Rep (IncS a) x
forall a.
(forall x. a -> Rep a x) -> (forall x. Rep a x -> a) -> Generic a
forall a x. Rep (IncS a) x -> IncS a
forall a x. IncS a -> Rep (IncS a) x
$cfrom :: forall a x. IncS a -> Rep (IncS a) x
from :: forall x. IncS a -> Rep (IncS a) x
$cto :: forall a x. Rep (IncS a) x -> IncS a
to :: forall x. Rep (IncS a) x -> IncS a
Generic, Bool -> SBool -> IncS a -> IncS a -> IncS a
(Bool -> SBool -> IncS a -> IncS a -> IncS a)
-> (forall b.
    (Ord b, SymVal b, Num b, Num (SBV b)) =>
    [IncS a] -> IncS a -> SBV b -> IncS a)
-> Mergeable (IncS a)
forall b.
(Ord b, SymVal b, Num b, Num (SBV b)) =>
[IncS a] -> IncS a -> SBV b -> IncS a
forall a.
Mergeable a =>
Bool -> SBool -> IncS a -> IncS a -> IncS a
forall a b.
(Mergeable a, Ord b, SymVal b, Num b, Num (SBV b)) =>
[IncS a] -> IncS a -> SBV b -> IncS a
forall a.
(Bool -> SBool -> a -> a -> a)
-> (forall b.
    (Ord b, SymVal b, Num b, Num (SBV b)) =>
    [a] -> a -> SBV b -> a)
-> Mergeable a
$csymbolicMerge :: forall a.
Mergeable a =>
Bool -> SBool -> IncS a -> IncS a -> IncS a
symbolicMerge :: Bool -> SBool -> IncS a -> IncS a -> IncS a
$cselect :: forall a b.
(Mergeable a, Ord b, SymVal b, Num b, Num (SBV b)) =>
[IncS a] -> IncS a -> SBV b -> IncS a
select :: forall b.
(Ord b, SymVal b, Num b, Num (SBV b)) =>
[IncS a] -> IncS a -> SBV b -> IncS a
Mergeable, Functor IncS
Foldable IncS
(Functor IncS, Foldable IncS) =>
(forall (f :: * -> *) a b.
 Applicative f =>
 (a -> f b) -> IncS a -> f (IncS b))
-> (forall (f :: * -> *) a.
    Applicative f =>
    IncS (f a) -> f (IncS a))
-> (forall (m :: * -> *) a b.
    Monad m =>
    (a -> m b) -> IncS a -> m (IncS b))
-> (forall (m :: * -> *) a. Monad m => IncS (m a) -> m (IncS a))
-> Traversable IncS
forall (t :: * -> *).
(Functor t, Foldable t) =>
(forall (f :: * -> *) a b.
 Applicative f =>
 (a -> f b) -> t a -> f (t b))
-> (forall (f :: * -> *) a. Applicative f => t (f a) -> f (t a))
-> (forall (m :: * -> *) a b.
    Monad m =>
    (a -> m b) -> t a -> m (t b))
-> (forall (m :: * -> *) a. Monad m => t (m a) -> m (t a))
-> Traversable t
forall (m :: * -> *) a. Monad m => IncS (m a) -> m (IncS a)
forall (f :: * -> *) a. Applicative f => IncS (f a) -> f (IncS a)
forall (m :: * -> *) a b.
Monad m =>
(a -> m b) -> IncS a -> m (IncS b)
forall (f :: * -> *) a b.
Applicative f =>
(a -> f b) -> IncS a -> f (IncS b)
$ctraverse :: forall (f :: * -> *) a b.
Applicative f =>
(a -> f b) -> IncS a -> f (IncS b)
traverse :: forall (f :: * -> *) a b.
Applicative f =>
(a -> f b) -> IncS a -> f (IncS b)
$csequenceA :: forall (f :: * -> *) a. Applicative f => IncS (f a) -> f (IncS a)
sequenceA :: forall (f :: * -> *) a. Applicative f => IncS (f a) -> f (IncS a)
$cmapM :: forall (m :: * -> *) a b.
Monad m =>
(a -> m b) -> IncS a -> m (IncS b)
mapM :: forall (m :: * -> *) a b.
Monad m =>
(a -> m b) -> IncS a -> m (IncS b)
$csequence :: forall (m :: * -> *) a. Monad m => IncS (m a) -> m (IncS a)
sequence :: forall (m :: * -> *) a. Monad m => IncS (m a) -> m (IncS a)
Traversable, (forall a b. (a -> b) -> IncS a -> IncS b)
-> (forall a b. a -> IncS b -> IncS a) -> Functor IncS
forall a b. a -> IncS b -> IncS a
forall a b. (a -> b) -> IncS a -> IncS b
forall (f :: * -> *).
(forall a b. (a -> b) -> f a -> f b)
-> (forall a b. a -> f b -> f a) -> Functor f
$cfmap :: forall a b. (a -> b) -> IncS a -> IncS b
fmap :: forall a b. (a -> b) -> IncS a -> IncS b
$c<$ :: forall a b. a -> IncS b -> IncS a
<$ :: forall a b. a -> IncS b -> IncS a
Functor, (forall m. Monoid m => IncS m -> m)
-> (forall m a. Monoid m => (a -> m) -> IncS a -> m)
-> (forall m a. Monoid m => (a -> m) -> IncS a -> m)
-> (forall a b. (a -> b -> b) -> b -> IncS a -> b)
-> (forall a b. (a -> b -> b) -> b -> IncS a -> b)
-> (forall b a. (b -> a -> b) -> b -> IncS a -> b)
-> (forall b a. (b -> a -> b) -> b -> IncS a -> b)
-> (forall a. (a -> a -> a) -> IncS a -> a)
-> (forall a. (a -> a -> a) -> IncS a -> a)
-> (forall a. IncS a -> [a])
-> (forall a. IncS a -> Bool)
-> (forall a. IncS a -> Int)
-> (forall a. Eq a => a -> IncS a -> Bool)
-> (forall a. Ord a => IncS a -> a)
-> (forall a. Ord a => IncS a -> a)
-> (forall a. Num a => IncS a -> a)
-> (forall a. Num a => IncS a -> a)
-> Foldable IncS
forall a. Eq a => a -> IncS a -> Bool
forall a. Num a => IncS a -> a
forall a. Ord a => IncS a -> a
forall m. Monoid m => IncS m -> m
forall a. IncS a -> Bool
forall a. IncS a -> Int
forall a. IncS a -> [a]
forall a. (a -> a -> a) -> IncS a -> a
forall m a. Monoid m => (a -> m) -> IncS a -> m
forall b a. (b -> a -> b) -> b -> IncS a -> b
forall a b. (a -> b -> b) -> b -> IncS a -> b
forall (t :: * -> *).
(forall m. Monoid m => t m -> m)
-> (forall m a. Monoid m => (a -> m) -> t a -> m)
-> (forall m a. Monoid m => (a -> m) -> t a -> m)
-> (forall a b. (a -> b -> b) -> b -> t a -> b)
-> (forall a b. (a -> b -> b) -> b -> t a -> b)
-> (forall b a. (b -> a -> b) -> b -> t a -> b)
-> (forall b a. (b -> a -> b) -> b -> t a -> b)
-> (forall a. (a -> a -> a) -> t a -> a)
-> (forall a. (a -> a -> a) -> t a -> a)
-> (forall a. t a -> [a])
-> (forall a. t a -> Bool)
-> (forall a. t a -> Int)
-> (forall a. Eq a => a -> t a -> Bool)
-> (forall a. Ord a => t a -> a)
-> (forall a. Ord a => t a -> a)
-> (forall a. Num a => t a -> a)
-> (forall a. Num a => t a -> a)
-> Foldable t
$cfold :: forall m. Monoid m => IncS m -> m
fold :: forall m. Monoid m => IncS m -> m
$cfoldMap :: forall m a. Monoid m => (a -> m) -> IncS a -> m
foldMap :: forall m a. Monoid m => (a -> m) -> IncS a -> m
$cfoldMap' :: forall m a. Monoid m => (a -> m) -> IncS a -> m
foldMap' :: forall m a. Monoid m => (a -> m) -> IncS a -> m
$cfoldr :: forall a b. (a -> b -> b) -> b -> IncS a -> b
foldr :: forall a b. (a -> b -> b) -> b -> IncS a -> b
$cfoldr' :: forall a b. (a -> b -> b) -> b -> IncS a -> b
foldr' :: forall a b. (a -> b -> b) -> b -> IncS a -> b
$cfoldl :: forall b a. (b -> a -> b) -> b -> IncS a -> b
foldl :: forall b a. (b -> a -> b) -> b -> IncS a -> b
$cfoldl' :: forall b a. (b -> a -> b) -> b -> IncS a -> b
foldl' :: forall b a. (b -> a -> b) -> b -> IncS a -> b
$cfoldr1 :: forall a. (a -> a -> a) -> IncS a -> a
foldr1 :: forall a. (a -> a -> a) -> IncS a -> a
$cfoldl1 :: forall a. (a -> a -> a) -> IncS a -> a
foldl1 :: forall a. (a -> a -> a) -> IncS a -> a
$ctoList :: forall a. IncS a -> [a]
toList :: forall a. IncS a -> [a]
$cnull :: forall a. IncS a -> Bool
null :: forall a. IncS a -> Bool
$clength :: forall a. IncS a -> Int
length :: forall a. IncS a -> Int
$celem :: forall a. Eq a => a -> IncS a -> Bool
elem :: forall a. Eq a => a -> IncS a -> Bool
$cmaximum :: forall a. Ord a => IncS a -> a
maximum :: forall a. Ord a => IncS a -> a
$cminimum :: forall a. Ord a => IncS a -> a
minimum :: forall a. Ord a => IncS a -> a
$csum :: forall a. Num a => IncS a -> a
sum :: forall a. Num a => IncS a -> a
$cproduct :: forall a. Num a => IncS a -> a
product :: forall a. Num a => IncS a -> a
Foldable)

-- | Show instance for t'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 {-# OVERLAPS #-} (SymVal a, Show a) => Show (IncS (SBV a)) where
   show :: IncS (SBV a) -> String
show (IncS SBV a
x SBV a
y) = String
"{x = " String -> ShowS
forall a. [a] -> [a] -> [a]
++ SBV a -> String
forall {a}. (Show a, SymVal a) => SBV a -> String
sh SBV a
x String -> ShowS
forall a. [a] -> [a] -> [a]
++ String
", y = " String -> ShowS
forall a. [a] -> [a] -> [a]
++ SBV a -> String
forall {a}. (Show a, SymVal a) => SBV a -> String
sh SBV a
y String -> ShowS
forall a. [a] -> [a] -> [a]
++ String
"}"
     where sh :: SBV a -> String
sh SBV a
v = String -> (a -> String) -> Maybe a -> String
forall b a. b -> (a -> b) -> Maybe a -> b
maybe String
"<symbolic>" a -> String
forall a. Show a => a -> String
show (SBV a -> Maybe a
forall a. SymVal a => SBV a -> Maybe a
unliteral SBV a
v)

-- | 'Queriable instance for our state
instance Queriable IO (IncS SInteger) where
  type QueryResult (IncS SInteger) = IncS Integer
  create :: QueryT IO I
create = SInteger -> SInteger -> I
forall a. a -> a -> IncS a
IncS (SInteger -> SInteger -> I)
-> QueryT IO SInteger -> QueryT IO (SInteger -> I)
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> QueryT IO SInteger
forall a (m :: * -> *).
(MonadIO m, MonadQuery m, SymVal a) =>
m (SBV a)
freshVar_ QueryT IO (SInteger -> I) -> QueryT IO SInteger -> QueryT IO I
forall a b. QueryT IO (a -> b) -> QueryT IO a -> QueryT IO b
forall (f :: * -> *) a b. Applicative f => f (a -> b) -> f a -> f b
<*> QueryT IO SInteger
forall a (m :: * -> *).
(MonadIO m, MonadQuery m, SymVal a) =>
m (SBV a)
freshVar_

-- | Helper type synonym
type I = IncS SInteger

-- * 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.
algorithm :: Stmt I -> Stmt I -> Stmt I
algorithm :: Stmt I -> Stmt I -> Stmt I
algorithm Stmt I
before Stmt I
after = [Stmt I] -> Stmt I
forall st. [Stmt st] -> Stmt st
Seq [ Stmt I
before
                             , (I -> I) -> Stmt I
forall st. (st -> st) -> Stmt st
Assign ((I -> I) -> Stmt I) -> (I -> I) -> Stmt I
forall a b. (a -> b) -> a -> b
$ \st :: I
st@IncS{SInteger
x :: forall a. IncS a -> a
x :: SInteger
x} -> I
st{y = x+1}
                             , Stmt I
after
                             ]

-- | 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.
pre :: I -> SBool
pre :: I -> SBool
pre IncS{SInteger
x :: forall a. IncS a -> a
x :: SInteger
x} = SInteger
x SInteger -> SInteger -> SBool
forall a. OrdSymbolic a => a -> a -> SBool
.>= SInteger
0

-- | Postcondition for our program: @y@ must equal @x+1@.
post :: I -> SBool
post :: I -> SBool
post IncS{SInteger
x :: forall a. IncS a -> a
x :: SInteger
x, SInteger
y :: forall a. IncS a -> a
y :: SInteger
y} = SInteger
y SInteger -> SInteger -> SBool
forall a. EqSymbolic a => a -> a -> SBool
.== SInteger
xSInteger -> SInteger -> SInteger
forall a. Num a => a -> a -> a
+SInteger
1

-- | Stability: @x@ must remain unchanged.
noChange :: Stable I
noChange :: Stable I
noChange = [String -> (I -> SInteger) -> I -> I -> (String, SBool)
forall a st.
EqSymbolic a =>
String -> (st -> a) -> st -> st -> (String, SBool)
stable String
"x" I -> SInteger
forall a. IncS a -> a
x]

-- | A program is the algorithm, together with its pre- and post-conditions.
imperativeInc :: Stmt I -> Stmt I -> Program I
imperativeInc :: Stmt I -> Stmt I -> Program I
imperativeInc Stmt I
before Stmt I
after = Program { setup :: Symbolic ()
setup         = () -> Symbolic ()
forall a. a -> SymbolicT IO a
forall (m :: * -> *) a. Monad m => a -> m a
return ()
                                     , precondition :: I -> SBool
precondition  = I -> SBool
pre
                                     , program :: Stmt I
program       = Stmt I -> Stmt I -> Stmt I
algorithm Stmt I
before Stmt I
after
                                     , postcondition :: I -> SBool
postcondition = I -> SBool
post
                                     , stability :: Stable I
stability     = Stable I
noChange
                                     }

-- * 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.
correctness :: Stmt I -> Stmt I -> IO (ProofResult (IncS Integer))
correctness :: Stmt I -> Stmt I -> IO (ProofResult (IncS Integer))
correctness Stmt I
before Stmt I
after = WPConfig -> Program I -> IO (ProofResult (IncS Integer))
forall st res.
(Show res, Mergeable st, Queriable IO st, res ~ QueryResult st) =>
WPConfig -> Program st -> IO (ProofResult res)
wpProveWith WPConfig
defaultWPCfg{wpVerbose=True} (Stmt I -> Stmt I -> Program I
imperativeInc Stmt I
before Stmt I
after)

-- * Example proof attempts
--
-- $examples

{- $examples
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!
-}