Data.SBV.Examples.Uninterpreted.Deduce

Demonstrates uninterpreted sorts and how they can be used for deduction. This example is inspired by the discussion at http://stackoverflow.com/questions/10635783/using-axioms-for-deductions-in-z3, essentially showing how to show the required deduction using SBV.

Representing uninterpreted booleans

newtype B

The uninterpreted sort B, corresponding to the carrier. To prevent SBV from translating it to an enumerated type, we simply attach an unused field

Constructors

B ()

Instances

Eq B

Methods

(==) :: B -> B -> Bool

(/=) :: B -> B -> Bool

Data B

Methods

gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> B -> c B

gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c B

toConstr :: B -> Constr

dataTypeOf :: B -> DataType

dataCast1 :: Typeable (* -> *) t => (forall d. Data d => c (t d)) -> Maybe (c B)

dataCast2 :: Typeable (* -> * -> *) t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c B)

gmapT :: (forall b. Data b => b -> b) -> B -> B

gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> B -> r

gmapQr :: (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> B -> r

gmapQ :: (forall d. Data d => d -> u) -> B -> [u]

gmapQi :: Int -> (forall d. Data d => d -> u) -> B -> u

gmapM :: Monad m => (forall d. Data d => d -> m d) -> B -> m B

gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> B -> m B

gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> B -> m B

Ord B

Methods

compare :: B -> B -> Ordering

(<) :: B -> B -> Bool

(<=) :: B -> B -> Bool

(>) :: B -> B -> Bool

(>=) :: B -> B -> Bool

max :: B -> B -> B

min :: B -> B -> B

Read B

Methods

readsPrec :: Int -> ReadS B

readList :: ReadS [B]

readPrec :: ReadPrec B

readListPrec :: ReadPrec [B]

Show B

Methods

showsPrec :: Int -> B -> ShowS

show :: B -> String

showList :: [B] -> ShowS

HasKind B

Methods

kindOf :: B -> Kind

hasSign :: B -> Bool

intSizeOf :: B -> Int

isBoolean :: B -> Bool

isBounded :: B -> Bool

isReal :: B -> Bool

isFloat :: B -> Bool

isDouble :: B -> Bool

isInteger :: B -> Bool

isUninterpreted :: B -> Bool

showType :: B -> String

SymWord B

Methods

forall :: String -> Symbolic (SBV B)

forall_ :: Symbolic (SBV B)

mkForallVars :: Int -> Symbolic [SBV B]

exists :: String -> Symbolic (SBV B)

exists_ :: Symbolic (SBV B)

mkExistVars :: Int -> Symbolic [SBV B]

free :: String -> Symbolic (SBV B)

free_ :: Symbolic (SBV B)

mkFreeVars :: Int -> Symbolic [SBV B]

symbolic :: String -> Symbolic (SBV B)

symbolics :: [String] -> Symbolic [SBV B]

literal :: B -> SBV B

unliteral :: SBV B -> Maybe B

fromCW :: CW -> B

isConcrete :: SBV B -> Bool

isSymbolic :: SBV B -> Bool

isConcretely :: SBV B -> (B -> Bool) -> Bool

mkSymWord :: Maybe Quantifier -> Maybe String -> Symbolic (SBV B)

type SB = SBV B

Handy shortcut for the type of symbolic values over B

Uninterpreted connectives over B

and :: SB -> SB -> SB

Uninterpreted logical connective and

or :: SB -> SB -> SB

Uninterpreted logical connective or

not :: SB -> SB

Uninterpreted logical connective not

Axioms of the logical system

ax1 :: [String]

Distributivity of OR over AND, as an axiom in terms of the uninterpreted functions we have introduced. Note how variables range over the uninterpreted sort B.

ax2 :: [String]

One of De Morgan's laws, again as an axiom in terms of our uninterpeted logical connectives.

ax3 :: [String]

Double negation axiom, similar to the above.

Demonstrated deduction

test :: IO ThmResult

Proves the equivalence NOT (p OR (q AND r)) == (NOT p AND NOT q) OR (NOT p AND NOT r), following from the axioms we have specified above. We have:

>>> test
Q.E.D.