Documentation.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

data B

The uninterpreted sort B, corresponding to the carrier.

Instances

Data B

Defined in Documentation.SBV.Examples.Uninterpreted.Deduce

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 :: forall r r'. (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

Read B

Defined in Documentation.SBV.Examples.Uninterpreted.Deduce

Methods

readsPrec :: Int -> ReadS B

readList :: ReadS [B]

readPrec :: ReadPrec B

readListPrec :: ReadPrec [B]

Show B

Defined in Documentation.SBV.Examples.Uninterpreted.Deduce

Methods

showsPrec :: Int -> B -> ShowS

show :: B -> String

showList :: [B] -> ShowS

HasKind B

Defined in Documentation.SBV.Examples.Uninterpreted.Deduce

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

isUnbounded :: B -> Bool

isUserSort :: B -> Bool

isChar :: B -> Bool

isString :: B -> Bool

isList :: B -> Bool

isSet :: B -> Bool

isTuple :: B -> Bool

isMaybe :: B -> Bool

isEither :: B -> Bool

showType :: B -> String

SymVal B

Defined in Documentation.SBV.Examples.Uninterpreted.Deduce

Methods

mkSymVal :: MonadSymbolic m => VarContext -> Maybe String -> m (SBV B)

literal :: B -> SBV B

fromCV :: CV -> B

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

forall :: MonadSymbolic m => String -> m (SBV B)

forall_ :: MonadSymbolic m => m (SBV B)

mkForallVars :: MonadSymbolic m => Int -> m [SBV B]

exists :: MonadSymbolic m => String -> m (SBV B)

exists_ :: MonadSymbolic m => m (SBV B)

mkExistVars :: MonadSymbolic m => Int -> m [SBV B]

free :: MonadSymbolic m => String -> m (SBV B)

free_ :: MonadSymbolic m => m (SBV B)

mkFreeVars :: MonadSymbolic m => Int -> m [SBV B]

symbolic :: MonadSymbolic m => String -> m (SBV B)

symbolics :: MonadSymbolic m => [String] -> m [SBV B]

unliteral :: SBV B -> Maybe B

isConcrete :: SBV B -> Bool

isSymbolic :: SBV B -> Bool

SatModel B

Defined in Documentation.SBV.Examples.Uninterpreted.Deduce

Methods

parseCVs :: [CV] -> Maybe (B, [CV])

cvtModel :: (B -> Maybe b) -> Maybe (B, [CV]) -> Maybe (b, [CV])

type SB = SBV B

Make this sort uninterpreted. This splice will automatically introduce the type SB into the environment, as a synonym for SBV 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.