Copyright	(c) Levent Erkok
License	BSD3
Maintainer	erkokl@gmail.com
Stability	experimental
Safe Haskell	None
Language	Haskell2010

Documentation.SBV.Examples.Uninterpreted.Deduce

Contents

Representing uninterpreted booleans
Uninterpreted connectives over B
Axioms of the logical system
Demonstrated deduction

Description

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.

Synopsis

newtype B = B ()
type SB = SBV B
and :: SB -> SB -> SB
or :: SB -> SB -> SB
not :: SB -> SB
ax1 :: [String]
ax2 :: [String]
ax3 :: [String]
test :: IO ThmResult

Representing uninterpreted booleans

newtype B Source #

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 Source #
Instance details Defined in Documentation.SBV.Examples.Uninterpreted.Deduce Methods (==) :: B -> B -> Bool # (/=) :: B -> B -> Bool #
Data B Source #
Instance details 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 :: (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 Source #
Instance details Defined in Documentation.SBV.Examples.Uninterpreted.Deduce 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 Source #
Instance details Defined in Documentation.SBV.Examples.Uninterpreted.Deduce Methods readsPrec :: Int -> ReadS B # readList :: ReadS [B] # readPrec :: ReadPrec B # readListPrec :: ReadPrec [B] #
Show B Source #
Instance details Defined in Documentation.SBV.Examples.Uninterpreted.Deduce Methods showsPrec :: Int -> B -> ShowS # show :: B -> String # showList :: [B] -> ShowS #
HasKind B Source #
Instance details Defined in Documentation.SBV.Examples.Uninterpreted.Deduce Methods kindOf :: B -> Kind Source # hasSign :: B -> Bool Source # intSizeOf :: B -> Int Source # isBoolean :: B -> Bool Source # isBounded :: B -> Bool Source # isReal :: B -> Bool Source # isFloat :: B -> Bool Source # isDouble :: B -> Bool Source # isInteger :: B -> Bool Source # isUninterpreted :: B -> Bool Source # isChar :: B -> Bool Source # isString :: B -> Bool Source # isList :: B -> Bool Source # showType :: B -> String Source #
SymWord B Source #
Instance details Defined in Documentation.SBV.Examples.Uninterpreted.Deduce Methods forall :: String -> Symbolic (SBV B) Source # forall_ :: Symbolic (SBV B) Source # mkForallVars :: Int -> Symbolic [SBV B] Source # exists :: String -> Symbolic (SBV B) Source # exists_ :: Symbolic (SBV B) Source # mkExistVars :: Int -> Symbolic [SBV B] Source # free :: String -> Symbolic (SBV B) Source # free_ :: Symbolic (SBV B) Source # mkFreeVars :: Int -> Symbolic [SBV B] Source # symbolic :: String -> Symbolic (SBV B) Source # symbolics :: [String] -> Symbolic [SBV B] Source # literal :: B -> SBV B Source # unliteral :: SBV B -> Maybe B Source # fromCW :: CW -> B Source # isConcrete :: SBV B -> Bool Source # isSymbolic :: SBV B -> Bool Source # isConcretely :: SBV B -> (B -> Bool) -> Bool Source # mkSymWord :: Maybe Quantifier -> Maybe String -> Symbolic (SBV B) Source #

type SB = SBV B Source #

Handy shortcut for the type of symbolic values over B

Uninterpreted connectives over `B`

and :: SB -> SB -> SB Source #

Uninterpreted logical connective and

or :: SB -> SB -> SB Source #

Uninterpreted logical connective or

not :: SB -> SB Source #

Uninterpreted logical connective not

Axioms of the logical system

ax1 :: [String] Source #

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] Source #

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

ax3 :: [String] Source #

Double negation axiom, similar to the above.

Demonstrated deduction

test :: IO ThmResult Source #

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.

Representing uninterpreted booleans

Uninterpreted connectives over B

Axioms of the logical system

Demonstrated deduction

Uninterpreted connectives over `B`